3.2.8 \(\int x (b+2 c x^2) (a+b x^2+c x^4)^p \, dx\)

Optimal. Leaf size=25 \[ \frac {\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]

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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1247, 629} \begin {gather*} \frac {\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^p,x]

[Out]

(a + b*x^2 + c*x^4)^(1 + p)/(2*(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 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int x \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^p \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2+c x^4\right )^{1+p}}{2 (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^p,x]

[Out]

(a + b*x^2 + c*x^4)^(1 + p)/(2*(1 + p))

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IntegrateAlgebraic [F]  time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^p,x]

[Out]

Defer[IntegrateAlgebraic][x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^p, x]

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fricas [A]  time = 0.78, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left (c x^{4} + b x^{2} + a\right )} {\left (c x^{4} + b x^{2} + a\right )}^{p}}{2 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^4 + b*x^2 + a)*(c*x^4 + b*x^2 + a)^p/(p + 1)

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giac [A]  time = 0.46, size = 23, normalized size = 0.92 \begin {gather*} \frac {{\left (c x^{4} + b x^{2} + a\right )}^{p + 1}}{2 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^4 + b*x^2 + a)^(p + 1)/(p + 1)

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maple [A]  time = 0.00, size = 24, normalized size = 0.96 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p +1}}{2 p +2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^4+b*x^2+a)^(p+1)/(p+1)

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maxima [A]  time = 0.60, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left (c x^{4} + b x^{2} + a\right )} {\left (c x^{4} + b x^{2} + a\right )}^{p}}{2 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^4 + b*x^2 + a)*(c*x^4 + b*x^2 + a)^p/(p + 1)

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mupad [B]  time = 2.09, size = 49, normalized size = 1.96 \begin {gather*} {\left (c\,x^4+b\,x^2+a\right )}^p\,\left (\frac {a}{2\,p+2}+\frac {b\,x^2}{2\,p+2}+\frac {c\,x^4}{2\,p+2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a + b*x^2 + c*x^4)^p*(a/(2*p + 2) + (b*x^2)/(2*p + 2) + (c*x^4)/(2*p + 2))

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sympy [F(-1)]  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*c*x**2+b)*(c*x**4+b*x**2+a)**p,x)

[Out]

Timed out

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