3.15 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx\)

Optimal. Leaf size=75 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (b*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 = 0.0637153, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (b*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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{3}\right )^{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**3+a)**2)**(1/2)/x**4,x)

[Out]

Integral(sqrt((a + b*x**3)**2)/x**4, x)

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Mathematica [A]  time = 0.016082, size = 39, normalized size = 0.52 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a-3 b x^3 \log (x)\right )}{3 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]

[Out]

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

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Maple [A]  time = 0.015, size = 38, normalized size = 0.5 \[{\frac{3\,b\ln \left ( x \right ){x}^{3}-a}{ \left ( 3\,b{x}^{3}+3\,a \right ){x}^{3}}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x^4,x)

[Out]

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

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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(sqrt((b*x^3 + a)^2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282301, size = 23, normalized size = 0.31 \[ \frac{3 \, b x^{3} \log \left (x\right ) - a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^4,x, algorithm="fricas")

[Out]

1/3*(3*b*x^3*log(x) - a)/x^3

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Sympy [A]  time = 1.19694, size = 10, normalized size = 0.13 \[ - \frac{a}{3 x^{3}} + b \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**3+a)**2)**(1/2)/x**4,x)

[Out]

-a/(3*x**3) + b*log(x)

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GIAC/XCAS [A]  time = 0.29695, size = 58, normalized size = 0.77 \[ b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) - \frac{b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a{\rm sign}\left (b x^{3} + a\right )}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^4,x, algorithm="giac")

[Out]

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