3.137 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^p}{x^5} \, dx\)

Optimal. Leaf size=60 \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (-\frac{4}{3},-2 p;-\frac{1}{3};-\frac{b x^3}{a}\right )}{4 x^4} \]

[Out]

-((a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[-4/3, -2*p, -1/3, -((b*x^3)/a)
])/(4*x^4*(1 + (b*x^3)/a)^(2*p))

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Rubi [A]  time = 0.04904, 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 \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (-\frac{4}{3},-2 p;-\frac{1}{3};-\frac{b x^3}{a}\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^5,x]

[Out]

-((a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[-4/3, -2*p, -1/3, -((b*x^3)/a)
])/(4*x^4*(1 + (b*x^3)/a)^(2*p))

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Rubi in Sympy [A]  time = 17.2184, size = 58, normalized size = 0.97 \[ - \frac{\left (1 + \frac{b x^{3}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, - \frac{4}{3} \\ - \frac{1}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**p/x**5,x)

[Out]

-(1 + b*x**3/a)**(-2*p)*(a**2 + 2*a*b*x**3 + b**2*x**6)**p*hyper((-2*p, -4/3), (
-1/3,), -b*x**3/a)/(4*x**4)

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Mathematica [A]  time = 0.0310892, size = 51, normalized size = 0.85 \[ -\frac{\left (\left (a+b x^3\right )^2\right )^p \left (\frac{b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (-\frac{4}{3},-2 p;-\frac{1}{3};-\frac{b x^3}{a}\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^5,x]

[Out]

-(((a + b*x^3)^2)^p*Hypergeometric2F1[-4/3, -2*p, -1/3, -((b*x^3)/a)])/(4*x^4*(1
+ (b*x^3)/a)^(2*p))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{ \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}}{{x}^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^p/x^5,x)

[Out]

int((b^2*x^6+2*a*b*x^3+a^2)^p/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{x^{5}}\,{d x} \]

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^5,x, algorithm="maxima")

[Out]

integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p/x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{x^{5}}, x\right ) \]

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^5,x, algorithm="fricas")

[Out]

integral((b^2*x^6 + 2*a*b*x^3 + a^2)^p/x^5, 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((b**2*x**6+2*a*b*x**3+a**2)**p/x**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{x^{5}}\,{d x} \]

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^5,x, algorithm="giac")

[Out]

integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p/x^5, x)