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

Optimal. Leaf size=79 \[ \frac{b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[Out]

(a*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b*x^7*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(7*(a + b*x^3))

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Rubi [A]  time = 0.0659783, antiderivative size = 79, 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 \[ \frac{b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b*x^7*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(7*(a + b*x^3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.012017, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (7 a x^4+4 b x^7\right )}{28 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(7*a*x^4 + 4*b*x^7))/(28*(a + b*x^3))

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Maple [A]  time = 0.004, size = 36, normalized size = 0.5 \[{\frac{{x}^{4} \left ( 4\,b{x}^{3}+7\,a \right ) }{28\,b{x}^{3}+28\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*((b*x^3+a)^2)^(1/2),x)

[Out]

1/28*x^4*(4*b*x^3+7*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

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Maxima [A]  time = 0.784532, size = 18, normalized size = 0.23 \[ \frac{1}{7} \, b x^{7} + \frac{1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*x^7 + 1/4*a*x^4

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Fricas [A]  time = 0.265921, size = 18, normalized size = 0.23 \[ \frac{1}{7} \, b x^{7} + \frac{1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*x^7 + 1/4*a*x^4

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Sympy [A]  time = 0.220402, size = 12, normalized size = 0.15 \[ \frac{a x^{4}}{4} + \frac{b x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/4 + b*x**7/7

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GIAC/XCAS [A]  time = 0.27898, size = 39, normalized size = 0.49 \[ \frac{1}{7} \, b x^{7}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{4} \, a x^{4}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*x^7*sign(b*x^3 + a) + 1/4*a*x^4*sign(b*x^3 + a)