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

Optimal. Leaf size=201 \[ -\frac{(2 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^2}+\frac{b (5 b c-3 a d)}{3 a^2 c \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)}+\frac{b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a^3 c \sqrt{a+\frac{b}{x}} (b c-a d)^2}-\frac{2 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 (b c-a d)^{5/2}}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(b*(5*b*c - 3*a*d))/(3*a^2*c*(b*c - a*d)*(a + b/x)^(3/2)) + (b*(5*b^2*c^2 - 8*a*
b*c*d + a^2*d^2))/(a^3*c*(b*c - a*d)^2*Sqrt[a + b/x]) + x/(a*c*(a + b/x)^(3/2))
- (2*d^(7/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c^2*(b*c - a*d)^(
5/2)) - ((5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^2)

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Rubi [A]  time = 0.946317, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(2 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^2}+\frac{b (5 b c-3 a d)}{3 a^2 c \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)}+\frac{b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a^3 c \sqrt{a+\frac{b}{x}} (b c-a d)^2}-\frac{2 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 (b c-a d)^{5/2}}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(b*(5*b*c - 3*a*d))/(3*a^2*c*(b*c - a*d)*(a + b/x)^(3/2)) + (b*(5*b^2*c^2 - 8*a*
b*c*d + a^2*d^2))/(a^3*c*(b*c - a*d)^2*Sqrt[a + b/x]) + x/(a*c*(a + b/x)^(3/2))
- (2*d^(7/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c^2*(b*c - a*d)^(
5/2)) - ((5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^2)

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Rubi in Sympy [A]  time = 106.153, size = 173, normalized size = 0.86 \[ \frac{2 d^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{2} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{x}{a c \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{b \left (3 a d - 5 b c\right )}{3 a^{2} c \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{b \left (a^{2} d^{2} - 8 a b c d + 5 b^{2} c^{2}\right )}{a^{3} c \sqrt{a + \frac{b}{x}} \left (a d - b c\right )^{2}} - \frac{\left (2 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d**(7/2)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**2*(a*d - b*c)**(5/2)
) + x/(a*c*(a + b/x)**(3/2)) + b*(3*a*d - 5*b*c)/(3*a**2*c*(a + b/x)**(3/2)*(a*d
- b*c)) + b*(a**2*d**2 - 8*a*b*c*d + 5*b**2*c**2)/(a**3*c*sqrt(a + b/x)*(a*d -
b*c)**2) - (2*a*d + 5*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/(a**(7/2)*c**2)

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Mathematica [A]  time = 0.963246, size = 250, normalized size = 1.24 \[ \frac{-\frac{3 (2 a d+5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{7/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (3 a^4 d^2 x^2+6 a^3 b d x (d-c x)+a^2 b^2 \left (3 c^2 x^2-32 c d x+3 d^2\right )+4 a b^3 c (5 c x-6 d)+15 b^4 c^2\right )}{a^3 (a x+b)^2 (b c-a d)^2}+\frac{6 d^{7/2} \log (c x+d)}{(a d-b c)^{5/2}}-\frac{6 d^{7/2} \log \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{a d-b c}-2 a d x+b c x-b d\right )}{(a d-b c)^{5/2}}}{6 c^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*Sqrt[a + b/x]*x*(15*b^4*c^2 + 3*a^4*d^2*x^2 + 6*a^3*b*d*x*(d - c*x) + 4*a*
b^3*c*(-6*d + 5*c*x) + a^2*b^2*(3*d^2 - 32*c*d*x + 3*c^2*x^2)))/(a^3*(b*c - a*d)
^2*(b + a*x)^2) + (6*d^(7/2)*Log[d + c*x])/(-(b*c) + a*d)^(5/2) - (3*(5*b*c + 2*
a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(7/2) - (6*d^(7/2)*Log[-(b*d)
+ b*c*x - 2*a*d*x + 2*Sqrt[d]*Sqrt[-(b*c) + a*d]*Sqrt[a + b/x]*x])/(-(b*c) + a*
d)^(5/2))/(6*c^2)

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Maple [B]  time = 0.024, size = 2490, normalized size = 12.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/(c+d/x),x)

[Out]

1/6*((a*x+b)/x)^(1/2)*x/a^(13/2)*(-54*(x*(a*x+b))^(1/2)*a^(17/2)*((a*d-b*c)*d/c^
2)^(1/2)*x^3*b*c^3*d^2+78*(x*(a*x+b))^(1/2)*a^(15/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3
*b^2*c^4*d-18*a^9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*
c)*d/c^2)^(1/2)*x^2*b*c*d^4+3*a^9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^3*b*c^2*d^3+36*(x*(a*x+b))^(3/2)*a^(15/2)*((a*
d-b*c)*d/c^2)^(1/2)*x*b*c^3*d^2-60*(x*(a*x+b))^(3/2)*a^(13/2)*((a*d-b*c)*d/c^2)^
(1/2)*x*b^2*c^4*d+18*(x*(a*x+b))^(1/2)*a^(17/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^
2*d^3+3*a^6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c
^2)^(1/2)*b^4*c^2*d^3+27*a^5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2
))*((a*d-b*c)*d/c^2)^(1/2)*b^5*c^3*d^2-39*a^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*b^6*c^4*d-6*a^10*ln(1/2*(2*(x*(a*x+b
))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^3*c*d^4+6*(x*(a*x+b
))^(1/2)*a^(19/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*c^2*d^3-30*(x*(a*x+b))^(1/2)*a^(13
/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*b^3*c^5+6*a^(19/2)*ln((2*(x*(a*x+b))^(1/2)*((a*d
-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^3*b*c*d^4+24*(x*(a*x+b))^(3/2
)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^5-90*(x*(a*x+b))^(1/2)*a^(11/2)*((a*d
-b*c)*d/c^2)^(1/2)*x^2*b^4*c^5+15*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^3*a^6*b^4*c^5+45*ln(1/2*(2*(x*(a*x+b))^(1/2)*a
^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^2*a^5*b^5*c^5+45*ln(1/2*(2*(x
*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*a^4*b^6*c^5-
162*(x*(a*x+b))^(1/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^3*d^2+234*(x*(a*x
+b))^(1/2)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^4*c^4*d+18*a^(17/2)*ln((2*(x*(a*
x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b^2*c*d^4+
32*(x*(a*x+b))^(3/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^3*d^2-52*(x*(a*x+b))
^(3/2)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*b^3*c^4*d-90*(x*(a*x+b))^(1/2)*a^(9/2)*(
(a*d-b*c)*d/c^2)^(1/2)*x*b^5*c^5-6*a^7*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x
+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*b^3*c*d^4+18*a^(15/2)*ln((2*(x*(a*x+b))^(1/
2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x*b^3*c*d^4+6*(x*(a*x+b
))^(1/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*b^3*c^2*d^3-54*(x*(a*x+b))^(1/2)*a^(11
/2)*((a*d-b*c)*d/c^2)^(1/2)*b^4*c^3*d^2+78*(x*(a*x+b))^(1/2)*a^(9/2)*((a*d-b*c)*
d/c^2)^(1/2)*b^5*c^4*d-6*a^(21/2)*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2
)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^3*d^5-6*a^(15/2)*ln((2*(x*(a*x+b))^(1/2)*((a*d
-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b^3*d^5+9*a^7*ln(1/2*(2*(x*(a*x
+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^2*d^3+81*a^
7*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*
x^2*b^3*c^3*d^2-39*a^7*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a
*d-b*c)*d/c^2)^(1/2)*x^3*b^3*c^4*d+81*a^6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*b^4*c^3*d^2-117*a^6*ln(1/2*(2*(x*(a*x+
b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^4*c^4*d-117*a^
5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*
x*b^5*c^4*d-162*(x*(a*x+b))^(1/2)*a^(15/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^3*d
^2+234*(x*(a*x+b))^(1/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^3*c^4*d-18*a^8*l
n(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*b
^2*c*d^4+9*a^8*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*
d/c^2)^(1/2)*x^2*b^2*c^2*d^3+27*a^8*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)
/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^3*b^2*c^3*d^2+18*(x*(a*x+b))^(1/2)*a^(15/2)*
((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^2*d^3-18*a^(19/2)*ln((2*(x*(a*x+b))^(1/2)*((a*d-
b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b*d^5+20*(x*(a*x+b))^(3/2)*a
^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*b^4*c^5-18*a^(17/2)*ln((2*(x*(a*x+b))^(1/2)*((a*d
-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x*b^2*d^5+6*a^(13/2)*ln((2*(x*(
a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b^4*c*d^4-30
*(x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*b^6*c^5+15*ln(1/2*(2*(x*(a*x+
b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*a^3*b^7*c^5)/(x*(a*x
+b))^(1/2)/(a*d-b*c)^3/c^3/((a*d-b*c)*d/c^2)^(1/2)/(a*x+b)^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.903203, 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)),x, algorithm="fricas")

[Out]

[1/6*(6*(a^4*d^3*x + a^3*b*d^3)*sqrt(a)*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x)*l
og(-(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)) + 3*(5*b^4*c^3 - 8*a*b^3*c^2*d + a^2*b^2*c*d^2 + 2*a^3*b*d^3 + (
5*a*b^3*c^3 - 8*a^2*b^2*c^2*d + a^3*b*c*d^2 + 2*a^4*d^3)*x)*sqrt((a*x + b)/x)*lo
g(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(15*b^4*c^3 - 24*a*b^3*c^2
*d + 3*a^2*b^2*c*d^2 + 3*(a^2*b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2)*x^2 + 2*(10*a
*b^3*c^3 - 16*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)*x)*sqrt(a))/((a^3*b^3*c^4 - 2*a^4*b
^2*c^3*d + a^5*b*c^2*d^2 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x)*sqrt(a
)*sqrt((a*x + b)/x)), 1/3*(3*(a^4*d^3*x + a^3*b*d^3)*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)) + 3*(5*b^4*c^3 - 8*a*b^3*c^2*d + a^2*b^2*c
*d^2 + 2*a^3*b*d^3 + (5*a*b^3*c^3 - 8*a^2*b^2*c^2*d + a^3*b*c*d^2 + 2*a^4*d^3)*x
)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) + (15*b^4*c^3 - 24*a*
b^3*c^2*d + 3*a^2*b^2*c*d^2 + 3*(a^2*b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2)*x^2 +
2*(10*a*b^3*c^3 - 16*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)*x)*sqrt(-a))/((a^3*b^3*c^4 -
2*a^4*b^2*c^3*d + a^5*b*c^2*d^2 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x
)*sqrt(-a)*sqrt((a*x + b)/x)), -1/6*(12*(a^4*d^3*x + a^3*b*d^3)*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))) - 3*(5*b^4*c^3 - 8*a*b^3*c^2*d + a^2*b^2*c*d^2 + 2*a^3*b*d^3 + (5*a
*b^3*c^3 - 8*a^2*b^2*c^2*d + a^3*b*c*d^2 + 2*a^4*d^3)*x)*sqrt((a*x + b)/x)*log(-
2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) - 2*(15*b^4*c^3 - 24*a*b^3*c^2*d
+ 3*a^2*b^2*c*d^2 + 3*(a^2*b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2)*x^2 + 2*(10*a*b^
3*c^3 - 16*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)*x)*sqrt(a))/((a^3*b^3*c^4 - 2*a^4*b^2*
c^3*d + a^5*b*c^2*d^2 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x)*sqrt(a)*s
qrt((a*x + b)/x)), -1/3*(6*(a^4*d^3*x + a^3*b*d^3)*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)))
- 3*(5*b^4*c^3 - 8*a*b^3*c^2*d + a^2*b^2*c*d^2 + 2*a^3*b*d^3 + (5*a*b^3*c^3 - 8
*a^2*b^2*c^2*d + a^3*b*c*d^2 + 2*a^4*d^3)*x)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a
)*sqrt((a*x + b)/x))) - (15*b^4*c^3 - 24*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 + 3*(a^2*
b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2)*x^2 + 2*(10*a*b^3*c^3 - 16*a^2*b^2*c^2*d +
3*a^3*b*c*d^2)*x)*sqrt(-a))/((a^3*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2 + (a
^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x)*sqrt(-a)*sqrt((a*x + b)/x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + \frac{b}{x}\right )^{\frac{5}{2}} \left (c x + d\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.258445, size = 332, normalized size = 1.65 \[ -\frac{1}{3} \,{\left (\frac{6 \, d^{4} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2}\right )} \sqrt{b c d - a d^{2}}} - \frac{2 \,{\left (a b^{2} c - a^{2} b d + \frac{6 \,{\left (a x + b\right )} b^{2} c}{x} - \frac{9 \,{\left (a x + b\right )} a b d}{x}\right )} x}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}} + \frac{3 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3} c} - \frac{3 \,{\left (5 \, b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b c^{2}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(6*d^4*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^3*c^4 - 2*a*b^2*
c^3*d + a^2*b*c^2*d^2)*sqrt(b*c*d - a*d^2)) - 2*(a*b^2*c - a^2*b*d + 6*(a*x + b)
*b^2*c/x - 9*(a*x + b)*a*b*d/x)*x/((a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*(a*x +
b)*sqrt((a*x + b)/x)) + 3*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a^3*c) - 3*(5*b*c
+ 2*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3*b*c^2))*b