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

Optimal. Leaf size=79 \[ \frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

[Out]

(a*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (b*x^5*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(5*(a + b*x^3))

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Rubi [A]  time = 0.0537178, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (b*x^5*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(5*(a + b*x^3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt((a + b*x**3)**2), x)

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Mathematica [A]  time = 0.0133414, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (5 a x^2+2 b x^5\right )}{10 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(5*a*x^2 + 2*b*x^5))/(10*(a + b*x^3))

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Maple [A]  time = 0.004, size = 36, normalized size = 0.5 \[{\frac{{x}^{2} \left ( 2\,b{x}^{3}+5\,a \right ) }{10\,b{x}^{3}+10\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*((b*x^3+a)^2)^(1/2),x)

[Out]

1/10*x^2*(2*b*x^3+5*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

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Maxima [A]  time = 0.816568, size = 18, normalized size = 0.23 \[ \frac{1}{5} \, b x^{5} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Fricas [A]  time = 0.26646, size = 18, normalized size = 0.23 \[ \frac{1}{5} \, b x^{5} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Sympy [A]  time = 0.207347, size = 12, normalized size = 0.15 \[ \frac{a x^{2}}{2} + \frac{b x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**2/2 + b*x**5/5

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GIAC/XCAS [A]  time = 0.277304, size = 39, normalized size = 0.49 \[ \frac{1}{5} \, b x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{2} \, a x^{2}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5*sign(b*x^3 + a) + 1/2*a*x^2*sign(b*x^3 + a)