3.801 \(\int \frac{1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=462 \[ \frac{7 (a d+b c)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2} c^{9/2}}+\frac{b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{12 a^3 c^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac{b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{4 a^4 c^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{d \sqrt{a+b x} \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right ) (a d+b c)}{12 a^4 c^4 \sqrt{c+d x} (b c-a d)^4}+\frac{d \sqrt{a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{12 a^4 c^3 (c+d x)^{3/2} (b c-a d)^3}-\frac{1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \]

[Out]

(b*(35*b^2*c^2 - 6*a*b*c*d - 21*a^2*d^2))/(12*a^3*c^2*(b*c - a*d)*(a + b*x)^(3/2
)*(c + d*x)^(3/2)) - 1/(2*a*c*x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (7*(b*c + a
*d))/(4*a^2*c^2*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*(35*b^3*c^3 - 55*a*b^2*c
^2*d - 3*a^2*b*c*d^2 + 7*a^3*d^3))/(4*a^4*c^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d
*x)^(3/2)) + (d*(105*b^4*c^4 - 200*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 48*a^3*b*c
*d^3 - 35*a^4*d^4)*Sqrt[a + b*x])/(12*a^4*c^3*(b*c - a*d)^3*(c + d*x)^(3/2)) + (
d*(b*c + a*d)*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^2*b^2*c^2*d^2 - 340*a^3*b*c
*d^3 + 105*a^4*d^4)*Sqrt[a + b*x])/(12*a^4*c^4*(b*c - a*d)^4*Sqrt[c + d*x]) - (5
*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/(4*a^(9/2)*c^(9/2))

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Rubi [A]  time = 1.73877, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{7 (a d+b c)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2} c^{9/2}}+\frac{b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{12 a^3 c^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac{b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{4 a^4 c^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{d \sqrt{a+b x} \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right ) (a d+b c)}{12 a^4 c^4 \sqrt{c+d x} (b c-a d)^4}+\frac{d \sqrt{a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{12 a^4 c^3 (c+d x)^{3/2} (b c-a d)^3}-\frac{1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(b*(35*b^2*c^2 - 6*a*b*c*d - 21*a^2*d^2))/(12*a^3*c^2*(b*c - a*d)*(a + b*x)^(3/2
)*(c + d*x)^(3/2)) - 1/(2*a*c*x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (7*(b*c + a
*d))/(4*a^2*c^2*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*(35*b^3*c^3 - 55*a*b^2*c
^2*d - 3*a^2*b*c*d^2 + 7*a^3*d^3))/(4*a^4*c^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d
*x)^(3/2)) + (d*(105*b^4*c^4 - 200*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 48*a^3*b*c
*d^3 - 35*a^4*d^4)*Sqrt[a + b*x])/(12*a^4*c^3*(b*c - a*d)^3*(c + d*x)^(3/2)) + (
d*(b*c + a*d)*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^2*b^2*c^2*d^2 - 340*a^3*b*c
*d^3 + 105*a^4*d^4)*Sqrt[a + b*x])/(12*a^4*c^4*(b*c - a*d)^4*Sqrt[c + d*x]) - (5
*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/(4*a^(9/2)*c^(9/2))

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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**3/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.52629, size = 291, normalized size = 0.63 \[ \frac{5 \log (x) \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right )}{8 a^{9/2} c^{9/2}}-\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{9/2} c^{9/2}}+\frac{1}{12} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{8 b^5 (9 b c-17 a d)}{a^4 (a+b x) (b c-a d)^4}+\frac{33 (a d+b c)}{a^4 c^4 x}-\frac{8 b^5}{a^3 (a+b x)^2 (a d-b c)^3}-\frac{6}{a^3 c^3 x^2}+\frac{8 d^5 (9 a d-17 b c)}{c^4 (c+d x) (b c-a d)^4}-\frac{8 d^5}{c^3 (c+d x)^2 (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-6/(a^3*c^3*x^2) + (33*(b*c + a*d))/(a^4*c^4*x) -
(8*b^5)/(a^3*(-(b*c) + a*d)^3*(a + b*x)^2) + (8*b^5*(9*b*c - 17*a*d))/(a^4*(b*c
- a*d)^4*(a + b*x)) - (8*d^5)/(c^3*(b*c - a*d)^3*(c + d*x)^2) + (8*d^5*(-17*b*c
+ 9*a*d))/(c^4*(b*c - a*d)^4*(c + d*x))))/12 + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^
2*d^2)*Log[x])/(8*a^(9/2)*c^(9/2)) - (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*Log
[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*a^(9
/2)*c^(9/2))

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Maple [B]  time = 0.092, size = 3392, normalized size = 7.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)

[Out]

-1/24/a^4/c^4*(-48*a^4*b^3*c^6*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+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^4*a^8*d^8+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^4*b^8*c^8+210*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^7*b*d^8+210*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^8*c^7*d+210*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^8*c*d^7+210*l
n((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^7*c^8+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^2*a^8*c^2*d^6
+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^2*a^2*b^6
*c^8-210*x^3*a^7*d^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^3*b^7*c^7*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^7*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
12*a^3*b^4*c^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+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^6*a^6*b^2*d^8+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^6*b^8*c^6*d^2+390*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^5*c^4*d^4-270*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^6*c^5*d^3-330*l
n((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^7*c^6*d^2
+150*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^7*b*c
*d^7-840*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^2*c^2*d^6+330*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^3*c^3*d^5+510*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^4*c^4*d^4+330*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^5*c^5*d^3-840*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^6*c^6*d^2+150*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^7*c^7*d-330*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*b*c^2*d^6-270*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^2*c^3*d^5+390*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^3*c^4*d^4+390*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^4*c^5*d^3-2
70*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^5*c
^6*d^2-330*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^6*c^7*d-270*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*b*c^3*d^5+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^2*a^6*b^2*c^4*d^4+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^5*b^3*c^5*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^2*a^4*b^4*c^6*d^2-270*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^5*c^7*d-48*a^6*b*c^4*d^3*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+72*a^5*b^2*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-270*
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^6*a^5*b^3*c*d^
7+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^6*a^4*b^
4*c^2*d^6+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^6
*a^3*b^5*c^3*d^5+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^6*a^2*b^6*c^4*d^4+372*x^4*a^4*b^3*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(
1/2)-456*x^4*a^3*b^4*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+372*x^4*a^2*b^5
*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+660*x^4*a*b^6*c^5*d^2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)-90*x^3*a^6*b*c*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10
98*x^3*a^5*b^2*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*x^3*a^4*b^3*c^3*d
^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*x^3*a^3*b^4*c^4*d^3*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2)+1098*x^3*a^2*b^5*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-
90*x^3*a*b^6*c^6*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+552*x^2*a^6*b*c^2*d^5*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*x^2*a^5*b^2*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2)-336*x^2*a^4*b^3*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*x^2*a
^3*b^4*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+552*x^2*a^2*b^5*c^6*d*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+126*x*a^6*b*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(
1/2)-84*x*a^5*b^2*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-84*x*a^4*b^3*c^5*d
^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+126*x*a^3*b^4*c^6*d*(a*c)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)+470*x^5*a^4*b^3*c*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-132*x^5
*a^3*b^4*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-132*x^5*a^2*b^5*c^3*d^4*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+470*x^5*a*b^6*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+660*x^4*a^5*b^2*c*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^5*a^5
*b^2*d^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^5*b^7*c^5*d^2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)-420*x^4*a^6*b*d^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-420*x
^4*b^7*c^6*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*x^2*a^7*c*d^6*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)-280*x^2*a*b^6*c^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*
x*a^7*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*x*a^2*b^5*c^7*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)-270*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^6*a*b^7*c^5*d^3-330*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^6*b^2*c*d^7-270*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^3*c^2*d^6+390*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^4*c^3*d^5)/((b*x+a)*(d*x+c))^(1/2)/(a*d
-b*c)^4/x^2/(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.66629, 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^3),x, algorithm="fricas")

[Out]

[-1/48*(4*(6*a^3*b^4*c^7 - 24*a^4*b^3*c^6*d + 36*a^5*b^2*c^5*d^2 - 24*a^6*b*c^4*
d^3 + 6*a^7*c^3*d^4 - (105*b^7*c^5*d^2 - 235*a*b^6*c^4*d^3 + 66*a^2*b^5*c^3*d^4
+ 66*a^3*b^4*c^2*d^5 - 235*a^4*b^3*c*d^6 + 105*a^5*b^2*d^7)*x^5 - 6*(35*b^7*c^6*
d - 55*a*b^6*c^5*d^2 - 31*a^2*b^5*c^4*d^3 + 38*a^3*b^4*c^3*d^4 - 31*a^4*b^3*c^2*
d^5 - 55*a^5*b^2*c*d^6 + 35*a^6*b*d^7)*x^4 - 3*(35*b^7*c^7 + 15*a*b^6*c^6*d - 18
3*a^2*b^5*c^5*d^2 + 69*a^3*b^4*c^4*d^3 + 69*a^4*b^3*c^3*d^4 - 183*a^5*b^2*c^2*d^
5 + 15*a^6*b*c*d^6 + 35*a^7*d^7)*x^3 - 4*(35*a*b^6*c^7 - 69*a^2*b^5*c^6*d - 3*a^
3*b^4*c^5*d^2 + 42*a^4*b^3*c^4*d^3 - 3*a^5*b^2*c^3*d^4 - 69*a^6*b*c^2*d^5 + 35*a
^7*c*d^6)*x^2 - 21*(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)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x +
c) - 15*((7*b^8*c^6*d^2 - 18*a*b^7*c^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*
d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^6 + 2*(7*b^8*c^7*d
- 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d^
5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^8)*x^5 + (7*b^8*c^8 + 10*a*
b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d^3 + 34*a^4*b^4*c^4*d^4 + 22*a^
5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*a^7*b*c*d^7 + 7*a^8*d^8)*x^4 + 2*(7*a*b^
7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c
^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b*c^2*d^6 + 7*a^8*c*d^7)*x^3 + (7*a^2*b^6*c^
8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*d^3 + 9*a^6*b^2*c^4*d^4
- 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6)*x^2)*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^4*b^6*c^8*d^2 - 4*a^5*b^5*
c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4*d^6)*x^6 + 2*(a^4*
b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7*b^3*c^6*d^4 - 3*a^8*b^
2*c^5*d^5 + a^9*b*c^4*d^6)*x^5 + (a^4*b^6*c^10 - 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*
c^7*d^3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^6)*x^4 + 2*(a^5*b^5*c^10 - 3*a^6*b^4*c^
9*d + 2*a^7*b^3*c^8*d^2 + 2*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*d^4 + a^10*c^5*d^5)*x^
3 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d + 6*a^8*b^2*c^8*d^2 - 4*a^9*b*c^7*d^3 + a^10
*c^6*d^4)*x^2)*sqrt(a*c)), -1/24*(2*(6*a^3*b^4*c^7 - 24*a^4*b^3*c^6*d + 36*a^5*b
^2*c^5*d^2 - 24*a^6*b*c^4*d^3 + 6*a^7*c^3*d^4 - (105*b^7*c^5*d^2 - 235*a*b^6*c^4
*d^3 + 66*a^2*b^5*c^3*d^4 + 66*a^3*b^4*c^2*d^5 - 235*a^4*b^3*c*d^6 + 105*a^5*b^2
*d^7)*x^5 - 6*(35*b^7*c^6*d - 55*a*b^6*c^5*d^2 - 31*a^2*b^5*c^4*d^3 + 38*a^3*b^4
*c^3*d^4 - 31*a^4*b^3*c^2*d^5 - 55*a^5*b^2*c*d^6 + 35*a^6*b*d^7)*x^4 - 3*(35*b^7
*c^7 + 15*a*b^6*c^6*d - 183*a^2*b^5*c^5*d^2 + 69*a^3*b^4*c^4*d^3 + 69*a^4*b^3*c^
3*d^4 - 183*a^5*b^2*c^2*d^5 + 15*a^6*b*c*d^6 + 35*a^7*d^7)*x^3 - 4*(35*a*b^6*c^7
- 69*a^2*b^5*c^6*d - 3*a^3*b^4*c^5*d^2 + 42*a^4*b^3*c^4*d^3 - 3*a^5*b^2*c^3*d^4
- 69*a^6*b*c^2*d^5 + 35*a^7*c*d^6)*x^2 - 21*(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)*sqrt(-a*
c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((7*b^8*c^6*d^2 - 18*a*b^7*c^5*d^3 + 9*a^2*b
^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^
2*d^8)*x^6 + 2*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*
c^4*d^4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^
8)*x^5 + (7*b^8*c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d^3 +
34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*a^7*b*c*d^7 +
7*a^8*d^8)*x^4 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13*a^4
*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b*c^2*d^6 + 7*a^8
*c*d^7)*x^3 + (7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*
c^5*d^3 + 9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6)*x^2)*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^4*b^
6*c^8*d^2 - 4*a^5*b^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*
c^4*d^6)*x^6 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7*
b^3*c^6*d^4 - 3*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6)*x^5 + (a^4*b^6*c^10 - 9*a^6*b^4
*c^8*d^2 + 16*a^7*b^3*c^7*d^3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^6)*x^4 + 2*(a^5*b
^5*c^10 - 3*a^6*b^4*c^9*d + 2*a^7*b^3*c^8*d^2 + 2*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*
d^4 + a^10*c^5*d^5)*x^3 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d + 6*a^8*b^2*c^8*d^2 -
4*a^9*b*c^7*d^3 + a^10*c^6*d^4)*x^2)*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**3/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.64649, 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^3),x, algorithm="giac")

[Out]

sage0*x