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

Optimal. Leaf size=100 \[ \frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0676623, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(2/3)*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.62149, size = 95, normalized size = 0.95 \[ \frac{3 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\log{\left (a + b x \right )}}{2 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{2}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0230276, size = 103, normalized size = 1.03 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2 a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 95, normalized size = 1. \[{\frac{1}{b}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{2\,b}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(2/3)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223692, size = 115, normalized size = 1.15 \[ \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right )}}{3 \, a}\right ) - \log \left (a^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right ) + 2 \, \log \left (a + \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right )}{2 \, \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.1814, size = 156, normalized size = 1.56 \[ - \frac{e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \sqrt [3]{b} \Gamma \left (\frac{4}{3}\right )} + \frac{\log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \sqrt [3]{b} \Gamma \left (\frac{4}{3}\right )} - \frac{e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \sqrt [3]{b} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(2/3)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217682, size = 158, normalized size = 1.58 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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