3.131 \(\int x^2 (b+2 c x^3) (a+b x^3+c x^6)^p \, dx\)

Optimal. Leaf size=25 \[ \frac {\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

[Out]

1/3*(c*x^6+b*x^3+a)^(1+p)/(1+p)

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1468, 629} \[ \frac {\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^p,x]

[Out]

(a + b*x^3 + c*x^6)^(1 + p)/(3*(1 + p))

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

\begin {align*} \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3+c x^6\right )^{1+p}}{3 (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^p,x]

[Out]

(a + b*x^3 + c*x^6)^(1 + p)/(3*(1 + p))

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fricas [A]  time = 0.79, size = 33, normalized size = 1.32 \[ \frac {{\left (c x^{6} + b x^{3} + a\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{3 \, {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^p,x, algorithm="fricas")

[Out]

1/3*(c*x^6 + b*x^3 + a)*(c*x^6 + b*x^3 + a)^p/(p + 1)

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giac [A]  time = 0.32, size = 23, normalized size = 0.92 \[ \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p + 1}}{3 \, {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^p,x, algorithm="giac")

[Out]

1/3*(c*x^6 + b*x^3 + a)^(p + 1)/(p + 1)

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maple [A]  time = 0.01, size = 24, normalized size = 0.96 \[ \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p +1}}{3 p +3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^p,x)

[Out]

1/3*(c*x^6+b*x^3+a)^(p+1)/(p+1)

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maxima [A]  time = 0.58, size = 33, normalized size = 1.32 \[ \frac {{\left (c x^{6} + b x^{3} + a\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{3 \, {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^p,x, algorithm="maxima")

[Out]

1/3*(c*x^6 + b*x^3 + a)*(c*x^6 + b*x^3 + a)^p/(p + 1)

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mupad [B]  time = 2.12, size = 49, normalized size = 1.96 \[ {\left (c\,x^6+b\,x^3+a\right )}^p\,\left (\frac {a}{3\,p+3}+\frac {b\,x^3}{3\,p+3}+\frac {c\,x^6}{3\,p+3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^p,x)

[Out]

(a + b*x^3 + c*x^6)^p*(a/(3*p + 3) + (b*x^3)/(3*p + 3) + (c*x^6)/(3*p + 3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(2*c*x**3+b)*(c*x**6+b*x**3+a)**p,x)

[Out]

Timed out

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