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

Optimal. Leaf size=409 \[ -\frac{(6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^4}-\frac{d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{9/2}}+\frac{d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{4 a c^3 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{12 a^2 c^3 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^3}+\frac{b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{4 a^3 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^4}+\frac{d (2 b c-3 a d)}{2 a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2} \]

[Out]

(b*(20*b^3*c^3 - 36*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 36*a^3*d^3))/(12*a^2*c^3*(b*c
- a*d)^3*(a + b/x)^(3/2)) + (b*(20*b^4*c^4 - 56*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^
2 - 35*a^3*b*c*d^3 + 12*a^4*d^4))/(4*a^3*c^3*(b*c - a*d)^4*Sqrt[a + b/x]) + (d*(
2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)^2) + (d*(4*b^2*c^
2 - 23*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*(a + b/x)^(3/2)*(c + d/x))
+ x/(a*c*(a + b/x)^(3/2)*(c + d/x)^2) - (d^(7/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*a
^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c - a*d)^(9/2
)) - ((5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^4)

_______________________________________________________________________________________

Rubi [A]  time = 2.07224, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{(6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^4}-\frac{d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{9/2}}+\frac{d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{4 a c^3 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{12 a^2 c^3 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^3}+\frac{b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{4 a^3 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^4}+\frac{d (2 b c-3 a d)}{2 a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(b*(20*b^3*c^3 - 36*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 36*a^3*d^3))/(12*a^2*c^3*(b*c
- a*d)^3*(a + b/x)^(3/2)) + (b*(20*b^4*c^4 - 56*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^
2 - 35*a^3*b*c*d^3 + 12*a^4*d^4))/(4*a^3*c^3*(b*c - a*d)^4*Sqrt[a + b/x]) + (d*(
2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)^2) + (d*(4*b^2*c^
2 - 23*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*(a + b/x)^(3/2)*(c + d/x))
+ x/(a*c*(a + b/x)^(3/2)*(c + d/x)^2) - (d^(7/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*a
^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c - a*d)^(9/2
)) - ((5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^4)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 3.81985, size = 465, normalized size = 1.14 \[ -\frac{\frac{12 (6 a d+5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{7/2}}+\frac{3 i d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \log \left (\frac{8 c^5 (b c-a d)^{7/2} \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d x-i b (d-c x)\right )}{d^{9/2} (c x+d) \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right )}\right )}{(b c-a d)^{9/2}}+\frac{2 \sqrt{a+\frac{b}{x}} \left (-30 a^5 d^5 (a x+b)^2 (c x+d)^2+6 a^4 d^6 (a x+b)^2 (b c-a d)+3 a^4 d^5 (a x+b)^2 (c x+d) (12 a d-23 b c)+63 a^4 b c d^4 (a x+b)^2 (c x+d)^2-8 b^6 c^4 (c x+d)^2 (b c-a d)-56 b^5 c^5 (a x+b)^2 (c x+d)^2+8 b^5 c^4 (a x+b) (c x+d)^2 (8 b c-17 a d)+128 a b^4 c^4 d (a x+b)^2 (c x+d)^2-12 a c x (a x+b)^2 (c x+d)^2 (b c-a d)^4\right )}{a^4 (a x+b)^2 (c x+d)^2 (b c-a d)^4}}{24 c^4} \]

Antiderivative was successfully verified.

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

[Out]

-((2*Sqrt[a + b/x]*(6*a^4*d^6*(b*c - a*d)*(b + a*x)^2 + 3*a^4*d^5*(-23*b*c + 12*
a*d)*(b + a*x)^2*(d + c*x) - 8*b^6*c^4*(b*c - a*d)*(d + c*x)^2 + 8*b^5*c^4*(8*b*
c - 17*a*d)*(b + a*x)*(d + c*x)^2 - 56*b^5*c^5*(b + a*x)^2*(d + c*x)^2 + 128*a*b
^4*c^4*d*(b + a*x)^2*(d + c*x)^2 + 63*a^4*b*c*d^4*(b + a*x)^2*(d + c*x)^2 - 30*a
^5*d^5*(b + a*x)^2*(d + c*x)^2 - 12*a*c*(b*c - a*d)^4*x*(b + a*x)^2*(d + c*x)^2)
)/(a^4*(b*c - a*d)^4*(b + a*x)^2*(d + c*x)^2) + (12*(5*b*c + 6*a*d)*Log[b + 2*a*
x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(7/2) + ((3*I)*d^(7/2)*(99*b^2*c^2 - 88*a*b*c*
d + 24*a^2*d^2)*Log[(8*c^5*(b*c - a*d)^(7/2)*((-2*I)*a*d*x + 2*Sqrt[d]*Sqrt[b*c
- a*d]*Sqrt[a + b/x]*x - I*b*(d - c*x)))/(d^(9/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*
a^2*d^2)*(d + c*x))])/(b*c - a*d)^(9/2))/(24*c^4)

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 7306, normalized size = 17.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

_______________________________________________________________________________________

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/((a + b/x)^(5/2)*(c + d/x)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 6.63689, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*(c + d/x)^3),x, algorithm="fricas")

[Out]

[1/24*(3*(99*a^3*b^3*c^2*d^5 - 88*a^4*b^2*c*d^6 + 24*a^5*b*d^7 + (99*a^4*b^2*c^4
*d^3 - 88*a^5*b*c^3*d^4 + 24*a^6*c^2*d^5)*x^3 + (99*a^3*b^3*c^4*d^3 + 110*a^4*b^
2*c^3*d^4 - 152*a^5*b*c^2*d^5 + 48*a^6*c*d^6)*x^2 + (198*a^3*b^3*c^3*d^4 - 77*a^
4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 24*a^6*d^7)*x)*sqrt(a)*sqrt(-d/(b*c - a*d))*sqr
t((a*x + b)/x)*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*
d + (b*c - 2*a*d)*x)/(c*x + d)) + 12*(5*b^6*c^5*d^2 - 14*a*b^5*c^4*d^3 + 6*a^2*b
^4*c^3*d^4 + 16*a^3*b^3*c^2*d^5 - 19*a^4*b^2*c*d^6 + 6*a^5*b*d^7 + (5*a*b^5*c^7
- 14*a^2*b^4*c^6*d + 6*a^3*b^3*c^5*d^2 + 16*a^4*b^2*c^4*d^3 - 19*a^5*b*c^3*d^4 +
6*a^6*c^2*d^5)*x^3 + (5*b^6*c^7 - 4*a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2 + 28*a^3*b
^3*c^4*d^3 + 13*a^4*b^2*c^3*d^4 - 32*a^5*b*c^2*d^5 + 12*a^6*c*d^6)*x^2 + (10*b^6
*c^6*d - 23*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3 + 38*a^3*b^3*c^3*d^4 - 22*a^4*b^2*
c^2*d^5 - 7*a^5*b*c*d^6 + 6*a^6*d^7)*x)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x +
b)/x) + (2*a*x + b)*sqrt(a)) + 2*(60*b^6*c^5*d^2 - 168*a*b^5*c^4*d^3 + 72*a^2*b
^4*c^3*d^4 - 105*a^3*b^3*c^2*d^5 + 36*a^4*b^2*c*d^6 + 12*(a^2*b^4*c^7 - 4*a^3*b^
3*c^6*d + 6*a^4*b^2*c^5*d^2 - 4*a^5*b*c^4*d^3 + a^6*c^3*d^4)*x^4 + (80*a*b^5*c^7
- 200*a^2*b^4*c^6*d + 48*a^3*b^3*c^5*d^2 + 48*a^4*b^2*c^4*d^3 - 135*a^5*b*c^3*d
^4 + 54*a^6*c^2*d^5)*x^3 + (60*b^6*c^7 - 8*a*b^5*c^6*d - 364*a^2*b^4*c^5*d^2 + 1
92*a^3*b^3*c^4*d^3 - 234*a^4*b^2*c^3*d^4 + 3*a^5*b*c^2*d^5 + 36*a^6*c*d^6)*x^2 +
(120*b^6*c^6*d - 256*a*b^5*c^5*d^2 - 80*a^2*b^4*c^4*d^3 - 15*a^3*b^3*c^3*d^4 -
156*a^4*b^2*c^2*d^5 + 72*a^5*b*c*d^6)*x)*sqrt(a))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*
c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6 + (a^4*b^4*c^10
- 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d^4)*x^3 + (a^
3*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c
^6*d^4 + 2*a^8*c^5*d^5)*x^2 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3*c
^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x)*sqrt(a)*sqrt((a*x
+ b)/x)), 1/24*(3*(99*a^3*b^3*c^2*d^5 - 88*a^4*b^2*c*d^6 + 24*a^5*b*d^7 + (99*a
^4*b^2*c^4*d^3 - 88*a^5*b*c^3*d^4 + 24*a^6*c^2*d^5)*x^3 + (99*a^3*b^3*c^4*d^3 +
110*a^4*b^2*c^3*d^4 - 152*a^5*b*c^2*d^5 + 48*a^6*c*d^6)*x^2 + (198*a^3*b^3*c^3*d
^4 - 77*a^4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 24*a^6*d^7)*x)*sqrt(-a)*sqrt(-d/(b*c
- a*d))*sqrt((a*x + b)/x)*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x +
b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 24*(5*b^6*c^5*d^2 - 14*a*b^5*c^4*d^
3 + 6*a^2*b^4*c^3*d^4 + 16*a^3*b^3*c^2*d^5 - 19*a^4*b^2*c*d^6 + 6*a^5*b*d^7 + (5
*a*b^5*c^7 - 14*a^2*b^4*c^6*d + 6*a^3*b^3*c^5*d^2 + 16*a^4*b^2*c^4*d^3 - 19*a^5*
b*c^3*d^4 + 6*a^6*c^2*d^5)*x^3 + (5*b^6*c^7 - 4*a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2
+ 28*a^3*b^3*c^4*d^3 + 13*a^4*b^2*c^3*d^4 - 32*a^5*b*c^2*d^5 + 12*a^6*c*d^6)*x^
2 + (10*b^6*c^6*d - 23*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3 + 38*a^3*b^3*c^3*d^4 -
22*a^4*b^2*c^2*d^5 - 7*a^5*b*c*d^6 + 6*a^6*d^7)*x)*sqrt((a*x + b)/x)*arctan(a/(s
qrt(-a)*sqrt((a*x + b)/x))) + 2*(60*b^6*c^5*d^2 - 168*a*b^5*c^4*d^3 + 72*a^2*b^4
*c^3*d^4 - 105*a^3*b^3*c^2*d^5 + 36*a^4*b^2*c*d^6 + 12*(a^2*b^4*c^7 - 4*a^3*b^3*
c^6*d + 6*a^4*b^2*c^5*d^2 - 4*a^5*b*c^4*d^3 + a^6*c^3*d^4)*x^4 + (80*a*b^5*c^7 -
200*a^2*b^4*c^6*d + 48*a^3*b^3*c^5*d^2 + 48*a^4*b^2*c^4*d^3 - 135*a^5*b*c^3*d^4
+ 54*a^6*c^2*d^5)*x^3 + (60*b^6*c^7 - 8*a*b^5*c^6*d - 364*a^2*b^4*c^5*d^2 + 192
*a^3*b^3*c^4*d^3 - 234*a^4*b^2*c^3*d^4 + 3*a^5*b*c^2*d^5 + 36*a^6*c*d^6)*x^2 + (
120*b^6*c^6*d - 256*a*b^5*c^5*d^2 - 80*a^2*b^4*c^4*d^3 - 15*a^3*b^3*c^3*d^4 - 15
6*a^4*b^2*c^2*d^5 + 72*a^5*b*c*d^6)*x)*sqrt(-a))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c
^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6 + (a^4*b^4*c^10 -
4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d^4)*x^3 + (a^3
*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^
6*d^4 + 2*a^8*c^5*d^5)*x^2 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3*c^
7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x)*sqrt(-a)*sqrt((a*x
+ b)/x)), -1/12*(3*(99*a^3*b^3*c^2*d^5 - 88*a^4*b^2*c*d^6 + 24*a^5*b*d^7 + (99*
a^4*b^2*c^4*d^3 - 88*a^5*b*c^3*d^4 + 24*a^6*c^2*d^5)*x^3 + (99*a^3*b^3*c^4*d^3 +
110*a^4*b^2*c^3*d^4 - 152*a^5*b*c^2*d^5 + 48*a^6*c*d^6)*x^2 + (198*a^3*b^3*c^3*
d^4 - 77*a^4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 24*a^6*d^7)*x)*sqrt(a)*sqrt(d/(b*c -
a*d))*sqrt((a*x + b)/x)*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x +
b)/x))) - 6*(5*b^6*c^5*d^2 - 14*a*b^5*c^4*d^3 + 6*a^2*b^4*c^3*d^4 + 16*a^3*b^3*c
^2*d^5 - 19*a^4*b^2*c*d^6 + 6*a^5*b*d^7 + (5*a*b^5*c^7 - 14*a^2*b^4*c^6*d + 6*a^
3*b^3*c^5*d^2 + 16*a^4*b^2*c^4*d^3 - 19*a^5*b*c^3*d^4 + 6*a^6*c^2*d^5)*x^3 + (5*
b^6*c^7 - 4*a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2 + 28*a^3*b^3*c^4*d^3 + 13*a^4*b^2*c
^3*d^4 - 32*a^5*b*c^2*d^5 + 12*a^6*c*d^6)*x^2 + (10*b^6*c^6*d - 23*a*b^5*c^5*d^2
- 2*a^2*b^4*c^4*d^3 + 38*a^3*b^3*c^3*d^4 - 22*a^4*b^2*c^2*d^5 - 7*a^5*b*c*d^6 +
6*a^6*d^7)*x)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt
(a)) - (60*b^6*c^5*d^2 - 168*a*b^5*c^4*d^3 + 72*a^2*b^4*c^3*d^4 - 105*a^3*b^3*c^
2*d^5 + 36*a^4*b^2*c*d^6 + 12*(a^2*b^4*c^7 - 4*a^3*b^3*c^6*d + 6*a^4*b^2*c^5*d^2
- 4*a^5*b*c^4*d^3 + a^6*c^3*d^4)*x^4 + (80*a*b^5*c^7 - 200*a^2*b^4*c^6*d + 48*a
^3*b^3*c^5*d^2 + 48*a^4*b^2*c^4*d^3 - 135*a^5*b*c^3*d^4 + 54*a^6*c^2*d^5)*x^3 +
(60*b^6*c^7 - 8*a*b^5*c^6*d - 364*a^2*b^4*c^5*d^2 + 192*a^3*b^3*c^4*d^3 - 234*a^
4*b^2*c^3*d^4 + 3*a^5*b*c^2*d^5 + 36*a^6*c*d^6)*x^2 + (120*b^6*c^6*d - 256*a*b^5
*c^5*d^2 - 80*a^2*b^4*c^4*d^3 - 15*a^3*b^3*c^3*d^4 - 156*a^4*b^2*c^2*d^5 + 72*a^
5*b*c*d^6)*x)*sqrt(a))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4
- 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b
^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d^4)*x^3 + (a^3*b^5*c^10 - 2*a^4*b^4*c^9*
d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4 + 2*a^8*c^5*d^5)*x^2
+ (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3*c^7*d^3 - 2*a^6*b^2*c^6*d^4
- 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x)*sqrt(a)*sqrt((a*x + b)/x)), -1/12*(3*(99*a^3
*b^3*c^2*d^5 - 88*a^4*b^2*c*d^6 + 24*a^5*b*d^7 + (99*a^4*b^2*c^4*d^3 - 88*a^5*b*
c^3*d^4 + 24*a^6*c^2*d^5)*x^3 + (99*a^3*b^3*c^4*d^3 + 110*a^4*b^2*c^3*d^4 - 152*
a^5*b*c^2*d^5 + 48*a^6*c*d^6)*x^2 + (198*a^3*b^3*c^3*d^4 - 77*a^4*b^2*c^2*d^5 -
40*a^5*b*c*d^6 + 24*a^6*d^7)*x)*sqrt(-a)*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)*a
rctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) - 12*(5*b^6*c^5*d^
2 - 14*a*b^5*c^4*d^3 + 6*a^2*b^4*c^3*d^4 + 16*a^3*b^3*c^2*d^5 - 19*a^4*b^2*c*d^6
+ 6*a^5*b*d^7 + (5*a*b^5*c^7 - 14*a^2*b^4*c^6*d + 6*a^3*b^3*c^5*d^2 + 16*a^4*b^
2*c^4*d^3 - 19*a^5*b*c^3*d^4 + 6*a^6*c^2*d^5)*x^3 + (5*b^6*c^7 - 4*a*b^5*c^6*d -
22*a^2*b^4*c^5*d^2 + 28*a^3*b^3*c^4*d^3 + 13*a^4*b^2*c^3*d^4 - 32*a^5*b*c^2*d^5
+ 12*a^6*c*d^6)*x^2 + (10*b^6*c^6*d - 23*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3 + 38
*a^3*b^3*c^3*d^4 - 22*a^4*b^2*c^2*d^5 - 7*a^5*b*c*d^6 + 6*a^6*d^7)*x)*sqrt((a*x
+ b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (60*b^6*c^5*d^2 - 168*a*b^5*c^4
*d^3 + 72*a^2*b^4*c^3*d^4 - 105*a^3*b^3*c^2*d^5 + 36*a^4*b^2*c*d^6 + 12*(a^2*b^4
*c^7 - 4*a^3*b^3*c^6*d + 6*a^4*b^2*c^5*d^2 - 4*a^5*b*c^4*d^3 + a^6*c^3*d^4)*x^4
+ (80*a*b^5*c^7 - 200*a^2*b^4*c^6*d + 48*a^3*b^3*c^5*d^2 + 48*a^4*b^2*c^4*d^3 -
135*a^5*b*c^3*d^4 + 54*a^6*c^2*d^5)*x^3 + (60*b^6*c^7 - 8*a*b^5*c^6*d - 364*a^2*
b^4*c^5*d^2 + 192*a^3*b^3*c^4*d^3 - 234*a^4*b^2*c^3*d^4 + 3*a^5*b*c^2*d^5 + 36*a
^6*c*d^6)*x^2 + (120*b^6*c^6*d - 256*a*b^5*c^5*d^2 - 80*a^2*b^4*c^4*d^3 - 15*a^3
*b^3*c^3*d^4 - 156*a^4*b^2*c^2*d^5 + 72*a^5*b*c*d^6)*x)*sqrt(-a))/((a^3*b^5*c^8*
d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6
+ (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^
6*d^4)*x^3 + (a^3*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7
*d^3 - 7*a^7*b*c^6*d^4 + 2*a^8*c^5*d^5)*x^2 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d
^2 + 8*a^5*b^3*c^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x)*s
qrt(-a)*sqrt((a*x + b)/x))]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266349, size = 703, normalized size = 1.72 \[ -\frac{1}{12} \, b{\left (\frac{3 \,{\left (99 \, b^{2} c^{2} d^{4} - 88 \, a b c d^{5} + 24 \, a^{2} d^{6}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{5} c^{8} - 4 \, a b^{4} c^{7} d + 6 \, a^{2} b^{3} c^{6} d^{2} - 4 \, a^{3} b^{2} c^{5} d^{3} + a^{4} b c^{4} d^{4}\right )} \sqrt{b c d - a d^{2}}} - \frac{8 \,{\left (a b^{4} c - a^{2} b^{3} d + \frac{6 \,{\left (a x + b\right )} b^{4} c}{x} - \frac{15 \,{\left (a x + b\right )} a b^{3} d}{x}\right )} x}{{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}} + \frac{3 \,{\left (21 \, b^{2} c^{2} d^{4} \sqrt{\frac{a x + b}{x}} - 29 \, a b c d^{5} \sqrt{\frac{a x + b}{x}} + 8 \, a^{2} d^{6} \sqrt{\frac{a x + b}{x}} + \frac{19 \,{\left (a x + b\right )} b c d^{5} \sqrt{\frac{a x + b}{x}}}{x} - \frac{8 \,{\left (a x + b\right )} a d^{6} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}\right )}{\left (b c - a d + \frac{{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac{12 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3} c^{3}} - \frac{12 \,{\left (5 \, b c + 6 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*(c + d/x)^3),x, algorithm="giac")

[Out]

-1/12*b*(3*(99*b^2*c^2*d^4 - 88*a*b*c*d^5 + 24*a^2*d^6)*arctan(d*sqrt((a*x + b)/
x)/sqrt(b*c*d - a*d^2))/((b^5*c^8 - 4*a*b^4*c^7*d + 6*a^2*b^3*c^6*d^2 - 4*a^3*b^
2*c^5*d^3 + a^4*b*c^4*d^4)*sqrt(b*c*d - a*d^2)) - 8*(a*b^4*c - a^2*b^3*d + 6*(a*
x + b)*b^4*c/x - 15*(a*x + b)*a*b^3*d/x)*x/((a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a
^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*(a*x + b)*sqrt((a*x + b)/x)) + 3*(21*b
^2*c^2*d^4*sqrt((a*x + b)/x) - 29*a*b*c*d^5*sqrt((a*x + b)/x) + 8*a^2*d^6*sqrt((
a*x + b)/x) + 19*(a*x + b)*b*c*d^5*sqrt((a*x + b)/x)/x - 8*(a*x + b)*a*d^6*sqrt(
(a*x + b)/x)/x)/((b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3
+ a^4*c^3*d^4)*(b*c - a*d + (a*x + b)*d/x)^2) + 12*sqrt((a*x + b)/x)/((a - (a*x
+ b)/x)*a^3*c^3) - 12*(5*b*c + 6*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-
a)*a^3*b*c^4))