3.2.31 \(\int x^2 (a+b x^2+c x^4)^p (3 a+b (5+2 p) x^2+c (7+4 p) x^4) \, dx\) [131]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {1602} \begin {gather*} x^3 \left (a+b x^2+c x^4\right )^{p+1} \end {gather*}

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 1602

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*}

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Mathematica [A]
time = 0.38, size = 20, normalized size = 1.00 \begin {gather*} x^3 \left (a+b x^2+c x^4\right )^{1+p} \end {gather*}

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 = 0.03, size = 21, normalized size = 1.05

method	result	size
gosper	\(x^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{1+p}\)	\(21\)
risch	\(\left (c \,x^{4}+b \,x^{2}+a \right )^{p} x^{3} \left (c \,x^{4}+b \,x^{2}+a \right )\)	\(31\)
norman	\(a \,x^{3} {\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}+b \,x^{5} {\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}+c \,x^{7} {\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}\)	\(65\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 31, normalized size = 1.55 \begin {gather*} {\left (c x^{7} + b x^{5} + a x^{3}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{p} \end {gather*}

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 = 0.40, size = 31, normalized size = 1.55 \begin {gather*} {\left (c x^{7} + b x^{5} + a x^{3}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{p} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (17) = 34\).
time = 174.57, size = 54, normalized size = 2.70 \begin {gather*} a x^{3} \left (a + b x^{2} + c x^{4}\right )^{p} + b x^{5} \left (a + b x^{2} + c x^{4}\right )^{p} + c x^{7} \left (a + b x^{2} + c x^{4}\right )^{p} \end {gather*}

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]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
time = 5.21, size = 58, normalized size = 2.90 \begin {gather*} {\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 {gather*}

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

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Mupad [B]
time = 1.10, size = 31, normalized size = 1.55 \begin {gather*} \left (c\,x^7+b\,x^5+a\,x^3\right )\,{\left (c\,x^4+b\,x^2+a\right )}^p \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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