3.478 \(\int \left (\frac{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2}-\frac{2 b^3 (1-2 p) (1-p) p \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{3 a^3 x}\right ) \, dx\)

Optimal. Leaf size=146 \[ \frac{b (1-p) \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a^2 x^{2/3}}-\frac{\left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a x}-\frac{b^2 (1-2 p) (1-p) \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a^3 \sqrt [3]{x}} \]

[Out]

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

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Rubi [C]  time = 0.199595, antiderivative size = 162, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 4, integrand size = 77, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.052 \[ \frac{2 b^3 (1-2 p) (1-p) p \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac{\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)}+\frac{3 b^3 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,2 p+1;2 (p+1);\frac{\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^3*(1 - 2*p)*(1 - p)*p*(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])/(a^3*(1 + 2*p)
) + (3*b^3*(1 + (b*x^(1/3))/a)*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*Hypergeomet
ric2F1[4, 1 + 2*p, 2*(1 + p), 1 + (b*x^(1/3))/a])/(a^3*(1 + 2*p))

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Rubi in Sympy [A]  time = 50.0171, size = 185, normalized size = 1.27 \[ \frac{2 b^{2} p \left (- 2 p + 1\right ) \left (- p + 1\right ) \left (a b + b^{2} \sqrt [3]{x}\right )^{- 2 p} \left (a b + b^{2} \sqrt [3]{x}\right )^{2 p + 1} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 1, 2 p + 1 \\ 2 p + 2 \end{matrix}\middle |{1 + \frac{b \sqrt [3]{x}}{a}} \right )}}{a^{4} \left (2 p + 1\right )} + \frac{3 b^{2} \left (a b + b^{2} \sqrt [3]{x}\right )^{- 2 p} \left (a b + b^{2} \sqrt [3]{x}\right )^{2 p + 1} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 4, 2 p + 1 \\ 2 p + 2 \end{matrix}\middle |{1 + \frac{b \sqrt [3]{x}}{a}} \right )}}{a^{4} \left (2 p + 1\right )} \]

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

[Out]

2*b**2*p*(-2*p + 1)*(-p + 1)*(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**4*(2*p + 1)) + 3*b**2*(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*hyp
er((4, 2*p + 1), (2*p + 2,), 1 + b*x**(1/3)/a)/(a**4*(2*p + 1))

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Mathematica [A]  time = 0.0844742, size = 64, normalized size = 0.44 \[ -\frac{\left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \left (a^2+a b (p-1) \sqrt [3]{x}+b^2 \left (2 p^2-3 p+1\right ) x^{2/3}\right )}{a^3 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ({a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}} \right ) ^{p}}-{\frac{2\,{b}^{3} \left ( 1-2\,p \right ) \left ( 1-p \right ) p}{3\,{a}^{3}x} \left ({a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}} \right ) ^{p}}\, dx \]

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

[Out]

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

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Maxima [A]  time = 1.21257, size = 85, normalized size = 0.58 \[ -\frac{{\left ({\left (2 \, p^{2} - 3 \, p + 1\right )} b^{3} x + 2 \,{\left (p^{2} - p\right )} a b^{2} x^{\frac{2}{3}} + a^{2} b p x^{\frac{1}{3}} + a^{3}\right )}{\left (b x^{\frac{1}{3}} + a\right )}^{2 \, p}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.375884, size = 111, normalized size = 0.76 \[ -\frac{{\left (a^{2} b p x^{\frac{1}{3}} + a^{3} +{\left (2 \, b^{3} p^{2} - 3 \, b^{3} p + b^{3}\right )} x + 2 \,{\left (a b^{2} p^{2} - a b^{2} p\right )} x^{\frac{2}{3}}\right )}{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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