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

Optimal. Leaf size=134 \[ \frac{a^{3/2} (5 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^2}+\frac{2 (b c-a d)^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 d^{3/2}}-\frac{b \sqrt{a+\frac{b}{x}} (a d+2 b c)}{c d}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c} \]

[Out]

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

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Rubi [A]  time = 0.639039, antiderivative size = 134, 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 \[ \frac{a^{3/2} (5 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^2}+\frac{2 (b c-a d)^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 d^{3/2}}-\frac{b \sqrt{a+\frac{b}{x}} (a d+2 b c)}{c d}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 62.0592, size = 114, normalized size = 0.85 \[ - \frac{a^{\frac{3}{2}} \left (2 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{c^{2}} + \frac{a x \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{c} - \frac{b \sqrt{a + \frac{b}{x}} \left (a d + 2 b c\right )}{c d} + \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{2} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.228124, size = 172, normalized size = 1.28 \[ -\frac{a^{3/2} (2 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 c^2}+\sqrt{a+\frac{b}{x}} \left (\frac{a^2 x}{c}-\frac{2 b^2}{d}\right )+\frac{(a d-b c)^{5/2} \log (c x+d)}{c^2 d^{3/2}}-\frac{(a d-b c)^{5/2} \log \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{a d-b c}-2 a d x+b c x-b d\right )}{c^2 d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 859, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/(c+d/x),x)

[Out]

1/2*((a*x+b)/x)^(1/2)/x*(-2*d^3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(
1/2))*a^3*x^2*c*((a*d-b*c)*d/c^2)^(1/2)+5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*a^2*b*x^2*d^2*c^2*((a*d-b*c)*d/c^2)^(1/2)+4*b^2*a*ln(1/2*(2*(a*x
^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*d*c^3*((a*d-b*c)*d/c^2)^(1/2)-4*ln(1
/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^2*x^2*d*c^3*((a*d-b*c)*d/c
^2)^(1/2)+2*(x*(a*x+b))^(1/2)*a^(5/2)*x^2*d^2*c^2*((a*d-b*c)*d/c^2)^(1/2)-4*(x*(
a*x+b))^(1/2)*a^(3/2)*b*x^2*d*c^3*((a*d-b*c)*d/c^2)^(1/2)+2*c^4*(x*(a*x+b))^(1/2
)*b^2*x^2*a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)-2*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))*a^(7/2)*x^2+6*d^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^(5/2)*b*x^2*c-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^
(3/2)*b^2*x^2*d^2*c^2+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*x^2*d*c^3*a^(1/2)+8*b*a^(3/2)*(a*x^2+b*x)^(1/2)*x^2*d*
c^3*((a*d-b*c)*d/c^2)^(1/2)-2*b^2*(a*x^2+b*x)^(1/2)*c^4*x^2*a^(1/2)*((a*d-b*c)*d
/c^2)^(1/2)-b^3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*c^4*x^2*((
a*d-b*c)*d/c^2)^(1/2)+c^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*
b^3*x^2*((a*d-b*c)*d/c^2)^(1/2)-4*b*(a*x^2+b*x)^(3/2)*d*c^3*a^(1/2)*((a*d-b*c)*d
/c^2)^(1/2))/(x*(a*x+b))^(1/2)/d^2/c^3/a^(1/2)/((a*d-b*c)*d/c^2)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.435569, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{d}} \log \left (\frac{2 \, d x \sqrt{-\frac{b c - a d}{d}} \sqrt{\frac{a x + b}{x}} + b d -{\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \,{\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt{\frac{a x + b}{x}}}{2 \, c^{2} d}, \frac{{\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{d}} \log \left (\frac{2 \, d x \sqrt{-\frac{b c - a d}{d}} \sqrt{\frac{a x + b}{x}} + b d -{\left (b c - 2 \, a d\right )} x}{c x + d}\right ) +{\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt{\frac{a x + b}{x}}}{c^{2} d}, \frac{4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{d}} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{\frac{b c - a d}{d}}}\right ) -{\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt{\frac{a x + b}{x}}}{2 \, c^{2} d}, \frac{{\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{d}} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{\frac{b c - a d}{d}}}\right ) +{\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt{\frac{a x + b}{x}}}{c^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((5*a*b*c*d - 2*a^2*d^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x)
+ b) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^2*c*
d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d), ((5*a*b*c*d - 2*a^2*d^2)*sqrt(-a)*a
rctan(sqrt((a*x + b)/x)/sqrt(-a)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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)) + (a^2*c*d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d), 1/2*(4*(b^2
*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((b*c - a*d)/d)*arctan(sqrt((a*x + b)/x)/sqrt((b
*c - a*d)/d)) - (5*a*b*c*d - 2*a^2*d^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*
x + b)/x) + b) + 2*(a^2*c*d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d), ((5*a*b*c
*d - 2*a^2*d^2)*sqrt(-a)*arctan(sqrt((a*x + b)/x)/sqrt(-a)) + 2*(b^2*c^2 - 2*a*b
*c*d + a^2*d^2)*sqrt((b*c - a*d)/d)*arctan(sqrt((a*x + b)/x)/sqrt((b*c - a*d)/d)
) + (a^2*c*d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError