3.384 \(\int \frac {1}{(d+e x)^{3/2} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=370 \[ -\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}-\frac {3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}+\frac {3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)^2} \]

[Out]

-3/4*(5*b^2*e^2+12*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(7/2)+3/4*c^(7/2)*(33*b^2*e^2-44*b
*c*d*e+16*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(7/2)+3/4*e*(-b^2*e^2-b*c*d*
e+c^2*d^2)*(5*b^2*e^2-8*b*c*d*e+8*c^2*d^2)/b^4/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)+1/2*(-b*(-b*e+c*d)-c*(-b*e+2*c*d
)*x)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)^2/(e*x+d)^(1/2)+1/4*(b*(5*b^3*e^3-17*b*c^2*d^2*e+12*c^3*d^3)+c*(-b*e+2*c*d)*
(-5*b^2*e^2-12*b*c*d*e+12*c^2*d^2)*x)/b^4/d^2/(-b*e+c*d)^2/(c*x^2+b*x)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.60, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {740, 822, 828, 826, 1166, 208} \[ \frac {c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)^2}+\frac {3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}-\frac {3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]

[Out]

(3*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2))/(4*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]
) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(2*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(12*c^3*d^3 -
17*b*c^2*d^2*e + 5*b^3*e^3) + c*(2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)
^2*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*
b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*
e]])/(4*b^5*(c*d - b*e)^(7/2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx &=-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c^2 d^2-5 b c d e-5 b^2 e^2\right )+\frac {7}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^2 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac {3}{4} c e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac {3}{4} c e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} c d e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )+\frac {3}{4} e (c d-b e)^3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac {3}{4} c e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^3 (c d-b e)^3}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^3}-\frac {\left (3 c^4 \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^3}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.29, size = 299, normalized size = 0.81 \[ \frac {2 b^4 d^2 (b e-c d)^3+b^3 d x (c d-b e)^3 (5 b e+8 c d)-x^2 \left (b^2 c d \left (5 b^2 e^2+5 b c d e-12 c^2 d^2\right ) (c d-b e)^2+(b+c x) \left (b c d (b e-c d) \left (5 b^3 e^3+2 b^2 c d e^2-36 b c^2 d^2 e+24 c^3 d^3\right )-(b+c x) \left (3 (c d-b e)^3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e x}{d}+1\right )-3 c^3 d^3 \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )\right )\right )\right )}{4 b^5 d^3 x^2 (b+c x)^2 \sqrt {d+e x} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]

[Out]

(2*b^4*d^2*(-(c*d) + b*e)^3 + b^3*d*(c*d - b*e)^3*(8*c*d + 5*b*e)*x - x^2*(b^2*c*d*(c*d - b*e)^2*(-12*c^2*d^2
+ 5*b*c*d*e + 5*b^2*e^2) + (b + c*x)*(b*c*d*(-(c*d) + b*e)*(24*c^3*d^3 - 36*b*c^2*d^2*e + 2*b^2*c*d*e^2 + 5*b^
3*e^3) - (b + c*x)*(-3*c^3*d^3*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*Hypergeometric2F1[-1/2, 1, 1/2, (c*(d +
e*x))/(c*d - b*e)] + 3*(c*d - b*e)^3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 +
(e*x)/d]))))/(4*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sqrt[d + e*x])

________________________________________________________________________________________

fricas [B]  time = 11.66, size = 4789, normalized size = 12.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(3*((16*c^7*d^6*e - 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*x^5 + (16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c
^5*d^5*e^2 + 66*b^3*c^4*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5*d^6*e + 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*
e^3)*x^3 + (16*b^2*c^5*d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*
d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 3*((16*c^7*d^5*e - 36*b*c^6*d^4*e^2 +
17*b^2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4 + 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*d^6 - 4*b*c^6*d^5*e -
55*b^2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3 + 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*e^6)*x^4 + (32*b*c^6*d
^6 - 56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*b^6*c*d*e^5 - 5*b^7*e^
6)*x^3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3*b^6*c*d^2*e^4 - 5*b^7
*d*e^5)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b^4*c^3*d^6 - 6*b^5*c^2*d^5*e + 6*b^6
*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b^2*c^5*d^4*e^2 + 5*b^3*c^4*d^3*e^3 + 3*b^4*c^3*d^2*e^4 - 5
*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5*e - 58*b^3*c^4*d^4*e^2 + 33*b^4*c^3*d^3*e^3 + 13*b^5*c^2*
d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65*b^3*c^4*d^5*e + 7*b^4*c^3*d^4*e^2 + 23*b^5*c^2*d^3*e^3 -
b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 - 19*b^4*c^3*d^5*e + 9*b^5*c^2*d^4*e^2 + 7*b^6*c*d^3*e^3 -
5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^
5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d
^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e
+ 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), 1/8*(6*((16*c^7*d^6*e - 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*x^5 + (
16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c^5*d^5*e^2 + 66*b^3*c^4*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5*d^6*e
+ 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*e^3)*x^3 + (16*b^2*c^5*d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*e^2)*x^2)
*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*((16*c^7*d^5*e
- 36*b*c^6*d^4*e^2 + 17*b^2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4 + 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*
d^6 - 4*b*c^6*d^5*e - 55*b^2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3 + 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*
e^6)*x^4 + (32*b*c^6*d^6 - 56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*
b^6*c*d*e^5 - 5*b^7*e^6)*x^3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3
*b^6*c*d^2*e^4 - 5*b^7*d*e^5)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^3*d^6 - 6
*b^5*c^2*d^5*e + 6*b^6*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b^2*c^5*d^4*e^2 + 5*b^3*c^4*d^3*e^3 +
3*b^4*c^3*d^2*e^4 - 5*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5*e - 58*b^3*c^4*d^4*e^2 + 33*b^4*c^3
*d^3*e^3 + 13*b^5*c^2*d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65*b^3*c^4*d^5*e + 7*b^4*c^3*d^4*e^2 +
23*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 - 19*b^4*c^3*d^5*e + 9*b^5*c^2*d^4*e^
2 + 7*b^6*c*d^3*e^3 - 5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3
- b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e
^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*
d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), 1/8*(6*((16*c^7*d^5*e - 36*b*c^6*d^4*e^2 + 17*b^
2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4 + 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*d^6 - 4*b*c^6*d^5*e - 55*b^
2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3 + 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*e^6)*x^4 + (32*b*c^6*d^6 -
56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*b^6*c*d*e^5 - 5*b^7*e^6)*x^
3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3*b^6*c*d^2*e^4 - 5*b^7*d*e^
5)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 3*((16*c^7*d^6*e - 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*
x^5 + (16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c^5*d^5*e^2 + 66*b^3*c^4*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5
*d^6*e + 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*e^3)*x^3 + (16*b^2*c^5*d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*e^
2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x +
b)) - 2*(2*b^4*c^3*d^6 - 6*b^5*c^2*d^5*e + 6*b^6*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b^2*c^5*d^
4*e^2 + 5*b^3*c^4*d^3*e^3 + 3*b^4*c^3*d^2*e^4 - 5*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5*e - 58*b
^3*c^4*d^4*e^2 + 33*b^4*c^3*d^3*e^3 + 13*b^5*c^2*d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65*b^3*c^4*
d^5*e + 7*b^4*c^3*d^4*e^2 + 23*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 - 19*b^4*c
^3*d^5*e + 9*b^5*c^2*d^4*e^2 + 7*b^6*c*d^3*e^3 - 5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*
d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*
c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^
10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), 1/4*(3*((16*c^7*d^6*e
- 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*x^5 + (16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c^5*d^5*e^2 + 66*b^3*c^4
*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5*d^6*e + 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*e^3)*x^3 + (16*b^2*c^5*
d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(
-c/(c*d - b*e))/(c*e*x + c*d)) + 3*((16*c^7*d^5*e - 36*b*c^6*d^4*e^2 + 17*b^2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4
+ 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*d^6 - 4*b*c^6*d^5*e - 55*b^2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3
+ 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*e^6)*x^4 + (32*b*c^6*d^6 - 56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e
^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*b^6*c*d*e^5 - 5*b^7*e^6)*x^3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d
^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3*b^6*c*d^2*e^4 - 5*b^7*d*e^5)*x^2)*sqrt(-d)*arctan(sqrt(e*x +
d)*sqrt(-d)/d) - (2*b^4*c^3*d^6 - 6*b^5*c^2*d^5*e + 6*b^6*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b
^2*c^5*d^4*e^2 + 5*b^3*c^4*d^3*e^3 + 3*b^4*c^3*d^2*e^4 - 5*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5
*e - 58*b^3*c^4*d^4*e^2 + 33*b^4*c^3*d^3*e^3 + 13*b^5*c^2*d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65
*b^3*c^4*d^5*e + 7*b^4*c^3*d^4*e^2 + 23*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 -
19*b^4*c^3*d^5*e + 9*b^5*c^2*d^4*e^2 + 7*b^6*c*d^3*e^3 - 5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3
*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2
+ 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5
*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2)]

________________________________________________________________________________________

giac [B]  time = 0.28, size = 787, normalized size = 2.13 \[ -\frac {3 \, {\left (16 \, c^{6} d^{2} - 44 \, b c^{5} d e + 33 \, b^{2} c^{4} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, e^{5}}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \sqrt {x e + d}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{6} d^{4} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{6} d^{5} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{6} d^{6} e - 24 \, \sqrt {x e + d} c^{6} d^{7} e - 48 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{5} d^{3} e^{2} + 180 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{5} d^{4} e^{2} - 216 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{5} d^{5} e^{2} + 84 \, \sqrt {x e + d} b c^{5} d^{6} e^{2} + 15 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{4} d^{2} e^{3} - 118 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{4} d^{3} e^{3} + 199 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{4} d^{4} e^{3} - 96 \, \sqrt {x e + d} b^{2} c^{4} d^{5} e^{3} + 9 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{3} d e^{4} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{3} d^{2} e^{4} - 38 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{3} d^{3} e^{4} + 30 \, \sqrt {x e + d} b^{3} c^{3} d^{4} e^{4} - 7 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} c^{2} e^{5} + 41 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} c^{2} d e^{5} - 58 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c^{2} d^{2} e^{5} + 30 \, \sqrt {x e + d} b^{4} c^{2} d^{3} e^{5} - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} c e^{6} + 41 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} c d e^{6} - 33 \, \sqrt {x e + d} b^{5} c d^{2} e^{6} - 7 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} e^{7} + 9 \, \sqrt {x e + d} b^{6} d e^{7}}{4 \, {\left (b^{4} c^{3} d^{6} - 3 \, b^{5} c^{2} d^{5} e + 3 \, b^{6} c d^{4} e^{2} - b^{7} d^{3} e^{3}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac {3 \, {\left (16 \, c^{2} d^{2} + 12 \, b c d e + 5 \, b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^6*d^2 - 44*b*c^5*d*e + 33*b^2*c^4*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^3*d^3 -
3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3)*sqrt(-c^2*d + b*c*e)) - 2*e^5/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*
d^4*e^2 - b^3*d^3*e^3)*sqrt(x*e + d)) + 1/4*(24*(x*e + d)^(7/2)*c^6*d^4*e - 72*(x*e + d)^(5/2)*c^6*d^5*e + 72*
(x*e + d)^(3/2)*c^6*d^6*e - 24*sqrt(x*e + d)*c^6*d^7*e - 48*(x*e + d)^(7/2)*b*c^5*d^3*e^2 + 180*(x*e + d)^(5/2
)*b*c^5*d^4*e^2 - 216*(x*e + d)^(3/2)*b*c^5*d^5*e^2 + 84*sqrt(x*e + d)*b*c^5*d^6*e^2 + 15*(x*e + d)^(7/2)*b^2*
c^4*d^2*e^3 - 118*(x*e + d)^(5/2)*b^2*c^4*d^3*e^3 + 199*(x*e + d)^(3/2)*b^2*c^4*d^4*e^3 - 96*sqrt(x*e + d)*b^2
*c^4*d^5*e^3 + 9*(x*e + d)^(7/2)*b^3*c^3*d*e^4 - 3*(x*e + d)^(5/2)*b^3*c^3*d^2*e^4 - 38*(x*e + d)^(3/2)*b^3*c^
3*d^3*e^4 + 30*sqrt(x*e + d)*b^3*c^3*d^4*e^4 - 7*(x*e + d)^(7/2)*b^4*c^2*e^5 + 41*(x*e + d)^(5/2)*b^4*c^2*d*e^
5 - 58*(x*e + d)^(3/2)*b^4*c^2*d^2*e^5 + 30*sqrt(x*e + d)*b^4*c^2*d^3*e^5 - 14*(x*e + d)^(5/2)*b^5*c*e^6 + 41*
(x*e + d)^(3/2)*b^5*c*d*e^6 - 33*sqrt(x*e + d)*b^5*c*d^2*e^6 - 7*(x*e + d)^(3/2)*b^6*e^7 + 9*sqrt(x*e + d)*b^6
*d*e^7)/((b^4*c^3*d^6 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 - b^7*d^3*e^3)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*
d^2 + (x*e + d)*b*e - b*d*e)^2) + 3/4*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^
5*sqrt(-d)*d^3)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 530, normalized size = 1.43 \[ \frac {21 \sqrt {e x +d}\, c^{4} e^{3}}{4 \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2} b^{2}}-\frac {33 \sqrt {e x +d}\, c^{5} d \,e^{2}}{4 \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2} b^{3}}+\frac {3 \sqrt {e x +d}\, c^{6} d^{2} e}{\left (b e -c d \right )^{3} \left (c e x +b e \right )^{2} b^{4}}+\frac {19 \left (e x +d \right )^{\frac {3}{2}} c^{5} e^{2}}{4 \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2} b^{3}}+\frac {99 c^{4} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{6} d e}{\left (b e -c d \right )^{3} \left (c e x +b e \right )^{2} b^{4}}-\frac {33 c^{5} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{4}}+\frac {12 c^{6} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{5}}+\frac {2 e^{5}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{3}}-\frac {15 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {7}{2}}}-\frac {9 c e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4} d^{\frac {5}{2}}}-\frac {12 c^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5} d^{\frac {3}{2}}}-\frac {9 \sqrt {e x +d}}{4 b^{3} d^{2} x^{2}}-\frac {3 \sqrt {e x +d}\, c}{b^{4} d e \,x^{2}}+\frac {7 \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} d^{3} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c}{b^{4} d^{2} e \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x)

[Out]

19/4*e^2*c^5/b^3/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)-3*e*c^6/b^4/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d
+21/4*e^3*c^4/b^2/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)-33/4*e^2*c^5/b^3/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(
1/2)*d+3*e*c^6/b^4/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2+99/4*e^2*c^4/b^3/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2
)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)-33*e*c^5/b^4/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2
)/((b*e-c*d)*c)^(1/2)*c)*d+12*c^6/b^5/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)
*c)*d^2+7/4/b^3/d^3/x^2*(e*x+d)^(3/2)+3/e/b^4/d^2/x^2*(e*x+d)^(3/2)*c-9/4/b^3/d^2/x^2*(e*x+d)^(1/2)-3/e/b^4/d/
x^2*(e*x+d)^(1/2)*c-15/4*e^2/b^3/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-9*e/b^4/d^(5/2)*arctanh((e*x+d)^(1/2)/
d^(1/2))*c-12/b^5/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2+2*e^5/(b*e-c*d)^3/d^3/(e*x+d)^(1/2)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

________________________________________________________________________________________

mupad [B]  time = 4.07, size = 9635, normalized size = 26.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^3*(d + e*x)^(3/2)),x)

[Out]

- ((2*e^5)/(c*d^2 - b*d*e) + (e*(d + e*x)^2*(15*b^6*e^6 - 72*c^6*d^6 - 199*b^2*c^4*d^4*e^2 + 38*b^3*c^3*d^3*e^
3 + 106*b^4*c^2*d^2*e^4 + 216*b*c^5*d^5*e - 89*b^5*c*d*e^5))/(4*b^4*(c*d^2 - b*d*e)^3) + (e*(d + e*x)*(25*b^5*
e^5 + 24*c^5*d^5 + 36*b^2*c^3*d^3*e^2 + 6*b^3*c^2*d^2*e^3 - 60*b*c^4*d^4*e - 56*b^4*c*d*e^4))/(4*b^4*(c*d^2 -
b*d*e)^2) - (3*e*(d + e*x)^4*(8*c^6*d^4 - 5*b^4*c^2*e^4 + 3*b^3*c^3*d*e^3 + 5*b^2*c^4*d^2*e^2 - 16*b*c^5*d^3*e
))/(4*b^4*(c*d^2 - b*d*e)^3) + (e*(d + e*x)^3*(72*c^6*d^5 + 30*b^5*c*e^5 - 73*b^4*c^2*d*e^4 + 118*b^2*c^4*d^3*
e^2 + 3*b^3*c^3*d^2*e^3 - 180*b*c^5*d^4*e))/(4*b^4*(c*d^2 - b*d*e)^3))/(c^2*(d + e*x)^(9/2) - (4*c^2*d - 2*b*c
*e)*(d + e*x)^(7/2) - (d + e*x)^(3/2)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) + (d + e*x)^(5/2)*(b^2*e^2 + 6*c
^2*d^2 - 6*b*c*d*e) + (d + e*x)^(1/2)*(c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e)) - (atan((((-c^7*(b*e - c*d)^7)^(1
/2)*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 2
10382848*b^15*c^19*d^25*e^5 + 564860160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c
^16*d^22*e^8 - 1667850624*b^19*c^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11
+ 438185088*b^22*c^12*d^18*e^12 - 201386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25
*c^9*d^15*e^15 + 15108480*b^26*c^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 13006
08*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(-c^7*(b*e - c*d)^7)^(1/2)*(
33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(356352*b^19*c^18*d^28*e^4 - 24576*b^18*c^19*d^29*e^3 - 2396160*b^20*c^1
7*d^27*e^5 + 9897984*b^21*c^16*d^26*e^6 - 28065792*b^22*c^15*d^25*e^7 + 57891840*b^23*c^14*d^24*e^8 - 90071040
*b^24*c^13*d^23*e^9 + 108810240*b^25*c^12*d^22*e^10 - 105566208*b^26*c^11*d^21*e^11 + 86406144*b^27*c^10*d^20*
e^12 - 63393792*b^28*c^9*d^19*e^13 + 43075584*b^29*c^8*d^18*e^14 - 26173440*b^30*c^7*d^17*e^15 + 13108224*b^31
*c^6*d^16*e^16 - 4964352*b^32*c^5*d^15*e^17 + 1302528*b^33*c^4*d^14*e^18 - 208896*b^34*c^3*d^13*e^19 + 15360*b
^35*c^2*d^12*e^20 + (3*(-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(1638
4*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 2
6091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*
d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 469647
36*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16
- 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*
c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6))))/(8*(b^12*e^7
- b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d
^2*e^5 - 7*b^11*c*d*e^6)))*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*3i)/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d
^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)) +
((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 533422
08*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 564860160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^2
3*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 85564
2240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 201386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*
d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b
^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(-c
^7*(b*e - c*d)^7)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(24576*b^18*c^19*d^29*e^3 - 356352*b^19*c^18*d^
28*e^4 + 2396160*b^20*c^17*d^27*e^5 - 9897984*b^21*c^16*d^26*e^6 + 28065792*b^22*c^15*d^25*e^7 - 57891840*b^23
*c^14*d^24*e^8 + 90071040*b^24*c^13*d^23*e^9 - 108810240*b^25*c^12*d^22*e^10 + 105566208*b^26*c^11*d^21*e^11 -
86406144*b^27*c^10*d^20*e^12 + 63393792*b^28*c^9*d^19*e^13 - 43075584*b^29*c^8*d^18*e^14 + 26173440*b^30*c^7*
d^17*e^15 - 13108224*b^31*c^6*d^16*e^16 + 4964352*b^32*c^5*d^15*e^17 - 1302528*b^33*c^4*d^14*e^18 + 208896*b^3
4*c^3*d^13*e^19 - 15360*b^35*c^2*d^12*e^20 + (3*(-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(33*b^2*e^2 + 16*c^
2*d^2 - 44*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 83148
80*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e
^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^
32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 11
05920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*(b^12*e^7 - b^5*c^7*d^7 +
7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*
c*d*e^6))))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^
3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)))*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*3i)/(8*(b^12*e^7 -
b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2
*e^5 - 7*b^11*c*d*e^6)))/(1769472*b^8*c^23*d^26*e^3 - 23003136*b^9*c^22*d^25*e^4 + 136138752*b^10*c^21*d^24*e^
5 - 483508224*b^11*c^20*d^23*e^6 + 1141579008*b^12*c^19*d^22*e^7 - 1869094656*b^13*c^18*d^21*e^8 + 2133106272*
b^14*c^17*d^20*e^9 - 1631703744*b^15*c^16*d^19*e^10 + 716335488*b^16*c^15*d^18*e^11 - 36390816*b^17*c^14*d^17*
e^12 - 153641664*b^18*c^13*d^16*e^13 + 89697024*b^19*c^12*d^15*e^14 - 40065408*b^20*c^11*d^14*e^15 + 43695936*
b^21*c^10*d^13*e^16 - 41388192*b^22*c^9*d^12*e^17 + 21843648*b^23*c^8*d^11*e^18 - 6082560*b^24*c^7*d^10*e^19 +
712800*b^25*c^6*d^9*e^20 - (3*(-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*e^2 - 825753
6*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 564860160*b^16*c^18*d^24*e
^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c^15*d^21*e^9 + 135825753
6*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 201386880*b^23*c^11*d^
17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c^8*d^14*e^16 - 6844032*b
^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10*e^20 - 2880
0*b^31*c^3*d^9*e^21) - (3*(-c^7*(b*e - c*d)^7)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(356352*b^19*c^18*
d^28*e^4 - 24576*b^18*c^19*d^29*e^3 - 2396160*b^20*c^17*d^27*e^5 + 9897984*b^21*c^16*d^26*e^6 - 28065792*b^22*
c^15*d^25*e^7 + 57891840*b^23*c^14*d^24*e^8 - 90071040*b^24*c^13*d^23*e^9 + 108810240*b^25*c^12*d^22*e^10 - 10
5566208*b^26*c^11*d^21*e^11 + 86406144*b^27*c^10*d^20*e^12 - 63393792*b^28*c^9*d^19*e^13 + 43075584*b^29*c^8*d
^18*e^14 - 26173440*b^30*c^7*d^17*e^15 + 13108224*b^31*c^6*d^16*e^16 - 4964352*b^32*c^5*d^15*e^17 + 1302528*b^
33*c^4*d^14*e^18 - 208896*b^34*c^3*d^13*e^19 + 15360*b^35*c^2*d^12*e^20 + (3*(-c^7*(b*e - c*d)^7)^(1/2)*(d + e
*x)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 18432
00*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7
+ 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^3
1*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 54
47680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/
(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 +
21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6))))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 3
5*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)))*(33*b^2*e^2 + 16*c^2*d^2 - 44
*b*c*d*e))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3
*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)) + (3*(-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(589824*b^1
2*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 56
4860160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c
^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12
- 201386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c
^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b
^30*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(-c^7*(b*e - c*d)^7)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c
*d*e)*(24576*b^18*c^19*d^29*e^3 - 356352*b^19*c^18*d^28*e^4 + 2396160*b^20*c^17*d^27*e^5 - 9897984*b^21*c^16*d
^26*e^6 + 28065792*b^22*c^15*d^25*e^7 - 57891840*b^23*c^14*d^24*e^8 + 90071040*b^24*c^13*d^23*e^9 - 108810240*
b^25*c^12*d^22*e^10 + 105566208*b^26*c^11*d^21*e^11 - 86406144*b^27*c^10*d^20*e^12 + 63393792*b^28*c^9*d^19*e^
13 - 43075584*b^29*c^8*d^18*e^14 + 26173440*b^30*c^7*d^17*e^15 - 13108224*b^31*c^6*d^16*e^16 + 4964352*b^32*c^
5*d^15*e^17 - 1302528*b^33*c^4*d^14*e^18 + 208896*b^34*c^3*d^13*e^19 - 15360*b^35*c^2*d^12*e^20 + (3*(-c^7*(b*
e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^
23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 603
83232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d
^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800
*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 81
92*b^38*c^2*d^15*e^18))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3
- 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6))))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e
- 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)))*(33*b
^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*
c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6))))*(-c^7*(b*e - c*d)^7)^(1/2)*(33*b^2
*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*3i)/(4*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8
*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)) - (atan(((((d + e*x)^(1/2)*(589824*
b^12*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 +
564860160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^1
9*c^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^
12 - 201386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^2
6*c^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 29376
0*b^30*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*(356352*b^19*c^18*d
^28*e^4 - 24576*b^18*c^19*d^29*e^3 - 2396160*b^20*c^17*d^27*e^5 + 9897984*b^21*c^16*d^26*e^6 - 28065792*b^22*c
^15*d^25*e^7 + 57891840*b^23*c^14*d^24*e^8 - 90071040*b^24*c^13*d^23*e^9 + 108810240*b^25*c^12*d^22*e^10 - 105
566208*b^26*c^11*d^21*e^11 + 86406144*b^27*c^10*d^20*e^12 - 63393792*b^28*c^9*d^19*e^13 + 43075584*b^29*c^8*d^
18*e^14 - 26173440*b^30*c^7*d^17*e^15 + 13108224*b^31*c^6*d^16*e^16 - 4964352*b^32*c^5*d^15*e^17 + 1302528*b^3
3*c^4*d^14*e^18 - 208896*b^34*c^3*d^13*e^19 + 15360*b^35*c^2*d^12*e^20 + (3*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^
2*d^2 + 12*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 83148
80*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e
^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^
32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 11
05920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*b^5*(d^7)^(1/2))))/(8*b^5*
(d^7)^(1/2)))*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*3i)/(8*b^5*(d^7)^(1/2)) + (((d + e*x)^(1/2)*(589824*b^12*c
^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 56486
0160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c^15
*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 2
01386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c^8*
d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30
*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*(24576*b^18*c^19*d^29*e^3
- 356352*b^19*c^18*d^28*e^4 + 2396160*b^20*c^17*d^27*e^5 - 9897984*b^21*c^16*d^26*e^6 + 28065792*b^22*c^15*d^
25*e^7 - 57891840*b^23*c^14*d^24*e^8 + 90071040*b^24*c^13*d^23*e^9 - 108810240*b^25*c^12*d^22*e^10 + 105566208
*b^26*c^11*d^21*e^11 - 86406144*b^27*c^10*d^20*e^12 + 63393792*b^28*c^9*d^19*e^13 - 43075584*b^29*c^8*d^18*e^1
4 + 26173440*b^30*c^7*d^17*e^15 - 13108224*b^31*c^6*d^16*e^16 + 4964352*b^32*c^5*d^15*e^17 - 1302528*b^33*c^4*
d^14*e^18 + 208896*b^34*c^3*d^13*e^19 - 15360*b^35*c^2*d^12*e^20 + (3*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2
+ 12*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^2
5*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 1
46432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8
*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*
b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*b^5*(d^7)^(1/2))))/(8*b^5*(d^7)^
(1/2)))*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*3i)/(8*b^5*(d^7)^(1/2)))/(1769472*b^8*c^23*d^26*e^3 - 23003136*b
^9*c^22*d^25*e^4 + 136138752*b^10*c^21*d^24*e^5 - 483508224*b^11*c^20*d^23*e^6 + 1141579008*b^12*c^19*d^22*e^7
- 1869094656*b^13*c^18*d^21*e^8 + 2133106272*b^14*c^17*d^20*e^9 - 1631703744*b^15*c^16*d^19*e^10 + 716335488*
b^16*c^15*d^18*e^11 - 36390816*b^17*c^14*d^17*e^12 - 153641664*b^18*c^13*d^16*e^13 + 89697024*b^19*c^12*d^15*e
^14 - 40065408*b^20*c^11*d^14*e^15 + 43695936*b^21*c^10*d^13*e^16 - 41388192*b^22*c^9*d^12*e^17 + 21843648*b^2
3*c^8*d^11*e^18 - 6082560*b^24*c^7*d^10*e^19 + 712800*b^25*c^6*d^9*e^20 - (3*((d + e*x)^(1/2)*(589824*b^12*c^2
2*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 5648601
60*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c^15*d
^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 201
386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c^8*d^
14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30*c
^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*(356352*b^19*c^18*d^28*e^4
- 24576*b^18*c^19*d^29*e^3 - 2396160*b^20*c^17*d^27*e^5 + 9897984*b^21*c^16*d^26*e^6 - 28065792*b^22*c^15*d^25
*e^7 + 57891840*b^23*c^14*d^24*e^8 - 90071040*b^24*c^13*d^23*e^9 + 108810240*b^25*c^12*d^22*e^10 - 105566208*b
^26*c^11*d^21*e^11 + 86406144*b^27*c^10*d^20*e^12 - 63393792*b^28*c^9*d^19*e^13 + 43075584*b^29*c^8*d^18*e^14
- 26173440*b^30*c^7*d^17*e^15 + 13108224*b^31*c^6*d^16*e^16 - 4964352*b^32*c^5*d^15*e^17 + 1302528*b^33*c^4*d^
14*e^18 - 208896*b^34*c^3*d^13*e^19 + 15360*b^35*c^2*d^12*e^20 + (3*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 +
12*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*
c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146
432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d
^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^
36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*b^5*(d^7)^(1/2))))/(8*b^5*(d^7)^(1
/2)))*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^7)^(1/2)) + (3*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*
e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 564860160*b^16
*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c^15*d^21*e^9
+ 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 201386880*
b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c^8*d^14*e^16
- 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10
*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*(24576*b^18*c^19*d^29*e^3 - 356352
*b^19*c^18*d^28*e^4 + 2396160*b^20*c^17*d^27*e^5 - 9897984*b^21*c^16*d^26*e^6 + 28065792*b^22*c^15*d^25*e^7 -
57891840*b^23*c^14*d^24*e^8 + 90071040*b^24*c^13*d^23*e^9 - 108810240*b^25*c^12*d^22*e^10 + 105566208*b^26*c^1
1*d^21*e^11 - 86406144*b^27*c^10*d^20*e^12 + 63393792*b^28*c^9*d^19*e^13 - 43075584*b^29*c^8*d^18*e^14 + 26173
440*b^30*c^7*d^17*e^15 - 13108224*b^31*c^6*d^16*e^16 + 4964352*b^32*c^5*d^15*e^17 - 1302528*b^33*c^4*d^14*e^18
+ 208896*b^34*c^3*d^13*e^19 - 15360*b^35*c^2*d^12*e^20 + (3*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*
d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^
28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*
b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^1
2 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*
d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*b^5*(d^7)^(1/2))))/(8*b^5*(d^7)^(1/2)))*(
5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^7)^(1/2))))*(5*b^2*e^2 + 16*c^2*d^2 + 12*b*c*d*e)*3i)/(4*b^5*(
d^7)^(1/2))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________