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

Optimal. Leaf size=352 \[ \frac{5 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{7/2} c^{7/2}}-\frac{b (5 b c-3 a d)}{3 a^2 c (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a^3 c \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac{d \sqrt{a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 a^3 c^2 (c+d x)^{3/2} (b c-a d)^3}-\frac{d \sqrt{a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{3 a^3 c^3 \sqrt{c+d x} (b c-a d)^4}-\frac{1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}} \]

[Out]

-(b*(5*b*c - 3*a*d))/(3*a^2*c*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - 1/(
a*c*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (b*(5*b^2*c^2 - 10*a*b*c*d + a^2*d^2))/
(a^3*c*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (d*(15*b^3*c^3 - 35*a*b^2*
c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[a + b*x])/(3*a^3*c^2*(b*c - a*d)^3*(c +
d*x)^(3/2)) - (d*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*
d^3 + 15*a^4*d^4)*Sqrt[a + b*x])/(3*a^3*c^3*(b*c - a*d)^4*Sqrt[c + d*x]) + (5*(b
*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(7/2)*c^(
7/2))

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Rubi [A]  time = 1.27408, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{5 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{7/2} c^{7/2}}-\frac{b (5 b c-3 a d)}{3 a^2 c (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a^3 c \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac{d \sqrt{a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 a^3 c^2 (c+d x)^{3/2} (b c-a d)^3}-\frac{d \sqrt{a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{3 a^3 c^3 \sqrt{c+d x} (b c-a d)^4}-\frac{1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(5*b*c - 3*a*d))/(3*a^2*c*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - 1/(
a*c*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (b*(5*b^2*c^2 - 10*a*b*c*d + a^2*d^2))/
(a^3*c*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (d*(15*b^3*c^3 - 35*a*b^2*
c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[a + b*x])/(3*a^3*c^2*(b*c - a*d)^3*(c +
d*x)^(3/2)) - (d*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*
d^3 + 15*a^4*d^4)*Sqrt[a + b*x])/(3*a^3*c^3*(b*c - a*d)^4*Sqrt[c + d*x]) + (5*(b
*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(7/2)*c^(
7/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.63752, size = 241, normalized size = 0.68 \[ -\frac{5 \log (x) (a d+b c)}{2 a^{7/2} c^{7/2}}+\frac{5 (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{7/2} c^{7/2}}+\frac{1}{3} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{4 b^4 (7 a d-3 b c)}{a^3 (a+b x) (b c-a d)^4}-\frac{3}{a^3 c^3 x}+\frac{2 b^4}{a^2 (a+b x)^2 (a d-b c)^3}+\frac{4 d^4 (7 b c-3 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac{2 d^4}{c^2 (c+d x)^2 (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-3/(a^3*c^3*x) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a
+ b*x)^2) + (4*b^4*(-3*b*c + 7*a*d))/(a^3*(b*c - a*d)^4*(a + b*x)) + (2*d^4)/(c^
2*(b*c - a*d)^3*(c + d*x)^2) + (4*d^4*(7*b*c - 3*a*d))/(c^3*(b*c - a*d)^4*(c + d
*x))))/3 - (5*(b*c + a*d)*Log[x])/(2*a^(7/2)*c^(7/2)) + (5*(b*c + a*d)*Log[2*a*c
+ b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*a^(7/2)*c^
(7/2))

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Maple [B]  time = 0.083, size = 2703, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)

[Out]

1/6/a^3/c^3*(80*x^4*a^3*b^3*c*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-36*x^4*a^2
*b^4*c^2*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-60*x^3*b^6*c^5*d*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2
*a*c)/x)*x^3*a^7*d^7+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*
a*c)/x)*x^3*b^7*c^7-30*x^2*a^6*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+80*x^4*a*
b^5*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+120*x^3*a^4*b^2*c*d^5*(a*c)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)+36*x^3*a^3*b^3*c^2*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(
1/2)+36*x^3*a^2*b^4*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+120*x^3*a*b^5*c^
4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+174*x^2*a^4*b^2*c^2*d^4*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)-96*x^2*a^3*b^3*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
+174*x^2*a^2*b^4*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+96*x*a^5*b*c^2*d^4*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-24*x*a^4*b^2*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)-24*x*a^3*b^3*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+96*x*a^2*b
^4*c^5*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+120*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^3*b^4*c^3*d^4-30*x^4*a^4*b^2*d^6*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^4*b^6*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)-60*x^3*a^5*b*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-60*ln((a*d*x+b*c*x+2*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a*b^6*c^5*d^2+15*ln((a*d*x+b*c*x+
2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^6*b*c*d^6-135*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^5*b^2*c^2*d^5+105*ln((
a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^4*b^3*c^3*d^4+
105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^3*b^4*
c^4*d^3-135*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*
a^2*b^5*c^5*d^2+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/
x)*x^3*a*b^6*c^6*d-60*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*
c)/x)*x^2*a^6*b*c^2*d^5-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
+2*a*c)/x)*x^2*a^5*b^2*c^3*d^4+120*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+2*a*c)/x)*x^2*a^4*b^3*c^4*d^3-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^3*b^4*c^5*d^2-60*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^2*b^5*c^6*d-45*ln((a*d*x+b*c*x+2*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^6*b*c^3*d^4+30*ln((a*d*x+b*c*x+2*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^5*b^2*c^4*d^3+30*ln((a*d*x+b*c*x+2*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^4*b^3*c^5*d^2-45*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^3*b^4*c^6*d+30*ln((a*d*x+
b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^2*b^5*c^3*d^4-45*ln(
(a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a*b^6*c^4*d^3-6
0*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^5*b^2*c*
d^6-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^4*b
^3*c^2*d^5-40*x*a^6*c*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-40*x*a*b^5*c^6*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*a^5*b*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)-36*a^4*b^2*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*a^3*b^3*c^5*d*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+2*a*c)/x)*x^5*a^4*b^3*c*d^6+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^3*b^4*c^2*d^5-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b^5*c^4*d^3-30*x^2*b^6*c^6*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)-6*a^6*c^2*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*a^2*
b^4*c^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^5*b^2*d^7+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*b^7*c^5*d^2+30*ln((a*d*x+b*c*x+2*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^6*b*d^7+30*ln((a*d*x+b*c*x+2*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*b^7*c^6*d+30*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^7*c*d^6+30*ln((a*d*x+b*c*x+2*(a*c)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a*b^6*c^7+15*ln((a*d*x+b*c*x+2*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^7*c^2*d^5+15*ln((a*d*x+b*c*x+2*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^2*b^5*c^7)/((b*x+a)*(d*x+c))^(1/2)
/(a*d-b*c)^4/x/(a*c)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79282, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/2)*x^2),x, algorithm="fricas")

[Out]

[-1/12*(4*(3*a^2*b^4*c^6 - 12*a^3*b^3*c^5*d + 18*a^4*b^2*c^4*d^2 - 12*a^5*b*c^3*
d^3 + 3*a^6*c^2*d^4 + (15*b^6*c^4*d^2 - 40*a*b^5*c^3*d^3 + 18*a^2*b^4*c^2*d^4 -
40*a^3*b^3*c*d^5 + 15*a^4*b^2*d^6)*x^4 + 6*(5*b^6*c^5*d - 10*a*b^5*c^4*d^2 - 3*a
^2*b^4*c^3*d^3 - 3*a^3*b^3*c^2*d^4 - 10*a^4*b^2*c*d^5 + 5*a^5*b*d^6)*x^3 + 3*(5*
b^6*c^6 - 29*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 29*a^4*b^2*c^2*d^4 + 5*a^6*d
^6)*x^2 + 4*(5*a*b^5*c^6 - 12*a^2*b^4*c^5*d + 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*
d^3 - 12*a^5*b*c^2*d^4 + 5*a^6*c*d^5)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) -
15*((b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*
a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^5 + 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - a^2*b^5*c^4*
d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x^4 + (
b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^
4 - 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^3 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^
6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3 - a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^5 +
a^7*c*d^6)*x^2 + (a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*
c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5)*x)*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c
*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2
)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2))/(((a^3*b^6*c^7*d^2 - 4*a^4*b^5
*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^3*c^4*d^5 + a^7*b^2*c^3*d^6)*x^5 + 2*(a^3
*b^6*c^8*d - 3*a^4*b^5*c^7*d^2 + 2*a^5*b^4*c^6*d^3 + 2*a^6*b^3*c^5*d^4 - 3*a^7*b
^2*c^4*d^5 + a^8*b*c^3*d^6)*x^4 + (a^3*b^6*c^9 - 9*a^5*b^4*c^7*d^2 + 16*a^6*b^3*
c^6*d^3 - 9*a^7*b^2*c^5*d^4 + a^9*c^3*d^6)*x^3 + 2*(a^4*b^5*c^9 - 3*a^5*b^4*c^8*
d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8*b*c^5*d^4 + a^9*c^4*d^5)*x^2 +
(a^5*b^4*c^9 - 4*a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6*d^3 + a^9*c^5*
d^4)*x)*sqrt(a*c)), -1/6*(2*(3*a^2*b^4*c^6 - 12*a^3*b^3*c^5*d + 18*a^4*b^2*c^4*d
^2 - 12*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + (15*b^6*c^4*d^2 - 40*a*b^5*c^3*d^3 + 18*
a^2*b^4*c^2*d^4 - 40*a^3*b^3*c*d^5 + 15*a^4*b^2*d^6)*x^4 + 6*(5*b^6*c^5*d - 10*a
*b^5*c^4*d^2 - 3*a^2*b^4*c^3*d^3 - 3*a^3*b^3*c^2*d^4 - 10*a^4*b^2*c*d^5 + 5*a^5*
b*d^6)*x^3 + 3*(5*b^6*c^6 - 29*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 29*a^4*b^2
*c^2*d^4 + 5*a^6*d^6)*x^2 + 4*(5*a*b^5*c^6 - 12*a^2*b^4*c^5*d + 3*a^3*b^3*c^4*d^
2 + 3*a^4*b^2*c^3*d^3 - 12*a^5*b*c^2*d^4 + 5*a^6*c*d^5)*x)*sqrt(-a*c)*sqrt(b*x +
a)*sqrt(d*x + c) - 15*((b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a
^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^5 + 2*(b^7*c^6*d - 2*a*b^6*c^5
*d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 +
a^6*b*d^7)*x^4 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3
+ 7*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^3 + 2*(a*b^6
*c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3 - a^5*b^2*c^3*d^4 -
2*a^6*b*c^2*d^5 + a^7*c*d^6)*x^2 + (a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c
^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5)*x)*arctan(1/2*(2*a*c
+ (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/(((a^3*b^6*c^7*
d^2 - 4*a^4*b^5*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^3*c^4*d^5 + a^7*b^2*c^3*d^
6)*x^5 + 2*(a^3*b^6*c^8*d - 3*a^4*b^5*c^7*d^2 + 2*a^5*b^4*c^6*d^3 + 2*a^6*b^3*c^
5*d^4 - 3*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6)*x^4 + (a^3*b^6*c^9 - 9*a^5*b^4*c^7*d^
2 + 16*a^6*b^3*c^6*d^3 - 9*a^7*b^2*c^5*d^4 + a^9*c^3*d^6)*x^3 + 2*(a^4*b^5*c^9 -
3*a^5*b^4*c^8*d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8*b*c^5*d^4 + a^9
*c^4*d^5)*x^2 + (a^5*b^4*c^9 - 4*a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6
*d^3 + a^9*c^5*d^4)*x)*sqrt(-a*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.00397, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/2)*x^2),x, algorithm="giac")

[Out]

sage0*x