3.636 \(\int x^2 (a+b x^4)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{3}{7} a^2 b x^7+\frac{a^3 x^3}{3}+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{15}}{15} \]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^7)/7 + (3*a*b^2*x^11)/11 + (b^3*x^15)/15

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Rubi [A]  time = 0.0138865, antiderivative size = 43, 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} \[ \frac{3}{7} a^2 b x^7+\frac{a^3 x^3}{3}+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^4)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^7)/7 + (3*a*b^2*x^11)/11 + (b^3*x^15)/15

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 )^3 \, dx &=\int \left (a^3 x^2+3 a^2 b x^6+3 a b^2 x^{10}+b^3 x^{14}\right ) \, dx\\ &=\frac{a^3 x^3}{3}+\frac{3}{7} a^2 b x^7+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{15}}{15}\\ \end{align*}

Mathematica [A]  time = 0.001695, size = 43, normalized size = 1. \[ \frac{3}{7} a^2 b x^7+\frac{a^3 x^3}{3}+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^4)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^7)/7 + (3*a*b^2*x^11)/11 + (b^3*x^15)/15

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Maple [A]  time = 0., size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{3}}{3}}+{\frac{3\,{a}^{2}b{x}^{7}}{7}}+{\frac{3\,a{b}^{2}{x}^{11}}{11}}+{\frac{{b}^{3}{x}^{15}}{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^4+a)^3,x)

[Out]

1/3*a^3*x^3+3/7*a^2*b*x^7+3/11*a*b^2*x^11+1/15*b^3*x^15

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Maxima [A]  time = 0.970673, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{15} \, b^{3} x^{15} + \frac{3}{11} \, a b^{2} x^{11} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^3*x^15 + 3/11*a*b^2*x^11 + 3/7*a^2*b*x^7 + 1/3*a^3*x^3

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Fricas [A]  time = 1.22121, size = 85, normalized size = 1.98 \begin{align*} \frac{1}{15} x^{15} b^{3} + \frac{3}{11} x^{11} b^{2} a + \frac{3}{7} x^{7} b a^{2} + \frac{1}{3} x^{3} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*x^15*b^3 + 3/11*x^11*b^2*a + 3/7*x^7*b*a^2 + 1/3*x^3*a^3

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Sympy [A]  time = 0.073989, size = 39, normalized size = 0.91 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{7}}{7} + \frac{3 a b^{2} x^{11}}{11} + \frac{b^{3} x^{15}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**4+a)**3,x)

[Out]

a**3*x**3/3 + 3*a**2*b*x**7/7 + 3*a*b**2*x**11/11 + b**3*x**15/15

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Giac [A]  time = 1.09585, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{15} \, b^{3} x^{15} + \frac{3}{11} \, a b^{2} x^{11} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^3*x^15 + 3/11*a*b^2*x^11 + 3/7*a^2*b*x^7 + 1/3*a^3*x^3