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

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 = 0.0133516, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]

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 = 6.06874, size = 19, normalized size = 0.7 \[ \frac{\left (- a + b x^{2} + c x^{4}\right )^{p + 1}}{2 \left (p + 1\right )} \]

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 = 0.0381282, size = 26, 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 = 26, 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 = 0.831103, size = 50, normalized size = 1.85 \[ \frac{{\left (c x^{4} + b x^{2} - a\right )}{\left (c x^{4} + b x^{2} - a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]

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 = 0.287448, size = 50, normalized size = 1.85 \[ \frac{{\left (c x^{4} + b x^{2} - a\right )}{\left (c x^{4} + b x^{2} - a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]

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 = 0.2727, size = 101, normalized size = 3.74 \[ \frac{c x^{4} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} - a\right )\right )} + b x^{2} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} - a\right )\right )} - a e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} - a\right )\right )}}{2 \,{\left (p + 1\right )}} \]

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)