3.2.31 \(\int x^2 (b+2 c x^3) (a+b x^3+c x^6)^p \, dx\) [131]

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

[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 = {1482, 643} \begin {gather*} \frac {\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \end {gather*}

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 643

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 1482

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} \text {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 = 0.05, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^3+c x^6\right )^{1+p}}{3 (1+p)} \end {gather*}

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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Maple [A]
time = 0.03, size = 24, normalized size = 0.96

method	result	size
gosper	\(\frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{1+p}}{3+3 p}\)	\(24\)
risch	\(\frac {\left (c \,x^{6}+b \,x^{3}+a \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{3+3 p}\)	\(34\)
norman	\(\frac {a \,{\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}+a \right )}}{3+3 p}+\frac {b \,x^{3} {\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}+a \right )}}{3+3 p}+\frac {c \,x^{6} {\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}+a \right )}}{3+3 p}\)	\(80\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 33, normalized size = 1.32 \begin {gather*} \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 )}} \end {gather*}

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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Fricas [A]
time = 0.35, size = 33, normalized size = 1.32 \begin {gather*} \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 )}} \end {gather*}

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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [A]
time = 2.34, size = 23, normalized size = 0.92 \begin {gather*} \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p + 1}}{3 \, {\left (p + 1\right )}} \end {gather*}

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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Mupad [B]
time = 2.12, size = 49, normalized size = 1.96 \begin {gather*} {\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 ) \end {gather*}

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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