3.624 \(\int x^2 (a+b x^4)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11} \]

[Out]

(a^2*x^3)/3 + (2*a*b*x^7)/7 + (b^2*x^11)/11

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Rubi [A]  time = 0.0097477, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^4)^2,x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^7)/7 + (b^2*x^11)/11

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^2 \left (a+b x^4\right )^2 \, dx &=\int \left (a^2 x^2+2 a b x^6+b^2 x^{10}\right ) \, dx\\ &=\frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0006658, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^4)^2,x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^7)/7 + (b^2*x^11)/11

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Maple [A]  time = 0., size = 25, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{{b}^{2}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^4+a)^2,x)

[Out]

1/3*x^3*a^2+2/7*a*b*x^7+1/11*b^2*x^11

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Maxima [A]  time = 0.964017, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{11} \, b^{2} x^{11} + \frac{2}{7} \, a b x^{7} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^4+a)^2,x, algorithm="maxima")

[Out]

1/11*b^2*x^11 + 2/7*a*b*x^7 + 1/3*a^2*x^3

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Fricas [A]  time = 1.21846, size = 58, normalized size = 1.93 \begin{align*} \frac{1}{11} x^{11} b^{2} + \frac{2}{7} x^{7} b a + \frac{1}{3} x^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/11*x^11*b^2 + 2/7*x^7*b*a + 1/3*x^3*a^2

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Sympy [A]  time = 0.060386, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**4+a)**2,x)

[Out]

a**2*x**3/3 + 2*a*b*x**7/7 + b**2*x**11/11

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Giac [A]  time = 1.07456, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{11} \, b^{2} x^{11} + \frac{2}{7} \, a b x^{7} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^4+a)^2,x, algorithm="giac")

[Out]

1/11*b^2*x^11 + 2/7*a*b*x^7 + 1/3*a^2*x^3