### 3.131 $$\int x^2 (a+b x^2+c x^4)^p (3 a+b (5+2 p) x^2+c (7+4 p) x^4) \, dx$$

Optimal. Leaf size=20 $x^3 \left (a+b x^2+c x^4\right )^{p+1}$

[Out]

x^3*(a + b*x^2 + c*x^4)^(1 + p)

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Rubi [A]  time = 0.0363838, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 42, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.024, Rules used = {1588} $x^3 \left (a+b x^2+c x^4\right )^{p+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2*(a + b*x^2 + c*x^4)^p*(3*a + b*(5 + 2*p)*x^2 + c*(7 + 4*p)*x^4),x]

[Out]

x^3*(a + b*x^2 + c*x^4)^(1 + p)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^2 \left (a+b x^2+c x^4\right )^p \left (3 a+b (5+2 p) x^2+c (7+4 p) x^4\right ) \, dx &=x^3 \left (a+b x^2+c x^4\right )^{1+p}\\ \end{align*}

Mathematica [A]  time = 0.147311, size = 20, normalized size = 1. $x^3 \left (a+b x^2+c x^4\right )^{p+1}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2*(a + b*x^2 + c*x^4)^p*(3*a + b*(5 + 2*p)*x^2 + c*(7 + 4*p)*x^4),x]

[Out]

x^3*(a + b*x^2 + c*x^4)^(1 + p)

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Maple [A]  time = 0.014, size = 21, normalized size = 1.1 \begin{align*}{x}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{1+p} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*x^4+b*x^2+a)^p*(3*a+b*(5+2*p)*x^2+c*(7+4*p)*x^4),x)

[Out]

x^3*(c*x^4+b*x^2+a)^(1+p)

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Maxima [A]  time = 1.23919, size = 42, normalized size = 2.1 \begin{align*}{\left (c x^{7} + b x^{5} + a x^{3}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^4+b*x^2+a)^p*(3*a+b*(5+2*p)*x^2+c*(7+4*p)*x^4),x, algorithm="maxima")

[Out]

(c*x^7 + b*x^5 + a*x^3)*(c*x^4 + b*x^2 + a)^p

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Fricas [A]  time = 1.85655, size = 63, normalized size = 3.15 \begin{align*}{\left (c x^{7} + b x^{5} + a x^{3}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^4+b*x^2+a)^p*(3*a+b*(5+2*p)*x^2+c*(7+4*p)*x^4),x, algorithm="fricas")

[Out]

(c*x^7 + b*x^5 + a*x^3)*(c*x^4 + b*x^2 + a)^p

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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**4+b*x**2+a)**p*(3*a+b*(5+2*p)*x**2+c*(7+4*p)*x**4),x)

[Out]

Timed out

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Giac [B]  time = 1.1973, size = 78, normalized size = 3.9 \begin{align*}{\left (c x^{4} + b x^{2} + a\right )}^{p} c x^{7} +{\left (c x^{4} + b x^{2} + a\right )}^{p} b x^{5} +{\left (c x^{4} + b x^{2} + a\right )}^{p} a x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^4+b*x^2+a)^p*(3*a+b*(5+2*p)*x^2+c*(7+4*p)*x^4),x, algorithm="giac")

[Out]

(c*x^4 + b*x^2 + a)^p*c*x^7 + (c*x^4 + b*x^2 + a)^p*b*x^5 + (c*x^4 + b*x^2 + a)^p*a*x^3