3.1699 \(\int \left (a+\frac{b}{x}\right )^{3/2} x^3 \, dx\)

Optimal. Leaf size=114 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{5/2}}-\frac{3 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^2}+\frac{b^2 x^2 \sqrt{a+\frac{b}{x}}}{32 a}+\frac{1}{4} x^4 \left (a+\frac{b}{x}\right )^{3/2}+\frac{1}{8} b x^3 \sqrt{a+\frac{b}{x}} \]

[Out]

(-3*b^3*Sqrt[a + b/x]*x)/(64*a^2) + (b^2*Sqrt[a + b/x]*x^2)/(32*a) + (b*Sqrt[a +
b/x]*x^3)/8 + ((a + b/x)^(3/2)*x^4)/4 + (3*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/
(64*a^(5/2))

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Rubi [A]  time = 0.162232, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{5/2}}-\frac{3 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^2}+\frac{b^2 x^2 \sqrt{a+\frac{b}{x}}}{32 a}+\frac{1}{4} x^4 \left (a+\frac{b}{x}\right )^{3/2}+\frac{1}{8} b x^3 \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)*x^3,x]

[Out]

(-3*b^3*Sqrt[a + b/x]*x)/(64*a^2) + (b^2*Sqrt[a + b/x]*x^2)/(32*a) + (b*Sqrt[a +
b/x]*x^3)/8 + ((a + b/x)^(3/2)*x^4)/4 + (3*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/
(64*a^(5/2))

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Rubi in Sympy [A]  time = 15.6965, size = 94, normalized size = 0.82 \[ \frac{b x^{3} \sqrt{a + \frac{b}{x}}}{8} + \frac{x^{4} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{4} + \frac{b^{2} x^{2} \sqrt{a + \frac{b}{x}}}{32 a} - \frac{3 b^{3} x \sqrt{a + \frac{b}{x}}}{64 a^{2}} + \frac{3 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{64 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**3*sqrt(a + b/x)/8 + x**4*(a + b/x)**(3/2)/4 + b**2*x**2*sqrt(a + b/x)/(32*a
) - 3*b**3*x*sqrt(a + b/x)/(64*a**2) + 3*b**4*atanh(sqrt(a + b/x)/sqrt(a))/(64*a
**(5/2))

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Mathematica [A]  time = 0.126748, size = 90, normalized size = 0.79 \[ \frac{2 \sqrt{a} x \sqrt{a+\frac{b}{x}} \left (16 a^3 x^3+24 a^2 b x^2+2 a b^2 x-3 b^3\right )+3 b^4 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{128 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*Sqrt[a + b/x]*x*(-3*b^3 + 2*a*b^2*x + 24*a^2*b*x^2 + 16*a^3*x^3) + 3*
b^4*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(128*a^(5/2))

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Maple [A]  time = 0.012, size = 137, normalized size = 1.2 \[{\frac{x}{128}\sqrt{{\frac{ax+b}{x}}} \left ( 32\,x \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{9/2}+16\,b \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}-12\,{b}^{2}\sqrt{a{x}^{2}+bx}x{a}^{7/2}-6\,{b}^{3}\sqrt{a{x}^{2}+bx}{a}^{5/2}+3\,{b}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*x^3,x)

[Out]

1/128*((a*x+b)/x)^(1/2)*x*(32*x*(a*x^2+b*x)^(3/2)*a^(9/2)+16*b*(a*x^2+b*x)^(3/2)
*a^(7/2)-12*b^2*(a*x^2+b*x)^(1/2)*x*a^(7/2)-6*b^3*(a*x^2+b*x)^(1/2)*a^(5/2)+3*b^
4*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2)/(x*(a*x+b))^(1/2)/a
^(9/2)

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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)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23794, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{4} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (16 \, a^{3} x^{4} + 24 \, a^{2} b x^{3} + 2 \, a b^{2} x^{2} - 3 \, b^{3} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{128 \, a^{\frac{5}{2}}}, -\frac{3 \, b^{4} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (16 \, a^{3} x^{4} + 24 \, a^{2} b x^{3} + 2 \, a b^{2} x^{2} - 3 \, b^{3} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{64 \, \sqrt{-a} a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/128*(3*b^4*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(16*a^3*x^4
+ 24*a^2*b*x^3 + 2*a*b^2*x^2 - 3*b^3*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(5/2), -1/
64*(3*b^4*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (16*a^3*x^4 + 24*a^2*b*x^3 +
2*a*b^2*x^2 - 3*b^3*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*a^2)]

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Sympy [A]  time = 28.8789, size = 153, normalized size = 1.34 \[ \frac{a^{2} x^{\frac{9}{2}}}{4 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{5 a \sqrt{b} x^{\frac{7}{2}}}{8 \sqrt{\frac{a x}{b} + 1}} + \frac{13 b^{\frac{3}{2}} x^{\frac{5}{2}}}{32 \sqrt{\frac{a x}{b} + 1}} - \frac{b^{\frac{5}{2}} x^{\frac{3}{2}}}{64 a \sqrt{\frac{a x}{b} + 1}} - \frac{3 b^{\frac{7}{2}} \sqrt{x}}{64 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{3 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{64 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**(9/2)/(4*sqrt(b)*sqrt(a*x/b + 1)) + 5*a*sqrt(b)*x**(7/2)/(8*sqrt(a*x/b +
1)) + 13*b**(3/2)*x**(5/2)/(32*sqrt(a*x/b + 1)) - b**(5/2)*x**(3/2)/(64*a*sqrt(
a*x/b + 1)) - 3*b**(7/2)*sqrt(x)/(64*a**2*sqrt(a*x/b + 1)) + 3*b**4*asinh(sqrt(a
)*sqrt(x)/sqrt(b))/(64*a**(5/2))

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GIAC/XCAS [A]  time = 0.258776, size = 143, normalized size = 1.25 \[ -\frac{3 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{128 \, a^{\frac{5}{2}}} + \frac{3 \, b^{4}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{128 \, a^{\frac{5}{2}}} + \frac{1}{64} \, \sqrt{a x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \, a x{\rm sign}\left (x\right ) + 3 \, b{\rm sign}\left (x\right )\right )} x + \frac{b^{2}{\rm sign}\left (x\right )}{a}\right )} x - \frac{3 \, b^{3}{\rm sign}\left (x\right )}{a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/128*b^4*ln(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sign(x)/a^(5/
2) + 3/128*b^4*ln(abs(b))*sign(x)/a^(5/2) + 1/64*sqrt(a*x^2 + b*x)*(2*(4*(2*a*x*
sign(x) + 3*b*sign(x))*x + b^2*sign(x)/a)*x - 3*b^3*sign(x)/a^2)