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

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))

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Rubi [A]  time = 0.0138469, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]

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))

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Rubi in Sympy [A]  time = 5.55645, size = 19, normalized size = 0.76 \[ \frac{\left (a + b x^{2} + c x^{4}\right )^{p + 1}}{2 \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2+a)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0336139, size = 24, normalized size = 0.96 \[ \frac{\left (a+b x^2+c x^4\right )^{p+1}}{2 p+2} \]

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 + 2*p)

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Maple [A]  time = 0.005, size = 24, normalized size = 1. \[{\frac{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{1+p}}{2+2\,p}} \]

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 = 0.856237, size = 45, normalized size = 1.8 \[ \frac{{\left (c x^{4} + b x^{2} + a\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)*(c*x^4 + b*x^2 + a)^p*x,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 = 0.278075, size = 45, normalized size = 1.8 \[ \frac{{\left (c x^{4} + b x^{2} + a\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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/XCAS [A]  time = 0.272383, size = 92, normalized size = 3.68 \[ \frac{c x^{4} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} + b x^{2} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} + a e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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