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

Optimal. Leaf size=179 \[ \frac{6 a^2 b \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}+\frac{6 a b^2 \log \left (\sqrt{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}-\frac{2 b^3 \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{\sqrt{x} \left (a+\frac{b}{\sqrt{x}}\right )}+\frac{a^3 x \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}} \]

[Out]

(-2*b^3*Sqrt[a^2 + b^2/x + (2*a*b)/Sqrt[x]])/((a + b/Sqrt[x])*Sqrt[x]) + (6*a^2*
b*Sqrt[a^2 + b^2/x + (2*a*b)/Sqrt[x]]*Sqrt[x])/(a + b/Sqrt[x]) + (a^3*Sqrt[a^2 +
b^2/x + (2*a*b)/Sqrt[x]]*x)/(a + b/Sqrt[x]) + (6*a*b^2*Sqrt[a^2 + b^2/x + (2*a*
b)/Sqrt[x]]*Log[Sqrt[x]])/(a + b/Sqrt[x])

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Rubi [A]  time = 0.200567, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{6 a^2 b \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}+\frac{6 a b^2 \log \left (\sqrt{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}-\frac{2 b^3 \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{\sqrt{x} \left (a+\frac{b}{\sqrt{x}}\right )}+\frac{a^3 x \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b^3*Sqrt[a^2 + b^2/x + (2*a*b)/Sqrt[x]])/((a + b/Sqrt[x])*Sqrt[x]) + (6*a^2*
b*Sqrt[a^2 + b^2/x + (2*a*b)/Sqrt[x]]*Sqrt[x])/(a + b/Sqrt[x]) + (a^3*Sqrt[a^2 +
b^2/x + (2*a*b)/Sqrt[x]]*x)/(a + b/Sqrt[x]) + (6*a*b^2*Sqrt[a^2 + b^2/x + (2*a*
b)/Sqrt[x]]*Log[Sqrt[x]])/(a + b/Sqrt[x])

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Rubi in Sympy [A]  time = 22.7061, size = 134, normalized size = 0.75 \[ - \frac{6 a b^{2} \sqrt{a^{2} + \frac{2 a b}{\sqrt{x}} + \frac{b^{2}}{x}} \log{\left (\frac{1}{\sqrt{x}} \right )}}{a + \frac{b}{\sqrt{x}}} - 6 b^{2} \sqrt{a^{2} + \frac{2 a b}{\sqrt{x}} + \frac{b^{2}}{x}} + 3 b \sqrt{x} \left (a + \frac{b}{\sqrt{x}}\right ) \sqrt{a^{2} + \frac{2 a b}{\sqrt{x}} + \frac{b^{2}}{x}} + x \left (a^{2} + \frac{2 a b}{\sqrt{x}} + \frac{b^{2}}{x}\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*a*b**2*sqrt(a**2 + 2*a*b/sqrt(x) + b**2/x)*log(1/sqrt(x))/(a + b/sqrt(x)) - 6
*b**2*sqrt(a**2 + 2*a*b/sqrt(x) + b**2/x) + 3*b*sqrt(x)*(a + b/sqrt(x))*sqrt(a**
2 + 2*a*b/sqrt(x) + b**2/x) + x*(a**2 + 2*a*b/sqrt(x) + b**2/x)**(3/2)

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Mathematica [A]  time = 0.0519086, size = 66, normalized size = 0.37 \[ \frac{\sqrt{\frac{\left (a \sqrt{x}+b\right )^2}{x}} \left (a^3 x^{3/2}+6 a^2 b x+3 a b^2 \sqrt{x} \log (x)-2 b^3\right )}{a \sqrt{x}+b} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(b + a*Sqrt[x])^2/x]*(-2*b^3 + 6*a^2*b*x + a^3*x^(3/2) + 3*a*b^2*Sqrt[x]*L
og[x]))/(b + a*Sqrt[x])

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Maple [A]  time = 0.036, size = 68, normalized size = 0.4 \[{1\sqrt{{1 \left ({a}^{2}{x}^{{\frac{3}{2}}}+{b}^{2}\sqrt{x}+2\,abx \right ){x}^{-{\frac{3}{2}}}}} \left ({x}^{{\frac{3}{2}}}{a}^{3}+6\,{a}^{2}bx+3\,\sqrt{x}\ln \left ( x \right ) a{b}^{2}-2\,{b}^{3} \right ) \left ( \sqrt{x}a+b \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((a^2*x^(3/2)+b^2*x^(1/2)+2*a*b*x)/x^(3/2))^(1/2)*(x^(3/2)*a^3+6*a^2*b*x+3*x^(1/
2)*ln(x)*a*b^2-2*b^3)/(x^(1/2)*a+b)

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Maxima [A]  time = 0.786421, size = 42, normalized size = 0.23 \[ a^{3} x + 3 \, a b^{2} \log \left (x\right ) + 6 \, a^{2} b \sqrt{x} - \frac{2 \, b^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*x + 3*a*b^2*log(x) + 6*a^2*b*sqrt(x) - 2*b^3/sqrt(x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**2+b**2/x+2*a*b/x**(1/2))**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.291724, size = 108, normalized size = 0.6 \[ a^{3} x{\rm sign}\left (a x + b \sqrt{x}\right ){\rm sign}\left (x\right ) + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (a x + b \sqrt{x}\right ){\rm sign}\left (x\right ) + 6 \, a^{2} b \sqrt{x}{\rm sign}\left (a x + b \sqrt{x}\right ){\rm sign}\left (x\right ) - \frac{2 \, b^{3}{\rm sign}\left (a x + b \sqrt{x}\right ){\rm sign}\left (x\right )}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*x*sign(a*x + b*sqrt(x))*sign(x) + 3*a*b^2*ln(abs(x))*sign(a*x + b*sqrt(x))*s
ign(x) + 6*a^2*b*sqrt(x)*sign(a*x + b*sqrt(x))*sign(x) - 2*b^3*sign(a*x + b*sqrt
(x))*sign(x)/sqrt(x)