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

Optimal. Leaf size=224 \[ -\frac{(4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^3}+\frac{b \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^2}+\frac{d^{5/2} (7 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{5/2}}+\frac{d (b c-2 a d)}{a c^2 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.96161, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{(4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^3}+\frac{b \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^2}+\frac{d^{5/2} (7 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{5/2}}+\frac{d (b c-2 a d)}{a c^2 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 110.149, size = 194, normalized size = 0.87 \[ - \frac{d x}{c \sqrt{a + \frac{b}{x}} \left (c + \frac{d}{x}\right ) \left (a d - b c\right )} + \frac{d^{\frac{5}{2}} \left (4 a d - 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{3} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{x \left (2 a d - b c\right )}{a c^{2} \sqrt{a + \frac{b}{x}} \left (a d - b c\right )} + \frac{b \left (2 a^{2} d^{2} - 2 a b c d + 3 b^{2} c^{2}\right )}{a^{2} c^{2} \sqrt{a + \frac{b}{x}} \left (a d - b c\right )^{2}} - \frac{\left (4 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*x/(c*sqrt(a + b/x)*(c + d/x)*(a*d - b*c)) + d**(5/2)*(4*a*d - 7*b*c)*atanh(sq
rt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**3*(a*d - b*c)**(5/2)) + x*(2*a*d - b*c)
/(a*c**2*sqrt(a + b/x)*(a*d - b*c)) + b*(2*a**2*d**2 - 2*a*b*c*d + 3*b**2*c**2)/
(a**2*c**2*sqrt(a + b/x)*(a*d - b*c)**2) - (4*a*d + 3*b*c)*atanh(sqrt(a + b/x)/s
qrt(a))/(a**(5/2)*c**3)

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Mathematica [C]  time = 1.10813, size = 290, normalized size = 1.29 \[ \frac{-\frac{(4 a d+3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{5/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (a^3 d^2 x (c x+2 d)+a^2 b d \left (-2 c^2 x^2-c d x+2 d^2\right )+a b^2 c \left (c^2 x^2-c d x-2 d^2\right )+3 b^3 c^2 (c x+d)\right )}{a^2 (a x+b) (c x+d) (b c-a d)^2}+\frac{i d^{5/2} (7 b c-4 a d) \log \left (-\frac{2 i c^4 (b c-a d)^{3/2} \left (-2 i \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 a d x+b c x-b d\right )}{d^{7/2} (c x+d) (7 b c-4 a d)}\right )}{(b c-a d)^{5/2}}}{2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 3121, normalized size = 13.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((a*x+b)/x)^(1/2)*x/a^(9/2)*(-6*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^2)^
(1/2)*b^5*c^5*d+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*a^3*b^5*c^6+3*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^2*b^6*c^6+3*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^2*b^6*c^5*d-4*ln(1/2*(2*(x*(a*x+
b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^8*((a*d-b*c)*d/c^2)^(1/2)*x^2*c*d^5+2*(x*(
a*x+b))^(3/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*c^4*d^2+2*(x*(a*x+b))^(1/2)*a
^(15/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*c^3*d^3-6*(x*(a*x+b))^(1/2)*a^(9/2)*((a*d-b*
c)*d/c^2)^(1/2)*x^3*b^3*c^6+11*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))*a^(15/2)*x^3*b*c^2*d^4-7*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))*a^(13/2)*x^3*b^2*c^3*d^3+4*(
x*(a*x+b))^(3/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^6+4*(x*(a*x+b))^(1/2)*a
^(15/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*c^2*d^4+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*a^7*((a*d-b*c)*d/c^2)^(1/2)*x^3*b*c^3*d^3+14*(x*(a*x+b))^(1/
2)*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^3*c^5*d+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a
^(1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^3*d^3-11*ln(1/2*(2*
(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^3*
c^4*d^2-5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d
/c^2)^(1/2)*x^3*b^3*c^5*d-12*(x*(a*x+b))^(1/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*
x^3*b*c^4*d^2+ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*
c)*d/c^2)^(1/2)*x*b^5*c^5*d-7*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/
2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^4*c^5*d-8*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1
/2)+2*a*x+b)/a^(1/2))*a^7*((a*d-b*c)*d/c^2)^(1/2)*x*b*c*d^5+14*ln(1/2*(2*(x*(a*x
+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^2*d^4+1
5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*((a*d-b*c)*d/c^2)^(1
/2)*x^2*b^2*c^3*d^3-3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*
((a*d-b*c)*d/c^2)^(1/2)*x^3*b^2*c^4*d^2+4*(x*(a*x+b))^(3/2)*a^(11/2)*((a*d-b*c)*
d/c^2)^(1/2)*x*b*c^4*d^2+10*(x*(a*x+b))^(1/2)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*x
^3*b^2*c^5*d-4*(x*(a*x+b))^(3/2)*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^5*d-4*(
x*(a*x+b))^(1/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^3*d^3-8*(x*(a*x+b))^(1
/2)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^4*d^2-5*ln(1/2*(2*(x*(a*x+b))^(1/
2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*b^5*c^4*d^2-4*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))*a^(17/2)*x^
3*c*d^5-8*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))*a^(15/2)*x*b*d^6+11*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))*a^(11/2)*b^3*c*d^5-7*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))*a^(9/2)*b^4*c^2*d^4-12*(x*(a*x+b))^
(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^4*c^6-4*ln(1/2*(2*(x*(a*x+b))^(1/2)*
a^(1/2)+2*a*x+b)/a^(1/2))*a^6*((a*d-b*c)*d/c^2)^(1/2)*b^2*c*d^5+3*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))*a^(15/2)*x^2*b*c
*d^5+15*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))*a^(13/2)*x^2*b^2*c^2*d^4-14*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))*a^(11/2)*x^2*b^3*c^3*d^3-2*(x*(a*x+b))^(3/2)*a^(
9/2)*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^4*d^2+4*(x*(a*x+b))^(3/2)*a^(7/2)*((a*d-b*c)*
d/c^2)^(1/2)*b^3*c^5*d+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a
^5*((a*d-b*c)*d/c^2)^(1/2)*b^3*c^2*d^4-4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a
*x+b)/a^(1/2))*a^8*((a*d-b*c)*d/c^2)^(1/2)*x^3*c^2*d^4-2*(x*(a*x+b))^(1/2)*a^(15
/2)*((a*d-b*c)*d/c^2)^(1/2)*x^4*c^4*d^2+18*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))*a^(13/2)*x*b^2*c*d^5-3*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))*a^(11/2)*x*b^3*c^2
*d^4-7*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))*a^(9/2)*x*b^4*c^3*d^3+4*(x*(a*x+b))^(1/2)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*
b^2*c^2*d^4-8*(x*(a*x+b))^(1/2)*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*b^3*c^3*d^3+10*(
x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*b^4*c^4*d^2-3*ln(1/2*(2*(x*(a*x
+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*b^4*c^3*d^3+8*(
x*(a*x+b))^(1/2)*a^(13/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^2*d^4-14*(x*(a*x+b))^(1/
2)*a^(11/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^3*d^3+12*(x*(a*x+b))^(1/2)*a^(9/2)*(
(a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^4*d^2-2*(x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^
2)^(1/2)*x*b^4*c^5*d-13*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^
4*((a*d-b*c)*d/c^2)^(1/2)*x*b^4*c^4*d^2+ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*
x+b)/a^(1/2))*a^7*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^2*d^4-6*(x*(a*x+b))^(1/2)*a^(5
/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^5*c^6+3*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^4*b^4*c^6-4*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))*a^(17/2)*x^2*d^6-4*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))*a^(13/2)*
b^2*d^6)/(x*(a*x+b))^(1/2)/(a*d-b*c)^3/c^4/((a*d-b*c)*d/c^2)^(1/2)/(a*x+b)^2/(c*
x+d)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.812128, 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)^(3/2)*(c + d/x)^2),x, algorithm="fricas")

[Out]

[-1/2*((7*a^2*b*c*d^3 - 4*a^3*d^4 + (7*a^2*b*c^2*d^2 - 4*a^3*c*d^3)*x)*sqrt(a)*s
qrt(-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*b^3*c^3*d - 2*a*b^2*
c^2*d^2 - 5*a^2*b*c*d^3 + 4*a^3*d^4 + (3*b^3*c^4 - 2*a*b^2*c^3*d - 5*a^2*b*c^2*d
^2 + 4*a^3*c*d^3)*x)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b
)*sqrt(a)) - 2*(3*b^3*c^3*d - 2*a*b^2*c^2*d^2 + 2*a^2*b*c*d^3 + (a*b^2*c^4 - 2*a
^2*b*c^3*d + a^3*c^2*d^2)*x^2 + (3*b^3*c^4 - a*b^2*c^3*d - a^2*b*c^2*d^2 + 2*a^3
*c*d^3)*x)*sqrt(a))/((a^2*b^2*c^5*d - 2*a^3*b*c^4*d^2 + a^4*c^3*d^3 + (a^2*b^2*c
^6 - 2*a^3*b*c^5*d + a^4*c^4*d^2)*x)*sqrt(a)*sqrt((a*x + b)/x)), 1/2*(2*(7*a^2*b
*c*d^3 - 4*a^3*d^4 + (7*a^2*b*c^2*d^2 - 4*a^3*c*d^3)*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))) + (3*b^3*c^3*d - 2*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 + 4*a^3*d^4 + (3*b^3*c^4 -
2*a*b^2*c^3*d - 5*a^2*b*c^2*d^2 + 4*a^3*c*d^3)*x)*sqrt((a*x + b)/x)*log(-2*a*x*
sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(3*b^3*c^3*d - 2*a*b^2*c^2*d^2 + 2*
a^2*b*c*d^3 + (a*b^2*c^4 - 2*a^2*b*c^3*d + a^3*c^2*d^2)*x^2 + (3*b^3*c^4 - a*b^2
*c^3*d - a^2*b*c^2*d^2 + 2*a^3*c*d^3)*x)*sqrt(a))/((a^2*b^2*c^5*d - 2*a^3*b*c^4*
d^2 + a^4*c^3*d^3 + (a^2*b^2*c^6 - 2*a^3*b*c^5*d + a^4*c^4*d^2)*x)*sqrt(a)*sqrt(
(a*x + b)/x)), -1/2*((7*a^2*b*c*d^3 - 4*a^3*d^4 + (7*a^2*b*c^2*d^2 - 4*a^3*c*d^3
)*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)) - 2*(3*b^3
*c^3*d - 2*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 + 4*a^3*d^4 + (3*b^3*c^4 - 2*a*b^2*c^3*
d - 5*a^2*b*c^2*d^2 + 4*a^3*c*d^3)*x)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt(
(a*x + b)/x))) - 2*(3*b^3*c^3*d - 2*a*b^2*c^2*d^2 + 2*a^2*b*c*d^3 + (a*b^2*c^4 -
2*a^2*b*c^3*d + a^3*c^2*d^2)*x^2 + (3*b^3*c^4 - a*b^2*c^3*d - a^2*b*c^2*d^2 + 2
*a^3*c*d^3)*x)*sqrt(-a))/((a^2*b^2*c^5*d - 2*a^3*b*c^4*d^2 + a^4*c^3*d^3 + (a^2*
b^2*c^6 - 2*a^3*b*c^5*d + a^4*c^4*d^2)*x)*sqrt(-a)*sqrt((a*x + b)/x)), ((7*a^2*b
*c*d^3 - 4*a^3*d^4 + (7*a^2*b*c^2*d^2 - 4*a^3*c*d^3)*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))) + (3*b^3*c^3*d - 2*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 + 4*a^3*d^4 + (3*b^3*c^4
- 2*a*b^2*c^3*d - 5*a^2*b*c^2*d^2 + 4*a^3*c*d^3)*x)*sqrt((a*x + b)/x)*arctan(a/(
sqrt(-a)*sqrt((a*x + b)/x))) + (3*b^3*c^3*d - 2*a*b^2*c^2*d^2 + 2*a^2*b*c*d^3 +
(a*b^2*c^4 - 2*a^2*b*c^3*d + a^3*c^2*d^2)*x^2 + (3*b^3*c^4 - a*b^2*c^3*d - a^2*b
*c^2*d^2 + 2*a^3*c*d^3)*x)*sqrt(-a))/((a^2*b^2*c^5*d - 2*a^3*b*c^4*d^2 + a^4*c^3
*d^3 + (a^2*b^2*c^6 - 2*a^3*b*c^5*d + a^4*c^4*d^2)*x)*sqrt(-a)*sqrt((a*x + b)/x)
)]

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Sympy [F(-2)]  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)**(3/2)/(c+d/x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.256391, size = 560, normalized size = 2.5 \[ b{\left (\frac{{\left (7 \, b c d^{3} - 4 \, a d^{4}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{3} c^{5} - 2 \, a b^{2} c^{4} d + a^{2} b c^{3} d^{2}\right )} \sqrt{b c d - a d^{2}}} + \frac{2 \, a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d - \frac{3 \,{\left (a x + b\right )} b^{3} c^{3}}{x} + \frac{7 \,{\left (a x + b\right )} a b^{2} c^{2} d}{x} - \frac{3 \,{\left (a x + b\right )} a^{2} b c d^{2}}{x} + \frac{2 \,{\left (a x + b\right )} a^{3} d^{3}}{x} - \frac{3 \,{\left (a x + b\right )}^{2} b^{2} c^{2} d}{x^{2}} + \frac{2 \,{\left (a x + b\right )}^{2} a b c d^{2}}{x^{2}} - \frac{2 \,{\left (a x + b\right )}^{2} a^{2} d^{3}}{x^{2}}}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )}{\left (a b c \sqrt{\frac{a x + b}{x}} - a^{2} d \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b c \sqrt{\frac{a x + b}{x}}}{x} + \frac{2 \,{\left (a x + b\right )} a d \sqrt{\frac{a x + b}{x}}}{x} - \frac{{\left (a x + b\right )}^{2} d \sqrt{\frac{a x + b}{x}}}{x^{2}}\right )}} + \frac{{\left (3 \, b c + 4 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*((7*b*c*d^3 - 4*a*d^4)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^3*c
^5 - 2*a*b^2*c^4*d + a^2*b*c^3*d^2)*sqrt(b*c*d - a*d^2)) + (2*a*b^3*c^3 - 2*a^2*
b^2*c^2*d - 3*(a*x + b)*b^3*c^3/x + 7*(a*x + b)*a*b^2*c^2*d/x - 3*(a*x + b)*a^2*
b*c*d^2/x + 2*(a*x + b)*a^3*d^3/x - 3*(a*x + b)^2*b^2*c^2*d/x^2 + 2*(a*x + b)^2*
a*b*c*d^2/x^2 - 2*(a*x + b)^2*a^2*d^3/x^2)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c
^2*d^2)*(a*b*c*sqrt((a*x + b)/x) - a^2*d*sqrt((a*x + b)/x) - (a*x + b)*b*c*sqrt(
(a*x + b)/x)/x + 2*(a*x + b)*a*d*sqrt((a*x + b)/x)/x - (a*x + b)^2*d*sqrt((a*x +
b)/x)/x^2)) + (3*b*c + 4*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2*
b*c^3))