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

Optimal. Leaf size=106 \[ -\frac{2 (b c-a d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 \sqrt{d}}+\frac{\sqrt{a} (3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^2}+\frac{a x \sqrt{a+\frac{b}{x}}}{c} \]

[Out]

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

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Rubi [A]  time = 0.384434, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 (b c-a d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 \sqrt{d}}+\frac{\sqrt{a} (3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^2}+\frac{a x \sqrt{a+\frac{b}{x}}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.9565, size = 92, normalized size = 0.87 \[ - \frac{\sqrt{a} \left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{c^{2}} + \frac{a x \sqrt{a + \frac{b}{x}}}{c} + \frac{2 \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.49446, size = 156, normalized size = 1.47 \[ \frac{\frac{2 (a d-b c)^{3/2} \log (c x+d)}{\sqrt{d}}-\frac{2 (a d-b c)^{3/2} \log \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{a d-b c}-2 a d x+b c x-b d\right )}{\sqrt{d}}-\sqrt{a} (2 a d-3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )+2 a c x \sqrt{a+\frac{b}{x}}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 528, normalized size = 5. \[{\frac{x}{2\,d{c}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}c\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) abd{c}^{2}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}+2\,\sqrt{x \left ( ax+b \right ) }{a}^{3/2}d{c}^{2}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}-2\,\sqrt{x \left ( ax+b \right ) }b\sqrt{a}{c}^{3}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}-2\,{d}^{3}\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ){a}^{5/2}+4\,{d}^{2}\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ){a}^{3/2}bc-2\,\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ){b}^{2}d\sqrt{a}{c}^{2}+2\,b\sqrt{a{x}^{2}+bx}\sqrt{a}{c}^{3}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){c}^{3}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{2}{c}^{3}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/(c+d/x),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296247, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a c x \sqrt{\frac{a x + b}{x}} -{\left (3 \, b c - 2 \, a d\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{d}} \log \left (\frac{2 \, d x \sqrt{-\frac{b c - a d}{d}} \sqrt{\frac{a x + b}{x}} + b d -{\left (b c - 2 \, a d\right )} x}{c x + d}\right )}{2 \, c^{2}}, \frac{a c x \sqrt{\frac{a x + b}{x}} +{\left (3 \, b c - 2 \, a d\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) -{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{d}} \log \left (\frac{2 \, d x \sqrt{-\frac{b c - a d}{d}} \sqrt{\frac{a x + b}{x}} + b d -{\left (b c - 2 \, a d\right )} x}{c x + d}\right )}{c^{2}}, \frac{2 \, a c x \sqrt{\frac{a x + b}{x}} - 4 \,{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{d}} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{\frac{b c - a d}{d}}}\right ) -{\left (3 \, b c - 2 \, a d\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right )}{2 \, c^{2}}, \frac{a c x \sqrt{\frac{a x + b}{x}} +{\left (3 \, b c - 2 \, a d\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) - 2 \,{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{d}} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{\frac{b c - a d}{d}}}\right )}{c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError