3.378 \(\int \frac {(d+e x)^{9/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=300 \[ -\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}}+\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {3 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 c^2} \]

[Out]

-1/2*(e*x+d)^(7/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^2+1/4*(e*x+d)^(3/2)*(b*c*d^2*(-13*b*e+12*c*d)+(-b*e+2*
c*d)*(-b^2*e^2-12*b*c*d*e+12*c^2*d^2)*x)/b^4/c/(c*x^2+b*x)-3/4*d^(5/2)*(21*b^2*e^2-36*b*c*d*e+16*c^2*d^2)*arct
anh((e*x+d)^(1/2)/d^(1/2))/b^5+3/4*(-b*e+c*d)^(5/2)*(b^2*e^2+4*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/c^(5/2)-3/4*e*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*(e*x+d)^(1/2)/b^4/c^2

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Rubi [A]  time = 0.52, antiderivative size = 300, 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 = {738, 818, 824, 826, 1166, 208} \[ \frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {3 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 c^2}-\frac {3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]

[Out]

(-3*e*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4*b^4*c^2) - ((d + e*x)^(7/2)*(b*d + (2*
c*d - b*e)*x))/(2*b^2*(b*x + c*x^2)^2) + ((d + e*x)^(3/2)*(b*c*d^2*(12*c*d - 13*b*e) + (2*c*d - b*e)*(12*c^2*d
^2 - 12*b*c*d*e - b^2*e^2)*x))/(4*b^4*c*(b*x + c*x^2)) - (3*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*Arc
Tanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

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 738

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

Rule 818

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

Rule 824

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

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 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 {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {(d+e x)^{5/2} \left (\frac {1}{2} d (12 c d-13 b e)-\frac {1}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (-\frac {3}{4} c d^2 \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )+\frac {3}{4} e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{2 b^4 c}\\ &=-\frac {3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c^2}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {3}{4} c^2 d^3 \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )-\frac {3}{4} e \left (8 c^4 d^4-16 b c^3 d^3 e+7 b^2 c^2 d^2 e^2+b^3 c d e^3+b^4 e^4\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^2}\\ &=-\frac {3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c^2}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} c^2 d^3 e \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )+\frac {3}{4} d e \left (8 c^4 d^4-16 b c^3 d^3 e+7 b^2 c^2 d^2 e^2+b^3 c d e^3+b^4 e^4\right )-\frac {3}{4} e \left (8 c^4 d^4-16 b c^3 d^3 e+7 b^2 c^2 d^2 e^2+b^3 c d e^3+b^4 e^4\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 c^2}\\ &=-\frac {3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c^2}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\left (3 (c d-b e)^3 \left (16 c^2 d^2+4 b c d e+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^2}+\frac {\left (3 c d^3 \left (16 c^2 d^2-36 b c d e+21 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}\\ &=-\frac {3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c^2}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {3 d^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 (c d-b e)^{5/2} \left (16 c^2 d^2+4 b c d e+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^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 275, normalized size = 0.92 \[ \frac {-3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{c^{5/2}}+\frac {b \sqrt {d+e x} \left (-3 b^5 e^4 x^2-5 b^4 c e^3 x^2 (d+e x)+b^3 c^2 d \left (-2 d^3-17 d^2 e x+33 d e^2 x^2+3 e^3 x^3\right )+b^2 c^3 d^2 x \left (8 d^2-73 d e x+21 e^2 x^2\right )+12 b c^4 d^3 x^2 (3 d-4 e x)+24 c^5 d^4 x^3\right )}{c^2 x^2 (b+c x)^2}}{4 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]

[Out]

((b*Sqrt[d + e*x]*(-3*b^5*e^4*x^2 + 24*c^5*d^4*x^3 + 12*b*c^4*d^3*x^2*(3*d - 4*e*x) - 5*b^4*c*e^3*x^2*(d + e*x
) + b^2*c^3*d^2*x*(8*d^2 - 73*d*e*x + 21*e^2*x^2) + b^3*c^2*d*(-2*d^3 - 17*d^2*e*x + 33*d*e^2*x^2 + 3*e^3*x^3)
))/(c^2*x^2*(b + c*x)^2) - 3*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (
3*(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(5/
2))/(4*b^5)

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fricas [B]  time = 3.95, size = 2373, normalized size = 7.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d
^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^
3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(
e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16
*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*
e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4
*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4
*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*
x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(6*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3
+ b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^
3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt(-(c*d - b*e)/c
)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2
*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e
+ 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*b*c^5
*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c
^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*
x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(6*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2
)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21
*b^4*c^2*d^2*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4
*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4
*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4
)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2
*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (3
6*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b
^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/4*(3*((16*c^6*d^4 - 28*b*c^5*d^
3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*
d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*
d*e^3 + b^6*e^4)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((16*
c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*
x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d)
- (2*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3
- (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 -
17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)]

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giac [B]  time = 0.28, size = 631, normalized size = 2.10 \[ \frac {3 \, {\left (16 \, c^{2} d^{5} - 36 \, b c d^{4} e + 21 \, b^{2} d^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} - \frac {3 \, {\left (16 \, c^{5} d^{5} - 44 \, b c^{4} d^{4} e + 37 \, b^{2} c^{3} d^{3} e^{2} - 7 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4} - b^{5} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5} c^{2}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{5} d^{4} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{5} d^{5} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{5} d^{6} e - 24 \, \sqrt {x e + d} c^{5} d^{7} e - 48 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{4} d^{3} e^{2} + 180 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{4} d^{4} e^{2} - 216 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{4} d^{5} e^{2} + 84 \, \sqrt {x e + d} b c^{4} d^{6} e^{2} + 21 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{3} d^{2} e^{3} - 136 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{3} d^{3} e^{3} + 217 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{3} d^{4} e^{3} - 102 \, \sqrt {x e + d} b^{2} c^{3} d^{5} e^{3} + 3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{2} d e^{4} + 24 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{2} d^{2} e^{4} - 74 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{2} d^{3} e^{4} + 45 \, \sqrt {x e + d} b^{3} c^{2} d^{4} e^{4} - 5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} c e^{5} + 10 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} c d e^{5} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c d^{2} e^{5} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} e^{6} + 6 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d e^{6} - 3 \, \sqrt {x e + d} b^{5} d^{2} e^{6}}{4 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2} b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

3/4*(16*c^2*d^5 - 36*b*c*d^4*e + 21*b^2*d^3*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) - 3/4*(16*c^5*d
^5 - 44*b*c^4*d^4*e + 37*b^2*c^3*d^3*e^2 - 7*b^3*c^2*d^2*e^3 - b^4*c*d*e^4 - b^5*e^5)*arctan(sqrt(x*e + d)*c/s
qrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c^2) + 1/4*(24*(x*e + d)^(7/2)*c^5*d^4*e - 72*(x*e + d)^(5/2)*c
^5*d^5*e + 72*(x*e + d)^(3/2)*c^5*d^6*e - 24*sqrt(x*e + d)*c^5*d^7*e - 48*(x*e + d)^(7/2)*b*c^4*d^3*e^2 + 180*
(x*e + d)^(5/2)*b*c^4*d^4*e^2 - 216*(x*e + d)^(3/2)*b*c^4*d^5*e^2 + 84*sqrt(x*e + d)*b*c^4*d^6*e^2 + 21*(x*e +
d)^(7/2)*b^2*c^3*d^2*e^3 - 136*(x*e + d)^(5/2)*b^2*c^3*d^3*e^3 + 217*(x*e + d)^(3/2)*b^2*c^3*d^4*e^3 - 102*sq
rt(x*e + d)*b^2*c^3*d^5*e^3 + 3*(x*e + d)^(7/2)*b^3*c^2*d*e^4 + 24*(x*e + d)^(5/2)*b^3*c^2*d^2*e^4 - 74*(x*e +
d)^(3/2)*b^3*c^2*d^3*e^4 + 45*sqrt(x*e + d)*b^3*c^2*d^4*e^4 - 5*(x*e + d)^(7/2)*b^4*c*e^5 + 10*(x*e + d)^(5/2
)*b^4*c*d*e^5 - 5*(x*e + d)^(3/2)*b^4*c*d^2*e^5 - 3*(x*e + d)^(5/2)*b^5*e^6 + 6*(x*e + d)^(3/2)*b^5*d*e^6 - 3*
sqrt(x*e + d)*b^5*d^2*e^6)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c^2)

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maple [B]  time = 0.08, size = 703, normalized size = 2.34 \[ -\frac {3 \sqrt {e x +d}\, b \,e^{6}}{4 \left (c e x +b e \right )^{2} c^{2}}+\frac {15 \sqrt {e x +d}\, d^{2} e^{4}}{2 \left (c e x +b e \right )^{2} b}-\frac {15 \sqrt {e x +d}\, c \,d^{3} e^{3}}{\left (c e x +b e \right )^{2} b^{2}}+\frac {45 \sqrt {e x +d}\, c^{2} d^{4} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, c^{3} d^{5} e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {3 d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b c}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} d \,e^{4}}{4 \left (c e x +b e \right )^{2} b}+\frac {21 \left (e x +d \right )^{\frac {3}{2}} c \,d^{2} e^{3}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {21 d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {31 \left (e x +d \right )^{\frac {3}{2}} c^{2} d^{3} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {111 c \,d^{3} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{3} d^{4} e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {33 c^{2} d^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {12 c^{3} d^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{5}}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} e^{5}}{4 \left (c e x +b e \right )^{2} c}+\frac {3 e^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, c^{2}}-\frac {63 d^{\frac {5}{2}} e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3}}+\frac {27 c \,d^{\frac {7}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}-\frac {12 c^{2} d^{\frac {9}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}+\frac {15 \sqrt {e x +d}\, d^{4}}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, c \,d^{5}}{b^{4} e \,x^{2}}-\frac {17 \left (e x +d \right )^{\frac {3}{2}} d^{3}}{4 b^{3} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c \,d^{4}}{b^{4} e \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(c*x^2+b*x)^3,x)

[Out]

-5/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(3/2)-3/4*e^6*b/(c*e*x+b*e)^2/c^2*(e*x+d)^(1/2)+3/4*e^5/c^2/((b*e-c*d)*c)^(1/
2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)-111/4*e^2/b^3*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2)*c)*d^3+33*e/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^4+21/4*e^3
/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^2-3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d
^5*c^3+3/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d+45/4*e^2/b^3/(c*e*x+b*e)^
2*c^2*(e*x+d)^(1/2)*d^4+3/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d+21/4*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(3/2)*d^2
-31/4*e^2/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(3/2)*d^3+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d^4*c^3+15/2*e^4/b/(c*e*
x+b*e)^2*(e*x+d)^(1/2)*d^2-15*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*d^3-12/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x
+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^5*c^3-17/4*d^3/b^3/x^2*(e*x+d)^(3/2)+3/e*d^4/b^4/x^2*(e*x+d)^(3/2)*c+15/4*d
^4/b^3/x^2*(e*x+d)^(1/2)-3/e*d^5/b^4/x^2*(e*x+d)^(1/2)*c-63/4*e^2*d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))+2
7*e*d^(7/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12*d^(9/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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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((e*x+d)^(9/2)/(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?

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mupad [B]  time = 1.47, size = 3946, normalized size = 13.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(9/2)/(b*x + c*x^2)^3,x)

[Out]

(((d + e*x)^(3/2)*(6*b^5*d*e^6 + 72*c^5*d^6*e - 216*b*c^4*d^5*e^2 - 5*b^4*c*d^2*e^5 + 217*b^2*c^3*d^4*e^3 - 74
*b^3*c^2*d^3*e^4))/(4*b^4*c^2) - (3*(d + e*x)^(1/2)*(8*c^5*d^7*e + b^5*d^2*e^6 - 28*b*c^4*d^6*e^2 + 34*b^2*c^3
*d^5*e^3 - 15*b^3*c^2*d^4*e^4))/(4*b^4*c^2) + (e*(d + e*x)^(7/2)*(24*c^4*d^4 - 5*b^4*e^4 + 21*b^2*c^2*d^2*e^2
- 48*b*c^3*d^3*e + 3*b^3*c*d*e^3))/(4*b^4*c) + ((b*e - 2*c*d)*(d + e*x)^(5/2)*(36*c^4*d^4*e - 3*b^4*e^5 - 72*b
*c^3*d^3*e^2 + 32*b^2*c^2*d^2*e^3 + 4*b^3*c*d*e^4))/(4*b^4*c^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b
^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^
2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) + (atan(((((3*(d^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^1
1*c^6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^12*c^3) - (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^
2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(64*b^13*c^3))*(21*b^2*e^2 + 16*c^2*d^2
- 36*b*c*d*e))/(8*b^5) - ((d + e*x)^(1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2
*c^8*d^8*e^4 - 44928*b^3*c^7*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 54
0*b^7*c^3*d^3*e^9 + 135*b^8*c^2*d^2*e^10 + 18*b^9*c*d*e^11))/(8*b^8*c^3))*(d^5)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2
- 36*b*c*d*e)*3i)/(8*b^5) - (((3*(d^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 -
42*b^12*c^5*d^3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^12*c^3) + (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d^5)^(1/2)*
(d + e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(64*b^13*c^3))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))
/(8*b^5) + ((d + e*x)^(1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e^4 -
44928*b^3*c^7*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*e
^9 + 135*b^8*c^2*d^2*e^10 + 18*b^9*c*d*e^11))/(8*b^8*c^3))*(d^5)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*
3i)/(8*b^5))/((3*((3*(d^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*
d^3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^12*c^3) - (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/
2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(64*b^13*c^3))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(8*b^5) - ((
d + e*x)^(1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e^4 - 44928*b^3*c^7
*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*e^9 + 135*b^8*
c^2*d^2*e^10 + 18*b^9*c*d*e^11))/(8*b^8*c^3))*(d^5)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(8*b^5) - ((
567*b^11*d^3*e^14)/32 - 1728*c^11*d^14*e^3 + 12096*b*c^10*d^13*e^4 + (81*b^10*c*d^4*e^13)/16 - 35748*b^2*c^9*d
^12*e^5 + 57240*b^3*c^8*d^11*e^6 - (211113*b^4*c^7*d^10*e^7)/4 + (109917*b^5*c^6*d^9*e^8)/4 - (253449*b^6*c^5*
d^8*e^9)/32 + (17901*b^7*c^4*d^7*e^10)/8 - (35829*b^8*c^3*d^6*e^11)/32 + (6993*b^9*c^2*d^5*e^12)/32)/(b^12*c^3
) + (3*((3*(d^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 +
3*b^13*c^4*d^2*e^6)/(b^12*c^3) + (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(21*b^2
*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(64*b^13*c^3))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(8*b^5) + ((d + e*x)^(
1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e^4 - 44928*b^3*c^7*d^7*e^5 +
21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*e^9 + 135*b^8*c^2*d^2*e^
10 + 18*b^9*c*d*e^11))/(8*b^8*c^3))*(d^5)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(8*b^5)))*(d^5)^(1/2)*
(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*3i)/(4*b^5) + (atan((((((d + e*x)^(1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e
^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e^4 - 44928*b^3*c^7*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^
5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*e^9 + 135*b^8*c^2*d^2*e^10 + 18*b^9*c*d*e^11))/(8*b^8*c^3) -
(3*(-c^5*(b*e - c*d)^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*d^
3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^12*c^3) - (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(-c^5*(b*e - c*d)^5)^(1/2)*
(d + e*x)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e))/(64*b^13*c^8))*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e))/(8*b^5*
c^5))*(-c^5*(b*e - c*d)^5)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e)*3i)/(8*b^5*c^5) + ((((d + e*x)^(1/2)*(9*b^
10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e^4 - 44928*b^3*c^7*d^7*e^5 + 21546*b^4
*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*e^9 + 135*b^8*c^2*d^2*e^10 + 18*b^
9*c*d*e^11))/(8*b^8*c^3) + (3*(-c^5*(b*e - c*d)^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^11*c^
6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^12*c^3) + (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(
-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e))/(64*b^13*c^8))*(b^2*e^2 + 16*c^2
*d^2 + 4*b*c*d*e))/(8*b^5*c^5))*(-c^5*(b*e - c*d)^5)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e)*3i)/(8*b^5*c^5))
/(((567*b^11*d^3*e^14)/32 - 1728*c^11*d^14*e^3 + 12096*b*c^10*d^13*e^4 + (81*b^10*c*d^4*e^13)/16 - 35748*b^2*c
^9*d^12*e^5 + 57240*b^3*c^8*d^11*e^6 - (211113*b^4*c^7*d^10*e^7)/4 + (109917*b^5*c^6*d^9*e^8)/4 - (253449*b^6*
c^5*d^8*e^9)/32 + (17901*b^7*c^4*d^7*e^10)/8 - (35829*b^8*c^3*d^6*e^11)/32 + (6993*b^9*c^2*d^5*e^12)/32)/(b^12
*c^3) + (3*(((d + e*x)^(1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e^4 -
44928*b^3*c^7*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*
e^9 + 135*b^8*c^2*d^2*e^10 + 18*b^9*c*d*e^11))/(8*b^8*c^3) - (3*(-c^5*(b*e - c*d)^5)^(1/2)*((3*b^14*c^3*d*e^7
- 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^12*c^3) - (3*(64*b^
11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e)
)/(64*b^13*c^8))*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e))/(8*b^5*c^5))*(-c^5*(b*e - c*d)^5)^(1/2)*(b^2*e^2 + 16*c^2
*d^2 + 4*b*c*d*e))/(8*b^5*c^5) - (3*(((d + e*x)^(1/2)*(9*b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3
+ 45792*b^2*c^8*d^8*e^4 - 44928*b^3*c^7*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 + 567*b^6*c^4*d
^4*e^8 - 540*b^7*c^3*d^3*e^9 + 135*b^8*c^2*d^2*e^10 + 18*b^9*c*d*e^11))/(8*b^8*c^3) + (3*(-c^5*(b*e - c*d)^5)^
(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 + 3*b^13*c^4*d^2*e^
6)/(b^12*c^3) + (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(b^2*e^2
+ 16*c^2*d^2 + 4*b*c*d*e))/(64*b^13*c^8))*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e))/(8*b^5*c^5))*(-c^5*(b*e - c*d)^5
)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4*b*c*d*e))/(8*b^5*c^5)))*(-c^5*(b*e - c*d)^5)^(1/2)*(b^2*e^2 + 16*c^2*d^2 + 4
*b*c*d*e)*3i)/(4*b^5*c^5)

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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((e*x+d)**(9/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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