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

Optimal. Leaf size=113 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{\sqrt [3]{x}}{a (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0958704, antiderivative size = 113, 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 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{\sqrt [3]{x}}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(2/3)*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5845, size = 107, normalized size = 0.95 \[ \frac{\sqrt [3]{x}}{a \left (a + b x\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{\log{\left (a + b x \right )}}{3 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.131675, size = 134, normalized size = 1.19 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{\sqrt [3]{b}}+\frac{3 a^{2/3} \sqrt [3]{x}}{a+b x}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{3 a^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 120, normalized size = 1.1 \[{\frac{1}{a \left ( bx+a \right ) }\sqrt [3]{x}}+{\frac{2}{3\,ab}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{3\,ab}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{3\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(2/3)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231429, size = 181, normalized size = 1.6 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b x + a\right )} \log \left (a^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right ) - 2 \, \sqrt{3}{\left (b x + a\right )} \log \left (a + \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right ) - 6 \,{\left (b x + a\right )} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}}{3 \, a}\right ) - 3 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right )}}{9 \, \left (a^{2} b\right )^{\frac{1}{3}}{\left (a b x + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.31162, size = 529, normalized size = 4.68 \[ - \frac{2 a^{\frac{4}{3}} b^{\frac{2}{3}} x e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{2 a^{\frac{4}{3}} b^{\frac{2}{3}} x \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} - \frac{2 a^{\frac{4}{3}} b^{\frac{2}{3}} x e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} - \frac{2 \sqrt [3]{a} b^{\frac{5}{3}} x^{2} e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{2 \sqrt [3]{a} b^{\frac{5}{3}} x^{2} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} - \frac{2 \sqrt [3]{a} b^{\frac{5}{3}} x^{2} e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{3 a b x^{\frac{4}{3}} \Gamma \left (\frac{1}{3}\right )}{9 a^{3} b x \Gamma \left (\frac{4}{3}\right ) + 9 a^{2} b^{2} x^{2} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(4/3)*b**(2/3)*x*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)
/a**(1/3))*gamma(1/3)/(9*a**3*b*x*gamma(4/3) + 9*a**2*b**2*x**2*gamma(4/3)) + 2*
a**(4/3)*b**(2/3)*x*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(1/
3)/(9*a**3*b*x*gamma(4/3) + 9*a**2*b**2*x**2*gamma(4/3)) - 2*a**(4/3)*b**(2/3)*x
*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(1/3)/
(9*a**3*b*x*gamma(4/3) + 9*a**2*b**2*x**2*gamma(4/3)) - 2*a**(1/3)*b**(5/3)*x**2
*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(1/3)/
(9*a**3*b*x*gamma(4/3) + 9*a**2*b**2*x**2*gamma(4/3)) + 2*a**(1/3)*b**(5/3)*x**2
*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(1/3)/(9*a**3*b*x*gamm
a(4/3) + 9*a**2*b**2*x**2*gamma(4/3)) - 2*a**(1/3)*b**(5/3)*x**2*exp(I*pi/3)*log
(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(1/3)/(9*a**3*b*x*gamm
a(4/3) + 9*a**2*b**2*x**2*gamma(4/3)) + 3*a*b*x**(4/3)*gamma(1/3)/(9*a**3*b*x*ga
mma(4/3) + 9*a**2*b**2*x**2*gamma(4/3))

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GIAC/XCAS [A]  time = 0.218408, size = 178, normalized size = 1.58 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b} + \frac{x^{\frac{1}{3}}}{{\left (b x + a\right )} a} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{3 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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