3.476 \(\int \frac{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x} \, dx\)

Optimal. Leaf size=69 \[ -\frac{3 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac{\sqrt [3]{x} b}{a}+1\right )}{2 p+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0681823, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{3 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac{\sqrt [3]{x} b}{a}+1\right )}{2 p+1} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p/x,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.8096, size = 85, normalized size = 1.23 \[ - \frac{3 \left (a b + b^{2} \sqrt [3]{x}\right )^{- 2 p} \left (a b + b^{2} \sqrt [3]{x}\right )^{2 p + 1} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 1, 2 p + 1 \\ 2 p + 2 \end{matrix}\middle |{1 + \frac{b \sqrt [3]{x}}{a}} \right )}}{a b \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**p/x,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0376492, size = 59, normalized size = 0.86 \[ \frac{3 \left (\frac{a}{b \sqrt [3]{x}}+1\right )^{-2 p} \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \, _2F_1\left (-2 p,-2 p;1-2 p;-\frac{a}{b \sqrt [3]{x}}\right )}{2 p} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p/x,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.012, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ({a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}} \right ) ^{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/x,x)

[Out]

int((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/x,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p/x,x, algorithm="maxima")

[Out]

integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p/x, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p/x,x, algorithm="fricas")

[Out]

integral((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p/x, x)

_______________________________________________________________________________________

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((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**p/x,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p/x,x, algorithm="giac")

[Out]

integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p/x, x)