3.376 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac {c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}+\frac {(5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)} \]

[Out]

-1/3*e*(5*b^2*e^2-6*b*c*d*e+6*c^2*d^2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)+(-b*(-b*e+c*d)-c*(-b*e+2*c*d)*x)/b^2
/d/(-b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x)+(5*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(7/2)-c^(7/2)*(-9*b
*e+4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(7/2)-e*(-b*e+2*c*d)*(5*b^2*e^2-b*c*d
*e+c^2*d^2)/b^2/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)

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Rubi [A]  time = 0.52, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {740, 828, 826, 1166, 208} \[ -\frac {e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}+\frac {(5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]

[Out]

-(e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2))/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (e*(2*c*d - b*e)*(c^2*d^2
- b*c*d*e + 5*b^2*e^2))/(b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(b^2*d*(c
*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2)) + ((4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(7/2)) - (c
^(7/2)*(4*c*d - 9*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(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 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)^{5/2} \left (b x+c x^2\right )^2} \, dx &=-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e) (4 c d+5 b e)+\frac {5}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^2 (4 c d+5 b e)+\frac {1}{2} c e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^3 (4 c d+5 b e)+\frac {1}{2} c e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} e (c d-b e)^3 (4 c d+5 b e)-\frac {1}{2} c d e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )+\frac {1}{2} c e (2 c d-b e) \left (c^2 d^2-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^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\left (c^4 (4 c d-9 b e)\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 )}{b^3 (c d-b e)^3}-\frac {(c (4 c d+5 b e)) \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 )}{b^3 d^3}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {(4 c d+5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 170, normalized size = 0.64 \[ \frac {c^2 d^2 x (b+c x) (4 c d-9 b e) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )-(c d-b e) \left (x (b+c x) \left (-5 b^2 e^2+b c d e+4 c^2 d^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e x}{d}+1\right )-3 b d \left (b^2 e+b c (e x-d)-2 c^2 d x\right )\right )}{3 b^3 d^2 x (b+c x) (d+e x)^{3/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]

[Out]

(c^2*d^2*(4*c*d - 9*b*e)*x*(b + c*x)*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e)
*(-3*b*d*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)) + (4*c^2*d^2 + b*c*d*e - 5*b^2*e^2)*x*(b + c*x)*Hypergeometric2F
1[-3/2, 1, -1/2, 1 + (e*x)/d]))/(3*b^3*d^2*(c*d - b*e)^2*x*(b + c*x)*(d + e*x)^(3/2))

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fricas [B]  time = 5.73, size = 3872, normalized size = 14.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

[1/6*(3*((4*c^5*d^5*e^2 - 9*b*c^4*d^4*e^3)*x^4 + (8*c^5*d^6*e - 14*b*c^4*d^5*e^2 - 9*b^2*c^3*d^4*e^3)*x^3 + (4
*c^5*d^7 - b*c^4*d^6*e - 18*b^2*c^3*d^5*e^2)*x^2 + (4*b*c^4*d^7 - 9*b^2*c^3*d^6*e)*x)*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*((4*c^5*d^4*e^2 - 7*b*c
^4*d^3*e^3 - 3*b^2*c^3*d^2*e^4 + 11*b^3*c^2*d*e^5 - 5*b^4*c*e^6)*x^4 + (8*c^5*d^5*e - 10*b*c^4*d^4*e^2 - 13*b^
2*c^3*d^3*e^3 + 19*b^3*c^2*d^2*e^4 + b^4*c*d*e^5 - 5*b^5*e^6)*x^3 + (4*c^5*d^6 + b*c^4*d^5*e - 17*b^2*c^3*d^4*
e^2 + 5*b^3*c^2*d^3*e^3 + 17*b^4*c*d^2*e^4 - 10*b^5*d*e^5)*x^2 + (4*b*c^4*d^6 - 7*b^2*c^3*d^5*e - 3*b^3*c^2*d^
4*e^2 + 11*b^4*c*d^3*e^3 - 5*b^5*d^2*e^4)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(3*b^2*c
^3*d^6 - 9*b^3*c^2*d^5*e + 9*b^4*c*d^4*e^2 - 3*b^5*d^3*e^3 + 3*(2*b*c^4*d^4*e^2 - 3*b^2*c^3*d^3*e^3 + 11*b^3*c
^2*d^2*e^4 - 5*b^4*c*d*e^5)*x^3 + (12*b*c^4*d^5*e - 15*b^2*c^3*d^4*e^2 + 35*b^3*c^2*d^3*e^3 + 13*b^4*c*d^2*e^4
- 15*b^5*d*e^5)*x^2 + (6*b*c^4*d^6 - 3*b^2*c^3*d^5*e - 9*b^3*c^2*d^4*e^2 + 41*b^4*c*d^3*e^3 - 20*b^5*d^2*e^4)
*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)*x^4 + (2*b^3*c^4
*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d^9 - b^4*c^3*d^8
*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*e + 3*b^6*c*d^7*e
^2 - b^7*d^6*e^3)*x), -1/6*(6*((4*c^5*d^5*e^2 - 9*b*c^4*d^4*e^3)*x^4 + (8*c^5*d^6*e - 14*b*c^4*d^5*e^2 - 9*b^2
*c^3*d^4*e^3)*x^3 + (4*c^5*d^7 - b*c^4*d^6*e - 18*b^2*c^3*d^5*e^2)*x^2 + (4*b*c^4*d^7 - 9*b^2*c^3*d^6*e)*x)*sq
rt(-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*((4*c^5*d^4*e^2 -
7*b*c^4*d^3*e^3 - 3*b^2*c^3*d^2*e^4 + 11*b^3*c^2*d*e^5 - 5*b^4*c*e^6)*x^4 + (8*c^5*d^5*e - 10*b*c^4*d^4*e^2 -
13*b^2*c^3*d^3*e^3 + 19*b^3*c^2*d^2*e^4 + b^4*c*d*e^5 - 5*b^5*e^6)*x^3 + (4*c^5*d^6 + b*c^4*d^5*e - 17*b^2*c^
3*d^4*e^2 + 5*b^3*c^2*d^3*e^3 + 17*b^4*c*d^2*e^4 - 10*b^5*d*e^5)*x^2 + (4*b*c^4*d^6 - 7*b^2*c^3*d^5*e - 3*b^3*
c^2*d^4*e^2 + 11*b^4*c*d^3*e^3 - 5*b^5*d^2*e^4)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(3
*b^2*c^3*d^6 - 9*b^3*c^2*d^5*e + 9*b^4*c*d^4*e^2 - 3*b^5*d^3*e^3 + 3*(2*b*c^4*d^4*e^2 - 3*b^2*c^3*d^3*e^3 + 11
*b^3*c^2*d^2*e^4 - 5*b^4*c*d*e^5)*x^3 + (12*b*c^4*d^5*e - 15*b^2*c^3*d^4*e^2 + 35*b^3*c^2*d^3*e^3 + 13*b^4*c*d
^2*e^4 - 15*b^5*d*e^5)*x^2 + (6*b*c^4*d^6 - 3*b^2*c^3*d^5*e - 9*b^3*c^2*d^4*e^2 + 41*b^4*c*d^3*e^3 - 20*b^5*d^
2*e^4)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)*x^4 + (2*b
^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d^9 - b^4*c
^3*d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*e + 3*b^6*c
*d^7*e^2 - b^7*d^6*e^3)*x), -1/6*(6*((4*c^5*d^4*e^2 - 7*b*c^4*d^3*e^3 - 3*b^2*c^3*d^2*e^4 + 11*b^3*c^2*d*e^5 -
5*b^4*c*e^6)*x^4 + (8*c^5*d^5*e - 10*b*c^4*d^4*e^2 - 13*b^2*c^3*d^3*e^3 + 19*b^3*c^2*d^2*e^4 + b^4*c*d*e^5 -
5*b^5*e^6)*x^3 + (4*c^5*d^6 + b*c^4*d^5*e - 17*b^2*c^3*d^4*e^2 + 5*b^3*c^2*d^3*e^3 + 17*b^4*c*d^2*e^4 - 10*b^5
*d*e^5)*x^2 + (4*b*c^4*d^6 - 7*b^2*c^3*d^5*e - 3*b^3*c^2*d^4*e^2 + 11*b^4*c*d^3*e^3 - 5*b^5*d^2*e^4)*x)*sqrt(-
d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 3*((4*c^5*d^5*e^2 - 9*b*c^4*d^4*e^3)*x^4 + (8*c^5*d^6*e - 14*b*c^4*d^5*e
^2 - 9*b^2*c^3*d^4*e^3)*x^3 + (4*c^5*d^7 - b*c^4*d^6*e - 18*b^2*c^3*d^5*e^2)*x^2 + (4*b*c^4*d^7 - 9*b^2*c^3*d^
6*e)*x)*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*(3*b^2*c^3*d^6 - 9*b^3*c^2*d^5*e + 9*b^4*c*d^4*e^2 - 3*b^5*d^3*e^3 + 3*(2*b*c^4*d^4*e^2 - 3*b^2*c^3*d
^3*e^3 + 11*b^3*c^2*d^2*e^4 - 5*b^4*c*d*e^5)*x^3 + (12*b*c^4*d^5*e - 15*b^2*c^3*d^4*e^2 + 35*b^3*c^2*d^3*e^3 +
13*b^4*c*d^2*e^4 - 15*b^5*d*e^5)*x^2 + (6*b*c^4*d^6 - 3*b^2*c^3*d^5*e - 9*b^3*c^2*d^4*e^2 + 41*b^4*c*d^3*e^3
- 20*b^5*d^2*e^4)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)
*x^4 + (2*b^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*
d^9 - b^4*c^3*d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*
e + 3*b^6*c*d^7*e^2 - b^7*d^6*e^3)*x), -1/3*(3*((4*c^5*d^5*e^2 - 9*b*c^4*d^4*e^3)*x^4 + (8*c^5*d^6*e - 14*b*c^
4*d^5*e^2 - 9*b^2*c^3*d^4*e^3)*x^3 + (4*c^5*d^7 - b*c^4*d^6*e - 18*b^2*c^3*d^5*e^2)*x^2 + (4*b*c^4*d^7 - 9*b^2
*c^3*d^6*e)*x)*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*
((4*c^5*d^4*e^2 - 7*b*c^4*d^3*e^3 - 3*b^2*c^3*d^2*e^4 + 11*b^3*c^2*d*e^5 - 5*b^4*c*e^6)*x^4 + (8*c^5*d^5*e - 1
0*b*c^4*d^4*e^2 - 13*b^2*c^3*d^3*e^3 + 19*b^3*c^2*d^2*e^4 + b^4*c*d*e^5 - 5*b^5*e^6)*x^3 + (4*c^5*d^6 + b*c^4*
d^5*e - 17*b^2*c^3*d^4*e^2 + 5*b^3*c^2*d^3*e^3 + 17*b^4*c*d^2*e^4 - 10*b^5*d*e^5)*x^2 + (4*b*c^4*d^6 - 7*b^2*c
^3*d^5*e - 3*b^3*c^2*d^4*e^2 + 11*b^4*c*d^3*e^3 - 5*b^5*d^2*e^4)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d)
+ (3*b^2*c^3*d^6 - 9*b^3*c^2*d^5*e + 9*b^4*c*d^4*e^2 - 3*b^5*d^3*e^3 + 3*(2*b*c^4*d^4*e^2 - 3*b^2*c^3*d^3*e^3
+ 11*b^3*c^2*d^2*e^4 - 5*b^4*c*d*e^5)*x^3 + (12*b*c^4*d^5*e - 15*b^2*c^3*d^4*e^2 + 35*b^3*c^2*d^3*e^3 + 13*b^4
*c*d^2*e^4 - 15*b^5*d*e^5)*x^2 + (6*b*c^4*d^6 - 3*b^2*c^3*d^5*e - 9*b^3*c^2*d^4*e^2 + 41*b^4*c*d^3*e^3 - 20*b^
5*d^2*e^4)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)*x^4 +
(2*b^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d^9 - b
^4*c^3*d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*e + 3*b
^6*c*d^7*e^2 - b^7*d^6*e^3)*x)]

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giac [A]  time = 0.27, size = 481, normalized size = 1.80 \[ \frac {{\left (4 \, c^{5} d - 9 \, b c^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{3} e - 2 \, \sqrt {x e + d} c^{4} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{2} e^{2} + 4 \, \sqrt {x e + d} b c^{3} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e^{3} - 6 \, \sqrt {x e + d} b^{2} c^{2} d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c e^{4} + 4 \, \sqrt {x e + d} b^{3} c d e^{4} - \sqrt {x e + d} b^{4} e^{5}}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}} - \frac {2 \, {\left (12 \, {\left (x e + d\right )} c d e^{3} + c d^{2} e^{3} - 6 \, {\left (x e + d\right )} b e^{4} - b d e^{4}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, c d + 5 \, b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(4*c^5*d - 9*b*c^4*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^3*d^3 - 3*b^4*c^2*d^2*e + 3*b^5*c*d
*e^2 - b^6*e^3)*sqrt(-c^2*d + b*c*e)) - (2*(x*e + d)^(3/2)*c^4*d^3*e - 2*sqrt(x*e + d)*c^4*d^4*e - 3*(x*e + d)
^(3/2)*b*c^3*d^2*e^2 + 4*sqrt(x*e + d)*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*b^2*c^2*d*e^3 - 6*sqrt(x*e + d)*b^2*c
^2*d^2*e^3 - (x*e + d)^(3/2)*b^3*c*e^4 + 4*sqrt(x*e + d)*b^3*c*d*e^4 - sqrt(x*e + d)*b^4*e^5)/((b^2*c^3*d^6 -
3*b^3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*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*(12*(x*e + d)*c*d*e^3 + c*d^2*e^3 - 6*(x*e + d)*b*e^4 - b*d*e^4)/((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)*(x*e + d)^(3/2)) - (4*c*d + 5*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^3)

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maple [A]  time = 0.07, size = 280, normalized size = 1.05 \[ \frac {9 c^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {4 c^{5} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {\sqrt {e x +d}\, c^{4} e}{\left (b e -c d \right )^{3} \left (c e x +b e \right ) b^{2}}-\frac {4 b \,e^{4}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{3}}+\frac {8 c \,e^{3}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{2}}-\frac {2 e^{3}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d^{2}}+\frac {5 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {7}{2}}}+\frac {4 c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} d^{\frac {5}{2}}}-\frac {\sqrt {e x +d}}{b^{2} d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x)

[Out]

e*c^4/b^2/(b*e-c*d)^3*(e*x+d)^(1/2)/(c*e*x+b*e)+9*e*c^4/b^2/(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)-4*c^5/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
)*d-1/b^2/d^3*(e*x+d)^(1/2)/x+5*e/b^2/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+4/b^3/d^(5/2)*arctanh((e*x+d)^(1/
2)/d^(1/2))*c-2/3*e^3/(b*e-c*d)^2/d^2/(e*x+d)^(3/2)-4*e^4/(b*e-c*d)^3/d^3/(e*x+d)^(1/2)*b+8*e^3/(b*e-c*d)^3/d^
2/(e*x+d)^(1/2)*c

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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(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,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 = 3.15, size = 5736, normalized size = 21.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^2*(d + e*x)^(5/2)),x)

[Out]

((10*e^3*(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2) - (2*e^3)/(3*(c*d^2 - b*d*e)) + (e*(d + e*x)^2*(15*b^4
*e^4 + 6*c^4*d^4 + 64*b^2*c^2*d^2*e^2 - 12*b*c^3*d^3*e - 58*b^3*c*d*e^3))/(3*b^2*(c*d^2 - b*d*e)^3) + (e*(b*e
- 2*c*d)*(d + e*x)^3*(c^3*d^2 + 5*b^2*c*e^2 - b*c^2*d*e))/(b^2*(c*d^2 - b*d*e)^3))/(c*(d + e*x)^(7/2) + (c*d^2
- b*d*e)*(d + e*x)^(3/2) + (b*e - 2*c*d)*(d + e*x)^(5/2)) + (atan((((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d
)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9*c^17*d^2
3*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^13*d^19*
e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*c^9*d^15*
e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*d^11*e^1
7 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) + ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b^10*c^18*
d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*d^24*e^7
+ 2528*b^15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d^20*e^11
+ 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^18*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^16*e^15 +
9280*b^23*c^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26*c^2*d^12*e^19 - ((-c^7*(
b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 + 1800*b^1
4*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104104*b^18*
c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 88088*b^2
2*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080*b^26*c^
4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 2
1*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(2*(b^10*
e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^
2*d^2*e^5 - 7*b^9*c*d*e^6)))*1i)/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^
4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)) + ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4
*c*d)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9*c^17
*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^13*d
^19*e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*c^9*d
^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*d^11
*e^17 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b^10*c
^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*d^24
*e^7 + 2528*b^15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d^20*e
^11 + 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^18*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^16*e^
15 + 9280*b^23*c^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26*c^2*d^12*e^19 + ((-c
^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 + 1800
*b^14*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104104*b
^18*c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 88088
*b^22*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080*b^2
6*c^4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e
- 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(2*(b
^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^
8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)))*1i)/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^
6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)))/(((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e
- 4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9*
c^17*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^
13*d^19*e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*c
^9*d^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*
d^11*e^17 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b^
10*c^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*
d^24*e^7 + 2528*b^15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d^
20*e^11 + 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^18*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^1
6*e^15 + 9280*b^23*c^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26*c^2*d^12*e^19 +
((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 +
1800*b^14*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 1041
04*b^18*c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 8
8088*b^22*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080
*b^26*c^4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d
^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(
2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 2
1*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b
^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)) - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*
e - 4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9
*c^17*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c
^13*d^19*e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*
c^9*d^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5
*d^11*e^17 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) + ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b
^10*c^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14
*d^24*e^7 + 2528*b^15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d
^20*e^11 + 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^18*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^
16*e^15 + 9280*b^23*c^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26*c^2*d^12*e^19 -
((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 +
1800*b^14*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104
104*b^18*c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 +
88088*b^22*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 108
0*b^26*c^4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*
d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/
(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 +
21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*
b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)) + 64*b^4*c^20*d^23*e^3 - 736*b^5*c
^19*d^22*e^4 + 4012*b^6*c^18*d^21*e^5 - 13790*b^7*c^17*d^20*e^6 + 32500*b^8*c^16*d^19*e^7 - 51528*b^9*c^15*d^1
8*e^8 + 45702*b^10*c^14*d^17*e^9 + 5916*b^11*c^13*d^16*e^10 - 83040*b^12*c^12*d^15*e^11 + 129800*b^13*c^11*d^1
4*e^12 - 115136*b^14*c^10*d^13*e^13 + 65234*b^15*c^9*d^12*e^14 - 23428*b^16*c^8*d^11*e^15 + 4880*b^17*c^7*d^10
*e^16 - 450*b^18*c^6*d^9*e^17))*(-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*1i)/(b^10*e^7 - b^3*c^7*d^7 + 7*b^4
*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6
) - (atan((b^19*d^15*e^19*(d + e*x)^(1/2)*125i - b^18*c*d^16*e^18*(d + e*x)^(1/2)*1700i + b^3*c^16*d^31*e^3*(d
+ e*x)^(1/2)*420i - b^4*c^15*d^30*e^4*(d + e*x)^(1/2)*4515i + b^5*c^14*d^29*e^5*(d + e*x)^(1/2)*20916i - b^6*
c^13*d^28*e^6*(d + e*x)^(1/2)*52836i + b^7*c^12*d^27*e^7*(d + e*x)^(1/2)*71070i - b^8*c^11*d^26*e^8*(d + e*x)^
(1/2)*19530i - b^9*c^10*d^25*e^9*(d + e*x)^(1/2)*107740i + b^10*c^9*d^24*e^10*(d + e*x)^(1/2)*212608i - b^11*c
^8*d^23*e^11*(d + e*x)^(1/2)*184563i + b^12*c^7*d^22*e^12*(d + e*x)^(1/2)*40965i + b^13*c^6*d^21*e^13*(d + e*x
)^(1/2)*91560i - b^14*c^5*d^20*e^14*(d + e*x)^(1/2)*126720i + b^15*c^4*d^19*e^15*(d + e*x)^(1/2)*87276i - b^16
*c^3*d^18*e^16*(d + e*x)^(1/2)*37776i + b^17*c^2*d^17*e^17*(d + e*x)^(1/2)*10440i)/(d^7*(d^7)^(1/2)*(d^7*(d^7*
(212608*b^10*c^9*e^10 - 107740*b^9*c^10*d*e^9 + 420*b^3*c^16*d^7*e^3 - 4515*b^4*c^15*d^6*e^4 + 20916*b^5*c^14*
d^5*e^5 - 52836*b^6*c^13*d^4*e^6 + 71070*b^7*c^12*d^3*e^7 - 19530*b^8*c^11*d^2*e^8) + 10440*b^17*c^2*e^17 - 37
776*b^16*c^3*d*e^16 - 184563*b^11*c^8*d^6*e^11 + 40965*b^12*c^7*d^5*e^12 + 91560*b^13*c^6*d^4*e^13 - 126720*b^
14*c^5*d^3*e^14 + 87276*b^15*c^4*d^2*e^15) + 125*b^19*d^5*e^19 - 1700*b^18*c*d^6*e^18)))*(5*b*e + 4*c*d)*1i)/(
b^3*(d^7)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**2,x)

[Out]

Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(5/2)), x)

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