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

Optimal. Leaf size=53 \[ \frac{x \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (1,2 p+\frac{4}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0378543, antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ x \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{1}{3},-2 p;\frac{4}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.03083, size = 49, normalized size = 0.92 \[ x \left (1 + \frac{b x^{3}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )} \]

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)

[Out]

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

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Mathematica [C]  time = 0.315211, size = 204, normalized size = 3.85 \[ \frac{4^{-p} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-2 p} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-2 p} \left (\left (a+b x^3\right )^2\right )^p F_1\left (2 p+1;-2 p,-2 p;2 (p+1);-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{b} (2 p+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*((a + b*x^3)^2)^p*AppellF1[1 + 2*p, -2*p, -2*p
, 2*(1 + p), ((-I)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(Sqrt[3]*a^(1/3)), (I + Sqr
t[3] - ((2*I)*b^(1/3)*x)/a^(1/3))/(3*I + Sqrt[3])])/(4^p*b^(1/3)*(1 + 2*p)*((a^(
1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^(2*p)*((I*(1 + (b^(1/3)
*x)/a^(1/3)))/(3*I + Sqrt[3]))^(2*p))

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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, algorithm="maxima")

[Out]

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

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

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, algorithm="fricas")

[Out]

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

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

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)

[Out]

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

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

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, algorithm="giac")

[Out]

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