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

Optimal. Leaf size=30 \[ \frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8} \]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + (b^2*x^8)/8

________________________________________________________________________________________

Rubi [A]  time = 0.0170845, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1584, 43} \[ \frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[x*(a*x^2 + b*x^3)^2,x]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + (b^2*x^8)/8

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \left (a x^2+b x^3\right )^2 \, dx &=\int x^5 (a+b x)^2 \, dx\\ &=\int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx\\ &=\frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.001907, size = 30, normalized size = 1. \[ \frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a*x^2 + b*x^3)^2,x]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + (b^2*x^8)/8

________________________________________________________________________________________

Maple [A]  time = 0., size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{{b}^{2}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^2*x^6+2/7*a*b*x^7+1/8*b^2*x^8

________________________________________________________________________________________

Maxima [A]  time = 0.979917, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{2}{7} \, a b x^{7} + \frac{1}{6} \, a^{2} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*x^8 + 2/7*a*b*x^7 + 1/6*a^2*x^6

________________________________________________________________________________________

Fricas [A]  time = 0.617166, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{8} x^{8} b^{2} + \frac{2}{7} x^{7} b a + \frac{1}{6} x^{6} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*b^2 + 2/7*x^7*b*a + 1/6*x^6*a^2

________________________________________________________________________________________

Sympy [A]  time = 0.242338, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{6}}{6} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**6/6 + 2*a*b*x**7/7 + b**2*x**8/8

________________________________________________________________________________________

Giac [A]  time = 1.13299, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{2}{7} \, a b x^{7} + \frac{1}{6} \, a^{2} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*x^8 + 2/7*a*b*x^7 + 1/6*a^2*x^6