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

Optimal. Leaf size=79 \[ \frac{b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]

[Out]

(a*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (b*x^8*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(8*(a + b*x^3))

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Rubi [A]  time = 0.0668339, 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 \[ \frac{b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (b*x^8*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(8*(a + b*x^3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \sqrt{\left (a + b x^{3}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0127001, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (8 a x^5+5 b x^8\right )}{40 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(8*a*x^5 + 5*b*x^8))/(40*(a + b*x^3))

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Maple [A]  time = 0.005, size = 36, normalized size = 0.5 \[{\frac{{x}^{5} \left ( 5\,b{x}^{3}+8\,a \right ) }{40\,b{x}^{3}+40\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*((b*x^3+a)^2)^(1/2),x)

[Out]

1/40*x^5*(5*b*x^3+8*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

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Maxima [A]  time = 0.781704, size = 18, normalized size = 0.23 \[ \frac{1}{8} \, b x^{8} + \frac{1}{5} \, a x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b*x^8 + 1/5*a*x^5

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Fricas [A]  time = 0.264422, size = 18, normalized size = 0.23 \[ \frac{1}{8} \, b x^{8} + \frac{1}{5} \, a x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b*x^8 + 1/5*a*x^5

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Sympy [A]  time = 0.214934, size = 12, normalized size = 0.15 \[ \frac{a x^{5}}{5} + \frac{b x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**5/5 + b*x**8/8

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GIAC/XCAS [A]  time = 0.27303, size = 39, normalized size = 0.49 \[ \frac{1}{8} \, b x^{8}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{5} \, a x^{5}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b*x^8*sign(b*x^3 + a) + 1/5*a*x^5*sign(b*x^3 + a)