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

Optimal. Leaf size=109 \[ -\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{4/3}}+\frac{\sqrt [3]{a} \log (a+b x)}{2 b^{4/3}}+\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{4/3}}+\frac{3 \sqrt [3]{x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0880264, antiderivative size = 109, 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 \[ -\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{4/3}}+\frac{\sqrt [3]{a} \log (a+b x)}{2 b^{4/3}}+\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{4/3}}+\frac{3 \sqrt [3]{x}}{b} \]

Antiderivative was successfully verified.

[In]  Int[x^(1/3)/(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.231, size = 104, normalized size = 0.95 \[ - \frac{3 \sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 b^{\frac{4}{3}}} + \frac{\sqrt [3]{a} \log{\left (a + b x \right )}}{2 b^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{b^{\frac{4}{3}}} + \frac{3 \sqrt [3]{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0276174, size = 126, normalized size = 1.16 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{b} \sqrt [3]{x}}{2 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(1/3)/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.008, size = 108, normalized size = 1. \[ 3\,{\frac{\sqrt [3]{x}}{b}}-{\frac{a}{{b}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{2\,{b}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}}{{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/3)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229134, size = 150, normalized size = 1.38 \[ -\frac{2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, x^{\frac{1}{3}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.57049, size = 180, normalized size = 1.65 \[ \frac{4 \sqrt [3]{a} e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 b^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} - \frac{4 \sqrt [3]{a} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 b^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{a} e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 b^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{x} \Gamma \left (\frac{4}{3}\right )}{b \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/3)/(b*x+a),x)

[Out]

4*a**(1/3)*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*g
amma(4/3)/(3*b**(4/3)*gamma(7/3)) - 4*a**(1/3)*log(1 - b**(1/3)*x**(1/3)*exp_pol
ar(I*pi)/a**(1/3))*gamma(4/3)/(3*b**(4/3)*gamma(7/3)) + 4*a**(1/3)*exp(I*pi/3)*l
og(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(3*b**(4/3)*ga
mma(7/3)) + 4*x**(1/3)*gamma(4/3)/(b*gamma(7/3))

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GIAC/XCAS [A]  time = 0.224201, size = 161, normalized size = 1.48 \[ \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{b^{2}} + \frac{3 \, x^{\frac{1}{3}}}{b} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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