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

Optimal. Leaf size=92 \[ \frac{3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{2} (a+b x)^{2/3} \]

[Out]

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

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Rubi [A]  time = 0.0843098, antiderivative size = 92, 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 \[ \frac{3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{2} (a+b x)^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(2/3)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.93584, size = 85, normalized size = 0.92 \[ - \frac{a^{\frac{2}{3}} \log{\left (x \right )}}{2} + \frac{3 a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{2} + \sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )} + \frac{3 \left (a + b x\right )^{\frac{2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0320994, size = 57, normalized size = 0.62 \[ \frac{3 (a+b x)-6 a \sqrt [3]{\frac{a}{b x}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x}\right )}{2 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(2/3)/x,x]

[Out]

(3*(a + b*x) - 6*a*(1 + a/(b*x))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -(a/(b*x
))])/(2*(a + b*x)^(1/3))

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Maple [A]  time = 0.008, size = 84, normalized size = 0.9 \[{\frac{3}{2} \left ( bx+a \right ) ^{{\frac{2}{3}}}}+{a}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ) -{\frac{1}{2}{a}^{{\frac{2}{3}}}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ) }+{a}^{{\frac{2}{3}}}\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(2/3)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220759, size = 146, normalized size = 1.59 \[ \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \,{\left (a^{2}\right )}^{\frac{2}{3}}}\right ) - \frac{1}{2} \,{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) +{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + \frac{3}{2} \,{\left (b x + a\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.93644, size = 182, normalized size = 1.98 \[ \frac{5 a^{\frac{2}{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{5}{3}\right )}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{5 a^{\frac{2}{3}} e^{\frac{8 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{5}{3}\right )}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{5 a^{\frac{2}{3}} e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{5}{3}\right )}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{5 b^{\frac{2}{3}} \left (\frac{a}{b} + x\right )^{\frac{2}{3}} \Gamma \left (\frac{5}{3}\right )}{2 \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(2/3)/x,x)

[Out]

5*a**(2/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(5/3)/(3*gamma(8/3))
+ 5*a**(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(5/3)/(3*gamma(8/3)) + 5*a**(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(5/3)/(3*gamma(8/3)) + 5*b
**(2/3)*(a/b + x)**(2/3)*gamma(5/3)/(2*gamma(8/3))

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GIAC/XCAS [A]  time = 0.506473, size = 116, normalized size = 1.26 \[ \sqrt{3} a^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{2} \, a^{\frac{2}{3}}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + a^{\frac{2}{3}}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + \frac{3}{2} \,{\left (b x + a\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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