3.681 \(\int \frac{1}{x^{5/3} (a+b x)} \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac{b^{2/3} \log (a+b x)}{2 a^{5/3}}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}-\frac{3}{2 a x^{2/3}} \]

[Out]

-3/(2*a*x^(2/3)) + (Sqrt[3]*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3
]*a^(1/3))])/a^(5/3) - (3*b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*a^(5/3)) +
(b^(2/3)*Log[a + b*x])/(2*a^(5/3))

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Rubi [A]  time = 0.090239, antiderivative size = 111, 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 \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac{b^{2/3} \log (a+b x)}{2 a^{5/3}}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}-\frac{3}{2 a x^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/3)*(a + b*x)),x]

[Out]

-3/(2*a*x^(2/3)) + (Sqrt[3]*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3
]*a^(1/3))])/a^(5/3) - (3*b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*a^(5/3)) +
(b^(2/3)*Log[a + b*x])/(2*a^(5/3))

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Rubi in Sympy [A]  time = 11.4381, size = 105, normalized size = 0.95 \[ - \frac{3}{2 a x^{\frac{2}{3}}} - \frac{3 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{5}{3}}} + \frac{b^{\frac{2}{3}} \log{\left (a + b x \right )}}{2 a^{\frac{5}{3}}} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/3)/(b*x+a),x)

[Out]

-3/(2*a*x**(2/3)) - 3*b**(2/3)*log(a**(1/3) + b**(1/3)*x**(1/3))/(2*a**(5/3)) +
b**(2/3)*log(a + b*x)/(2*a**(5/3)) + sqrt(3)*b**(2/3)*atan(sqrt(3)*(a**(1/3)/3 -
2*b**(1/3)*x**(1/3)/3)/a**(1/3))/a**(5/3)

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Mathematica [A]  time = 0.0534176, size = 126, normalized size = 1.14 \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-\frac{3 a^{2/3}}{x^{2/3}}-2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2 a^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/3)*(a + b*x)),x]

[Out]

((-3*a^(2/3))/x^(2/3) + 2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3
))/Sqrt[3]] - 2*b^(2/3)*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)])/(2*a^(5/3))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[ -{\frac{3}{2\,a}{x}^{-{\frac{2}{3}}}}-{\frac{1}{a}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{2\,a}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}}{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/3)/(b*x+a),x)

[Out]

-3/2/a/x^(2/3)-1/a/(a/b)^(2/3)*ln(x^(1/3)+(a/b)^(1/3))+1/2/a/(a/b)^(2/3)*ln(x^(2
/3)-x^(1/3)*(a/b)^(1/3)+(a/b)^(2/3))-1/a/(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^(5/3)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225254, size = 208, normalized size = 1.87 \[ -\frac{2 \, \sqrt{3} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, b x^{\frac{1}{3}} + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right )}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{\frac{1}{3}} - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 3}{2 \, a x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*x^(5/3)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.25006, size = 185, normalized size = 1.67 \[ \frac{\Gamma \left (- \frac{2}{3}\right )}{a x^{\frac{2}{3}} \Gamma \left (\frac{1}{3}\right )} - \frac{2 b^{\frac{2}{3}} e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{2}{3}\right )}{3 a^{\frac{5}{3}} \Gamma \left (\frac{1}{3}\right )} + \frac{2 b^{\frac{2}{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{2}{3}\right )}{3 a^{\frac{5}{3}} \Gamma \left (\frac{1}{3}\right )} - \frac{2 b^{\frac{2}{3}} e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{2}{3}\right )}{3 a^{\frac{5}{3}} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/3)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218496, size = 162, normalized size = 1.46 \[ \frac{b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a^{2}} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, a^{2}} - \frac{3}{2 \, a x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*x^(5/3)),x, algorithm="giac")

[Out]

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