3.404 \(\int \frac{1}{x^2 \sqrt [3]{-a+b x}} \, dx\)

Optimal. Leaf size=103 \[ \frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}+\frac{(b x-a)^{2/3}}{a x} \]

[Out]

(-a + b*x)^(2/3)/(a*x) - (b*ArcTan[(a^(1/3) - 2*(-a + b*x)^(1/3))/(Sqrt[3]*a^(1/
3))])/(Sqrt[3]*a^(4/3)) + (b*Log[x])/(6*a^(4/3)) - (b*Log[a^(1/3) + (-a + b*x)^(
1/3)])/(2*a^(4/3))

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Rubi [A]  time = 0.0872958, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}+\frac{(b x-a)^{2/3}}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(-a + b*x)^(1/3)),x]

[Out]

(-a + b*x)^(2/3)/(a*x) - (b*ArcTan[(a^(1/3) - 2*(-a + b*x)^(1/3))/(Sqrt[3]*a^(1/
3))])/(Sqrt[3]*a^(4/3)) + (b*Log[x])/(6*a^(4/3)) - (b*Log[a^(1/3) + (-a + b*x)^(
1/3)])/(2*a^(4/3))

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Rubi in Sympy [A]  time = 7.66705, size = 90, normalized size = 0.87 \[ \frac{\left (- a + b x\right )^{\frac{2}{3}}}{a x} + \frac{b \log{\left (x \right )}}{6 a^{\frac{4}{3}}} - \frac{b \log{\left (\sqrt [3]{a} + \sqrt [3]{- a + b x} \right )}}{2 a^{\frac{4}{3}}} - \frac{\sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{- a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a + b*x)**(2/3)/(a*x) + b*log(x)/(6*a**(4/3)) - b*log(a**(1/3) + (-a + b*x)**(
1/3))/(2*a**(4/3)) - sqrt(3)*b*atan(sqrt(3)*(a**(1/3)/3 - 2*(-a + b*x)**(1/3)/3)
/a**(1/3))/(3*a**(4/3))

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Mathematica [C]  time = 0.0431539, size = 62, normalized size = 0.6 \[ \frac{-b x \sqrt [3]{1-\frac{a}{b x}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{a}{b x}\right )-a+b x}{a x \sqrt [3]{b x-a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 103, normalized size = 1. \[{\frac{1}{ax} \left ( bx-a \right ) ^{{\frac{2}{3}}}}-{\frac{b}{3}\ln \left ( \sqrt [3]{a}+\sqrt [3]{bx-a} \right ){a}^{-{\frac{4}{3}}}}+{\frac{b}{6}\ln \left ( \left ( bx-a \right ) ^{{\frac{2}{3}}}-\sqrt [3]{bx-a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{4}{3}}}}+{\frac{b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx-a}}{\sqrt [3]{a}}}-1 \right ) } \right ){a}^{-{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x-a)^(1/3),x)

[Out]

(b*x-a)^(2/3)/a/x-1/3*b*ln(a^(1/3)+(b*x-a)^(1/3))/a^(4/3)+1/6*b/a^(4/3)*ln((b*x-
a)^(2/3)-(b*x-a)^(1/3)*a^(1/3)+a^(2/3))+1/3*b/a^(4/3)*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)^(1/3)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221938, size = 186, normalized size = 1.81 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x \log \left ({\left (b x - a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 2 \, \sqrt{3} b x \log \left ({\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + a\right ) + 6 \, b x \arctan \left (\frac{2 \, \sqrt{3}{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) - 6 \, \sqrt{3}{\left (b x - a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}}\right )}}{18 \, \left (-a\right )^{\frac{1}{3}} a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*b*x*log((b*x - a)^(2/3)*(-a)^(1/3) + (b*x - a)^(1/3)*(-a)
^(2/3) - a) - 2*sqrt(3)*b*x*log((b*x - a)^(1/3)*(-a)^(2/3) + a) + 6*b*x*arctan(1
/3*(2*sqrt(3)*(b*x - a)^(1/3)*(-a)^(2/3) - sqrt(3)*a)/a) - 6*sqrt(3)*(b*x - a)^(
2/3)*(-a)^(1/3))/((-a)^(1/3)*a*x)

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Sympy [A]  time = 6.57654, size = 646, normalized size = 6.27 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/3)*b**3*(-a/b + x)**2*exp(10*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)
*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*(-a/b + x)**2*gamma(5/3) +
9*a**2*b**3*(-a/b + x)**3*gamma(5/3)) - 2*a**(5/3)*b**3*(-a/b + x)**2*log(1 - b*
*(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*(-a/b
+ x)**2*gamma(5/3) + 9*a**2*b**3*(-a/b + x)**3*gamma(5/3)) - 2*a**(5/3)*b**3*(-
a/b + x)**2*exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)
/a**(1/3))*gamma(2/3)/(9*a**3*b**2*(-a/b + x)**2*gamma(5/3) + 9*a**2*b**3*(-a/b
+ x)**3*gamma(5/3)) - 2*a**(2/3)*b**4*(-a/b + x)**3*exp(10*I*pi/3)*log(1 - b**(1
/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*(-a/b
+ x)**2*gamma(5/3) + 9*a**2*b**3*(-a/b + x)**3*gamma(5/3)) - 2*a**(2/3)*b**4*(-a
/b + x)**3*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/
3)/(9*a**3*b**2*(-a/b + x)**2*gamma(5/3) + 9*a**2*b**3*(-a/b + x)**3*gamma(5/3))
- 2*a**(2/3)*b**4*(-a/b + x)**3*exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3
)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*(-a/b + x)**2*gamma(5/3)
+ 9*a**2*b**3*(-a/b + x)**3*gamma(5/3)) + 6*a*b**(11/3)*(-a/b + x)**(8/3)*gamma
(2/3)/(9*a**3*b**2*(-a/b + x)**2*gamma(5/3) + 9*a**2*b**3*(-a/b + x)**3*gamma(5/
3))

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GIAC/XCAS [A]  time = 0.599703, size = 194, normalized size = 1.88 \[ -\frac{\frac{2 \, \sqrt{3} \left (-a\right )^{\frac{2}{3}} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{a^{2}} - \frac{\left (-a\right )^{\frac{2}{3}} b^{2}{\rm ln}\left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right )}{a^{2}} + \frac{2 \, \left (-a\right )^{\frac{2}{3}} b^{2}{\rm ln}\left ({\left |{\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}} \right |}\right )}{a^{2}} - \frac{6 \,{\left (b x - a\right )}^{\frac{2}{3}} b}{a x}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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