3.411 \(\int \frac{1}{x^2 (a+b x)^{2/3}} \, dx\)

Optimal. Leaf size=98 \[ \frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{a x} \]

[Out]

-((a + b*x)^(1/3)/(a*x)) + (2*b*ArcTan[(a^(1/3) + 2*(a + b*x)^(1/3))/(Sqrt[3]*a^
(1/3))])/(Sqrt[3]*a^(5/3)) + (b*Log[x])/(3*a^(5/3)) - (b*Log[a^(1/3) - (a + b*x)
^(1/3)])/a^(5/3)

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Rubi [A]  time = 0.0834813, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^(2/3)),x]

[Out]

-((a + b*x)^(1/3)/(a*x)) + (2*b*ArcTan[(a^(1/3) + 2*(a + b*x)^(1/3))/(Sqrt[3]*a^
(1/3))])/(Sqrt[3]*a^(5/3)) + (b*Log[x])/(3*a^(5/3)) - (b*Log[a^(1/3) - (a + b*x)
^(1/3)])/a^(5/3)

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Rubi in Sympy [A]  time = 7.62188, size = 90, normalized size = 0.92 \[ - \frac{\sqrt [3]{a + b x}}{a x} + \frac{b \log{\left (x \right )}}{3 a^{\frac{5}{3}}} - \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{a^{\frac{5}{3}}} + \frac{2 \sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(1/3)/(a*x) + b*log(x)/(3*a**(5/3)) - b*log(a**(1/3) - (a + b*x)**(1
/3))/a**(5/3) + 2*sqrt(3)*b*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x)**(1/3)/3)/a**
(1/3))/(3*a**(5/3))

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Mathematica [C]  time = 0.0370604, size = 60, normalized size = 0.61 \[ \frac{b x \left (\frac{a}{b x}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )-a-b x}{a x (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 95, normalized size = 1. \[ -{\frac{1}{ax}\sqrt [3]{bx+a}}-{\frac{2\,b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{5}{3}}}}+{\frac{b}{3}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{5}{3}}}}+{\frac{2\,b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(2/3),x)

[Out]

-(b*x+a)^(1/3)/a/x-2/3*b/a^(5/3)*ln((b*x+a)^(1/3)-a^(1/3))+1/3*b/a^(5/3)*ln((b*x
+a)^(2/3)+(b*x+a)^(1/3)*a^(1/3)+a^(2/3))+2/3*b/a^(5/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/a^(1/3)*(b*x+a)^(1/3)+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246412, size = 190, normalized size = 1.94 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x \log \left (a^{2} - \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a + \left (-a^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b x \log \left (a + \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) - 6 \, b x \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 3 \, \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{9 \, \left (-a^{2}\right )^{\frac{1}{3}} a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*(sqrt(3)*b*x*log(a^2 - (-a^2)^(1/3)*(b*x + a)^(1/3)*a + (-a^2)^(2/3
)*(b*x + a)^(2/3)) - 2*sqrt(3)*b*x*log(a + (-a^2)^(1/3)*(b*x + a)^(1/3)) - 6*b*x
*arctan(-1/3*(sqrt(3)*a - 2*sqrt(3)*(-a^2)^(1/3)*(b*x + a)^(1/3))/a) + 3*sqrt(3)
*(-a^2)^(1/3)*(b*x + a)^(1/3))/((-a^2)^(1/3)*a*x)

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Sympy [A]  time = 6.67593, size = 610, normalized size = 6.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(4/3)*b**2*(a/b + x)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(1/3)
/(-9*a**3*b*(a/b + x)*gamma(4/3) + 9*a**2*b**2*(a/b + x)**2*gamma(4/3)) + 2*a**(
4/3)*b**2*(a/b + x)*exp(4*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*
I*pi/3)/a**(1/3))*gamma(1/3)/(-9*a**3*b*(a/b + x)*gamma(4/3) + 9*a**2*b**2*(a/b
+ x)**2*gamma(4/3)) + 2*a**(4/3)*b**2*(a/b + x)*exp(2*I*pi/3)*log(1 - b**(1/3)*(
a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(1/3)/(-9*a**3*b*(a/b + x)*ga
mma(4/3) + 9*a**2*b**2*(a/b + x)**2*gamma(4/3)) - 2*a**(1/3)*b**3*(a/b + x)**2*l
og(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(1/3)/(-9*a**3*b*(a/b + x)*gamma
(4/3) + 9*a**2*b**2*(a/b + x)**2*gamma(4/3)) - 2*a**(1/3)*b**3*(a/b + x)**2*exp(
4*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(
1/3)/(-9*a**3*b*(a/b + x)*gamma(4/3) + 9*a**2*b**2*(a/b + x)**2*gamma(4/3)) - 2*
a**(1/3)*b**3*(a/b + x)**2*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_p
olar(4*I*pi/3)/a**(1/3))*gamma(1/3)/(-9*a**3*b*(a/b + x)*gamma(4/3) + 9*a**2*b**
2*(a/b + x)**2*gamma(4/3)) - 3*a*b**(7/3)*(a/b + x)**(4/3)*gamma(1/3)/(-9*a**3*b
*(a/b + x)*gamma(4/3) + 9*a**2*b**2*(a/b + x)**2*gamma(4/3))

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GIAC/XCAS [A]  time = 0.554939, size = 146, normalized size = 1.49 \[ \frac{\frac{2 \, \sqrt{3} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{5}{3}}} + \frac{b^{2}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{5}{3}}} - \frac{2 \, b^{2}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}}} - \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} b}{a x}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(
5/3) + b^2*ln((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(5/3) - 2*b
^2*ln(abs((b*x + a)^(1/3) - a^(1/3)))/a^(5/3) - 3*(b*x + a)^(1/3)*b/(a*x))/b