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

Optimal. Leaf size=117 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} b^{4/3}}-\frac{\log (a+b x)}{6 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}-\frac{\sqrt [3]{x}}{b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0919778, antiderivative size = 117, 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 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} b^{4/3}}-\frac{\log (a+b x)}{6 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}-\frac{\sqrt [3]{x}}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^(1/3)/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5434, size = 107, normalized size = 0.91 \[ - \frac{\sqrt [3]{x}}{b \left (a + b x\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{2}{3}} b^{\frac{4}{3}}} - \frac{\log{\left (a + b x \right )}}{6 a^{\frac{2}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.144454, size = 134, normalized size = 1.15 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} \sqrt [3]{x}}{a+b x}}{6 b^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 112, normalized size = 1. \[ -{\frac{1}{b \left ( bx+a \right ) }\sqrt [3]{x}}+{\frac{1}{3\,{b}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,{b}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/3)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220809, size = 182, normalized size = 1.56 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b x + a\right )} \log \left (a^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right ) - 2 \, \sqrt{3}{\left (b x + a\right )} \log \left (a + \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right ) - 6 \,{\left (b x + a\right )} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}}{3 \, a}\right ) + 6 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right )}}{18 \, \left (a^{2} b\right )^{\frac{1}{3}}{\left (b^{2} x + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.97869, size = 520, normalized size = 4.44 \[ - \frac{4 a^{\frac{7}{3}} b e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} + \frac{4 a^{\frac{7}{3}} b \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} - \frac{4 a^{\frac{7}{3}} b e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} - \frac{4 a^{\frac{4}{3}} b^{2} x e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} + \frac{4 a^{\frac{4}{3}} b^{2} x \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} - \frac{4 a^{\frac{4}{3}} b^{2} x e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} - \frac{12 a^{2} b^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{4}{3}\right )}{9 a^{3} b^{\frac{7}{3}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{2} b^{\frac{10}{3}} x \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(7/3)*b*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3)
)*gamma(4/3)/(9*a**3*b**(7/3)*gamma(7/3) + 9*a**2*b**(10/3)*x*gamma(7/3)) + 4*a*
*(7/3)*b*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(4/3)/(9*a**3*
b**(7/3)*gamma(7/3) + 9*a**2*b**(10/3)*x*gamma(7/3)) - 4*a**(7/3)*b*exp(I*pi/3)*
log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(9*a**3*b**(7
/3)*gamma(7/3) + 9*a**2*b**(10/3)*x*gamma(7/3)) - 4*a**(4/3)*b**2*x*exp(5*I*pi/3
)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(4/3)/(9*a**3*b**(7
/3)*gamma(7/3) + 9*a**2*b**(10/3)*x*gamma(7/3)) + 4*a**(4/3)*b**2*x*log(1 - b**(
1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(4/3)/(9*a**3*b**(7/3)*gamma(7/3) +
9*a**2*b**(10/3)*x*gamma(7/3)) - 4*a**(4/3)*b**2*x*exp(I*pi/3)*log(1 - b**(1/3)
*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(9*a**3*b**(7/3)*gamma(7/3) +
9*a**2*b**(10/3)*x*gamma(7/3)) - 12*a**2*b**(4/3)*x**(1/3)*gamma(4/3)/(9*a**3*b
**(7/3)*gamma(7/3) + 9*a**2*b**(10/3)*x*gamma(7/3))

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GIAC/XCAS [A]  time = 0.221679, size = 184, normalized size = 1.57 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{x^{\frac{1}{3}}}{{\left (b x + a\right )} b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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