3.417 \(\int \frac{1}{x (a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=93 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{a \sqrt [3]{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0843766, antiderivative size = 93, 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 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{a \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)^(4/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.49621, size = 85, normalized size = 0.91 \[ \frac{3}{a \sqrt [3]{a + b x}} - \frac{\log{\left (x \right )}}{2 a^{\frac{4}{3}}} + \frac{3 \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{2 a^{\frac{4}{3}}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)**(4/3),x)

[Out]

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

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Mathematica [C]  time = 0.0357555, size = 50, normalized size = 0.54 \[ \frac{3-3 \sqrt [3]{\frac{a}{b x}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x}\right )}{a \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x)^(4/3)),x]

[Out]

(3 - 3*(1 + a/(b*x))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -(a/(b*x))])/(a*(a +
b*x)^(1/3))

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Maple [A]  time = 0.011, size = 87, normalized size = 0.9 \[{1\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{4}{3}}}}-{\frac{1}{2}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{4}{3}}}}+{\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{4}{3}}}}+3\,{\frac{1}{a\sqrt [3]{bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)^(4/3),x)

[Out]

1/a^(4/3)*ln((b*x+a)^(1/3)-a^(1/3))-1/2/a^(4/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a
^(1/3)+a^(2/3))+1/a^(4/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(1/3)+1)
)+3/a/(b*x+a)^(1/3)

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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)^(4/3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218735, size = 151, normalized size = 1.62 \[ \frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right )}}{3 \, a}\right ) -{\left (b x + a\right )}^{\frac{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) + 2 \,{\left (b x + a\right )}^{\frac{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right ) + 6 \, a^{\frac{1}{3}}}{2 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.2152, size = 184, normalized size = 1.98 \[ - \frac{\Gamma \left (- \frac{1}{3}\right )}{a \sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} \Gamma \left (\frac{2}{3}\right )} - \frac{\log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{e^{\frac{8 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)**(4/3),x)

[Out]

-gamma(-1/3)/(a*b**(1/3)*(a/b + x)**(1/3)*gamma(2/3)) - log(1 - b**(1/3)*(a/b +
x)**(1/3)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*gamma(2/3)) - exp(8*I*pi/3)*log(1 -
b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*
gamma(2/3)) - exp(4*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3
)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*gamma(2/3))

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GIAC/XCAS [A]  time = 0.507363, size = 120, normalized size = 1.29 \[ \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{4}{3}}} - \frac{{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{2 \, a^{\frac{4}{3}}} + \frac{{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{4}{3}}} + \frac{3}{{\left (b x + a\right )}^{\frac{1}{3}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*x),x, algorithm="giac")

[Out]

sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(4/3) - 1/2*
ln((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) + ln(abs((b*x +
a)^(1/3) - a^(1/3)))/a^(4/3) + 3/((b*x + a)^(1/3)*a)