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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0191148, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ \frac{\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]

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 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]

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]

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*}

Mathematica [A]  time = 0.0110264, size = 25, normalized size = 1. \[ \frac{\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 24, normalized size = 1. \begin{align*}{\frac{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{1+p}}{2+2\,p}} \end{align*}

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)^(1+p)/(1+p)

________________________________________________________________________________________

Maxima [A]  time = 1.16291, size = 45, normalized size = 1.8 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.18789, size = 74, normalized size = 2.96 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [B]  time = 1.14325, size = 84, normalized size = 3.36 \begin{align*} \frac{{\left (c x^{4} + b x^{2} + a\right )}^{p} c x^{4} +{\left (c x^{4} + b x^{2} + a\right )}^{p} b x^{2} +{\left (c x^{4} + b x^{2} + a\right )}^{p} a}{2 \,{\left (p + 1\right )}} \end{align*}

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