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

Optimal. Leaf size=156 \[ -\frac{(b c-4 a d) \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 \sqrt{d}}+\frac{\sqrt{a} (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^3}-\frac{\sqrt{a+\frac{b}{x}} (b c-2 a d)}{c^2 \left (c+\frac{d}{x}\right )}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.670228, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(b c-4 a d) \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 \sqrt{d}}+\frac{\sqrt{a} (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^3}-\frac{\sqrt{a+\frac{b}{x}} (b c-2 a d)}{c^2 \left (c+\frac{d}{x}\right )}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 68.3839, size = 129, normalized size = 0.83 \[ - \frac{\sqrt{a} \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{c^{3}} + \frac{a x \sqrt{a + \frac{b}{x}}}{c \left (c + \frac{d}{x}\right )} + \frac{\sqrt{a + \frac{b}{x}} \left (2 a d - b c\right )}{c^{2} \left (c + \frac{d}{x}\right )} + \frac{\sqrt{a d - b c} \left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{3} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*(4*a*d - 3*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/c**3 + a*x*sqrt(a + b/x)/(
c*(c + d/x)) + sqrt(a + b/x)*(2*a*d - b*c)/(c**2*(c + d/x)) + sqrt(a*d - b*c)*(4
*a*d - b*c)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**3*sqrt(d))

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Mathematica [C]  time = 0.478662, size = 231, normalized size = 1.48 \[ -\frac{\frac{i \left (4 a^2 d^2-5 a b c d+b^2 c^2\right ) \log \left (\frac{2 c^4 \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d x-i b (d-c x)\right )}{\sqrt{d} (c x+d) \sqrt{b c-a d} \left (4 a^2 d^2-5 a b c d+b^2 c^2\right )}\right )}{\sqrt{d} \sqrt{b c-a d}}-\frac{2 c x \sqrt{a+\frac{b}{x}} (a c x+2 a d-b c)}{c x+d}+\sqrt{a} (4 a d-3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 1202, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/(c+d/x)^2,x)

[Out]

1/2*((a*x+b)/x)^(1/2)*x*(-4*a^(5/2)*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)*c*d^4+2*c^4*(x*(a*x+b))^(3/2)*a*((a*d-b*c)*d/c
^2)^(1/2)*d+2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^5-4*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))*x*a^3*c*d^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))*x*b^
3*c^4*d+4*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^2*c^2*d^3+2*(x*(a*x+b))^(1
/2)*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^4*d+9*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^2*b*c*d^4-6*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*b^2*c^2*d^3-4*a^(5/2)*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*c^2*d^
3-2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*a^2*c^4*d+2*(x*(a*x+b))^(1/2)*
((a*d-b*c)*d/c^2)^(1/2)*x^2*a*b*c^5+7*a^(3/2)*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*c^2*d^3+2*(x*(a*x+b))^(1/2)*((a*d-
b*c)*d/c^2)^(1/2)*x*a^2*c^3*d^2-3*a^(1/2)*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^2*c^3*d^2+9*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*a^2*b*c^2*d^3-6*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*a*b^2*c^3*
d^2-6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a*b*c^3*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^3*d^5+7*a^(3/2)*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*
c^3*d^2-3*a^(1/2)*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^2*c^4*d-4*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a*b*c^
4*d-2*c^5*(x*(a*x+b))^(3/2)*b*((a*d-b*c)*d/c^2)^(1/2)+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*c^3*d^2)/(x*(a*x+b))^(1/
2)/c^4/((a*d-b*c)*d/c^2)^(1/2)/(a*d-b*c)/(c*x+d)/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((a + b/x)^(3/2)/(c + d/x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292952, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((3*b*c*d - 4*a*d^2 + (3*b*c^2 - 4*a*c*d)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)
*x*sqrt((a*x + b)/x) + b) + (b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt(-(b*c -
a*d)/d)*log((2*d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)
*x)/(c*x + d)) - 2*(a*c^2*x^2 - (b*c^2 - 2*a*c*d)*x)*sqrt((a*x + b)/x))/(c^4*x +
c^3*d), -1/2*(2*(b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt((b*c - a*d)/d)*arc
tan(sqrt((a*x + b)/x)/sqrt((b*c - a*d)/d)) + (3*b*c*d - 4*a*d^2 + (3*b*c^2 - 4*a
*c*d)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(a*c^2*x^2 -
(b*c^2 - 2*a*c*d)*x)*sqrt((a*x + b)/x))/(c^4*x + c^3*d), 1/2*(2*(3*b*c*d - 4*a*
d^2 + (3*b*c^2 - 4*a*c*d)*x)*sqrt(-a)*arctan(sqrt((a*x + b)/x)/sqrt(-a)) - (b*c*
d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt(-(b*c - a*d)/d)*log((2*d*x*sqrt(-(b*c -
a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) + 2*(a*c^2*x^2 - (
b*c^2 - 2*a*c*d)*x)*sqrt((a*x + b)/x))/(c^4*x + c^3*d), ((3*b*c*d - 4*a*d^2 + (3
*b*c^2 - 4*a*c*d)*x)*sqrt(-a)*arctan(sqrt((a*x + b)/x)/sqrt(-a)) - (b*c*d - 4*a*
d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt((b*c - a*d)/d)*arctan(sqrt((a*x + b)/x)/sqrt((b*
c - a*d)/d)) + (a*c^2*x^2 - (b*c^2 - 2*a*c*d)*x)*sqrt((a*x + b)/x))/(c^4*x + c^3
*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{\left (c x + d\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError