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

Optimal. Leaf size=116 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac{\log (a+b x)}{6 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} b^{2/3}}+\frac{x^{2/3}}{a (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0909379, antiderivative size = 116, 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 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac{\log (a+b x)}{6 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} b^{2/3}}+\frac{x^{2/3}}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(1/3)*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.2629, size = 107, normalized size = 0.92 \[ \frac{x^{\frac{2}{3}}}{a \left (a + b x\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{4}{3}} b^{\frac{2}{3}}} + \frac{\log{\left (a + b x \right )}}{6 a^{\frac{4}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{4}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.12354, size = 133, normalized size = 1.15 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} x^{2/3}}{a+b x}}{6 a^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 120, normalized size = 1. \[{\frac{1}{a \left ( bx+a \right ) }{x}^{{\frac{2}{3}}}}-{\frac{1}{3\,ab}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{6\,ab}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{3\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/3)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222477, size = 198, normalized size = 1.71 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b x + a\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) - \sqrt{3}{\left (b x + a\right )} \log \left (-a b + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{\frac{2}{3}} + \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) - 6 \,{\left (b x + a\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}}{3 \, a b}\right ) + 6 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} x^{\frac{2}{3}}\right )}}{18 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\left (a b x + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.95525, size = 563, normalized size = 4.85 \[ - \frac{2 a^{\frac{5}{3}} b^{\frac{4}{3}} x^{2} e^{\frac{10 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} - \frac{2 a^{\frac{5}{3}} b^{\frac{4}{3}} x^{2} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} - \frac{2 a^{\frac{5}{3}} b^{\frac{4}{3}} x^{2} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} - \frac{2 a^{\frac{2}{3}} b^{\frac{7}{3}} x^{3} e^{\frac{10 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} - \frac{2 a^{\frac{2}{3}} b^{\frac{7}{3}} x^{3} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} - \frac{2 a^{\frac{2}{3}} b^{\frac{7}{3}} x^{3} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} + \frac{6 a b^{2} x^{\frac{8}{3}} \Gamma \left (\frac{2}{3}\right )}{9 a^{3} b^{2} x^{2} \Gamma \left (\frac{5}{3}\right ) + 9 a^{2} b^{3} x^{3} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/3)*b**(4/3)*x**2*exp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*p
i/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b**3*x**3*gamma(
5/3)) - 2*a**(5/3)*b**(4/3)*x**2*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1
/3))*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b**3*x**3*gamma(5/3)) - 2*
a**(5/3)*b**(4/3)*x**2*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/
3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b**3*x**3*gamma(5/
3)) - 2*a**(2/3)*b**(7/3)*x**3*exp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_pola
r(I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b**3*x**3*g
amma(5/3)) - 2*a**(2/3)*b**(7/3)*x**3*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/
a**(1/3))*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b**3*x**3*gamma(5/3))
- 2*a**(2/3)*b**(7/3)*x**3*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*
I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b**3*x**3*gam
ma(5/3)) + 6*a*b**2*x**(8/3)*gamma(2/3)/(9*a**3*b**2*x**2*gamma(5/3) + 9*a**2*b*
*3*x**3*gamma(5/3))

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GIAC/XCAS [A]  time = 0.222316, size = 178, normalized size = 1.53 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{x^{\frac{2}{3}}}{{\left (b x + a\right )} a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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