3.39 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=162 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.125499, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.0262, size = 128, normalized size = 0.79 \[ \frac{3 a b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} + \frac{a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{2 x^{6}} + b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} - \frac{2 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{3 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0278203, size = 61, normalized size = 0.38 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a^3+6 a^2 b x^3-18 a b^2 x^6 \log (x)-2 b^3 x^9\right )}{6 x^6 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^7,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(a^3 + 6*a^2*b*x^3 - 2*b^3*x^9 - 18*a*b^2*x^6*Log[x]))/(6*
x^6*(a + b*x^3))

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Maple [A]  time = 0.018, size = 60, normalized size = 0.4 \[{\frac{2\,{b}^{3}{x}^{9}+18\,a{b}^{2}\ln \left ( x \right ){x}^{6}-6\,{x}^{3}{a}^{2}b-{a}^{3}}{6\, \left ( b{x}^{3}+a \right ) ^{3}{x}^{6}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^7,x)

[Out]

1/6*((b*x^3+a)^2)^(3/2)*(2*b^3*x^9+18*a*b^2*ln(x)*x^6-6*x^3*a^2*b-a^3)/(b*x^3+a)
^3/x^6

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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((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290712, size = 53, normalized size = 0.33 \[ \frac{2 \, b^{3} x^{9} + 18 \, a b^{2} x^{6} \log \left (x\right ) - 6 \, a^{2} b x^{3} - a^{3}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^7,x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^9 + 18*a*b^2*x^6*log(x) - 6*a^2*b*x^3 - a^3)/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**7,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.272545, size = 116, normalized size = 0.72 \[ \frac{1}{3} \, b^{3} x^{3}{\rm sign}\left (b x^{3} + a\right ) + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) - \frac{9 \, a b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 6 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{3}{\rm sign}\left (b x^{3} + a\right )}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

1/3*b^3*x^3*sign(b*x^3 + a) + 3*a*b^2*ln(abs(x))*sign(b*x^3 + a) - 1/6*(9*a*b^2*
x^6*sign(b*x^3 + a) + 6*a^2*b*x^3*sign(b*x^3 + a) + a^3*sign(b*x^3 + a))/x^6