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

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

[Out]

(b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (a*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 = 0.0611676, 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 \[ \frac{b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (a*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. \[ \int \frac{\sqrt{\left (a + b x^{3}\right )^{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((a + b*x**3)**2)/x, x)

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Mathematica [A]  time = 0.0155816, size = 37, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (3 a \log (x)+b x^3\right )}{3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 34, normalized size = 0.5 \[{\frac{b{x}^{3}+3\,a\ln \left ( x \right ) }{3\,b{x}^{3}+3\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x,x)

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273561, size = 15, normalized size = 0.2 \[ \frac{1}{3} \, b x^{3} + a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b*x^3 + a*log(x)

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Sympy [A]  time = 0.277418, size = 10, normalized size = 0.13 \[ a \log{\left (x \right )} + \frac{b x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*log(x) + b*x**3/3

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GIAC/XCAS [A]  time = 0.269077, size = 38, normalized size = 0.51 \[ \frac{1}{3} \, b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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