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

Optimal. Leaf size=58 \[ \frac{x^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (1,2 p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0433727, antiderivative size = 60, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{2} x^2 \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.6771, size = 53, normalized size = 0.91 \[ \frac{x^{2} \left (1 + \frac{b x^{3}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0182643, size = 51, normalized size = 0.88 \[ \frac{1}{2} x^2 \left (\left (a+b x^3\right )^2\right )^p \left (\frac{b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int x \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x*(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. \[ \int{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x\,{d x} \]

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,x, algorithm="maxima")

[Out]

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

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

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,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*((a + b*x**3)**2)**p, x)

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

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,x, algorithm="giac")

[Out]

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