3.128 \(\int x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx\)

Optimal. Leaf size=60 \[ \frac{1}{5} x^5 \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{5}{3},-2 p;\frac{8}{3};-\frac{b x^3}{a}\right ) \]

[Out]

(x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[5/3, -2*p, 8/3, -((b*x^3)/a
)])/(5*(1 + (b*x^3)/a)^(2*p))

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Rubi [A]  time = 0.0549996, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{5} x^5 \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{5}{3},-2 p;\frac{8}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]

[Out]

(x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[5/3, -2*p, 8/3, -((b*x^3)/a
)])/(5*(1 + (b*x^3)/a)^(2*p))

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Rubi in Sympy [A]  time = 17.4698, size = 53, normalized size = 0.88 \[ \frac{x^{5} \left (1 + \frac{b x^{3}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**p,x)

[Out]

x**5*(1 + b*x**3/a)**(-2*p)*(a**2 + 2*a*b*x**3 + b**2*x**6)**p*hyper((-2*p, 5/3)
, (8/3,), -b*x**3/a)/5

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Mathematica [A]  time = 0.0286049, size = 51, normalized size = 0.85 \[ \frac{1}{5} x^5 \left (\left (a+b x^3\right )^2\right )^p \left (\frac{b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{5}{3},-2 p;\frac{8}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]

[Out]

(x^5*((a + b*x^3)^2)^p*Hypergeometric2F1[5/3, -2*p, 8/3, -((b*x^3)/a)])/(5*(1 +
(b*x^3)/a)^(2*p))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{x}^{4} \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b^2*x^6+2*a*b*x^3+a^2)^p,x)

[Out]

int(x^4*(b^2*x^6+2*a*b*x^3+a^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^4,x, algorithm="maxima")

[Out]

integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^4,x, algorithm="fricas")

[Out]

integral((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^4, x)

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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**4*(b**2*x**6+2*a*b*x**3+a**2)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^4,x, algorithm="giac")

[Out]

integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^4, x)