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

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

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {b^2 x^8}{8} \end {gather*}

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 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])

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]

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*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {b^2 x^8}{8} \end {gather*}

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a x^2+b x^3\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(a*x^2 + b*x^3)^2,x]

[Out]

IntegrateAlgebraic[x*(a*x^2 + b*x^3)^2, x]

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fricas [A]  time = 0.34, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} x^{8} b^{2} + \frac {2}{7} x^{7} b a + \frac {1}{6} x^{6} a^{2} \end {gather*}

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

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giac [A]  time = 0.15, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {2}{7} \, a b x^{7} + \frac {1}{6} \, a^{2} x^{6} \end {gather*}

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

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maple [A]  time = 0.06, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{8} b^{2} x^{8}+\frac {2}{7} a b \,x^{7}+\frac {1}{6} a^{2} x^{6} \end {gather*}

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

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maxima [A]  time = 1.20, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {2}{7} \, a b x^{7} + \frac {1}{6} \, a^{2} x^{6} \end {gather*}

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

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mupad [B]  time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^6}{6}+\frac {2\,a\,b\,x^7}{7}+\frac {b^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a*x^2 + b*x^3)^2,x)

[Out]

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

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sympy [A]  time = 0.07, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{6}}{6} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{8}}{8} \end {gather*}

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

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