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

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 = 0.0209339, 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 \[ x^3 \left (a+b x^2+c x^4\right )^{p+1} \]

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)

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Rubi in Sympy [A]  time = 13.6066, size = 17, normalized size = 0.85 \[ x^{3} \left (a + b x^{2} + c x^{4}\right )^{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

x**3*(a + b*x**2 + c*x**4)**(p + 1)

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Mathematica [A]  time = 0.0533136, 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 = 0.011, size = 21, normalized size = 1.1 \[{x}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{1+p} \]

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 = 0.806387, size = 42, normalized size = 2.1 \[{\left (c x^{7} + b x^{5} + a x^{3}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*(4*p + 7)*x^4 + b*(2*p + 5)*x^2 + 3*a)*(c*x^4 + b*x^2 + a)^p*x^2,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 = 0.279548, size = 42, normalized size = 2.1 \[{\left (c x^{7} + b x^{5} + a x^{3}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*(4*p + 7)*x^4 + b*(2*p + 5)*x^2 + 3*a)*(c*x^4 + b*x^2 + a)^p*x^2,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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.302884, size = 86, normalized size = 4.3 \[ c x^{7} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} + b x^{5} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} + a x^{3} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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