### 3.134 $$\int x (b+2 c x^2) (-a+b x^2+c x^4)^p \, dx$$

Optimal. Leaf size=27 $\frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)}$

[Out]

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

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Rubi [A]  time = 0.0195965, antiderivative size = 27, normalized size of antiderivative = 1., 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 = {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 = 0.0140538, size = 27, 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))

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Maple [A]  time = 0.003, size = 26, 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)

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Maxima [A]  time = 1.16771, size = 50, normalized size = 1.85 \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)

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Fricas [A]  time = 1.07853, size = 74, normalized size = 2.74 \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)

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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

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Giac [B]  time = 1.15072, size = 93, normalized size = 3.44 \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)