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

Optimal. Leaf size=123 \[ \frac{3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}-\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{7/3}}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b} \]

[Out]

(-3*a*x^(1/3))/b^2 + (3*x^(4/3))/(4*b) - (Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/b^(7/3) + (3*a^(4/3)*Log[a^(1/3) + b^(1/3)*x^
(1/3)])/(2*b^(7/3)) - (a^(4/3)*Log[a + b*x])/(2*b^(7/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.124793, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}-\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{7/3}}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(4/3)/(a + b*x),x]

[Out]

(-3*a*x^(1/3))/b^2 + (3*x^(4/3))/(4*b) - (Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/b^(7/3) + (3*a^(4/3)*Log[a^(1/3) + b^(1/3)*x^
(1/3)])/(2*b^(7/3)) - (a^(4/3)*Log[a + b*x])/(2*b^(7/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.8885, size = 117, normalized size = 0.95 \[ \frac{3 a^{\frac{4}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 b^{\frac{7}{3}}} - \frac{a^{\frac{4}{3}} \log{\left (a + b x \right )}}{2 b^{\frac{7}{3}}} - \frac{\sqrt{3} a^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{b^{\frac{7}{3}}} - \frac{3 a \sqrt [3]{x}}{b^{2}} + \frac{3 x^{\frac{4}{3}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**(4/3)*log(a**(1/3) + b**(1/3)*x**(1/3))/(2*b**(7/3)) - a**(4/3)*log(a + b*x
)/(2*b**(7/3)) - sqrt(3)*a**(4/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(1/3)
/3)/a**(1/3))/b**(7/3) - 3*a*x**(1/3)/b**2 + 3*x**(4/3)/(4*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0395493, size = 140, normalized size = 1.14 \[ \frac{-2 a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )+4 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-4 \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-12 a \sqrt [3]{b} \sqrt [3]{x}+3 b^{4/3} x^{4/3}}{4 b^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(4/3)/(a + b*x),x]

[Out]

(-12*a*b^(1/3)*x^(1/3) + 3*b^(4/3)*x^(4/3) - 4*Sqrt[3]*a^(4/3)*ArcTan[(1 - (2*b^
(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] + 4*a^(4/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] - 2*
a^(4/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(4*b^(7/3))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 121, normalized size = 1. \[{\frac{3}{4\,b}{x}^{{\frac{4}{3}}}}-3\,{\frac{a\sqrt [3]{x}}{{b}^{2}}}+{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{2\,{b}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}}{{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(4/3)/(b*x+a),x)

[Out]

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

_______________________________________________________________________________________

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^(4/3)/(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.218596, size = 155, normalized size = 1.26 \[ \frac{4 \, \sqrt{3} a \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} - \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) - 2 \, a \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{2}{3}} - x^{\frac{1}{3}} \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 4 \, a \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 3 \,{\left (b x - 4 \, a\right )} x^{\frac{1}{3}}}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.98978, size = 204, normalized size = 1.66 \[ - \frac{7 a^{\frac{4}{3}} e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 b^{\frac{7}{3}} \Gamma \left (\frac{10}{3}\right )} + \frac{7 a^{\frac{4}{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 b^{\frac{7}{3}} \Gamma \left (\frac{10}{3}\right )} - \frac{7 a^{\frac{4}{3}} e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 b^{\frac{7}{3}} \Gamma \left (\frac{10}{3}\right )} - \frac{7 a \sqrt [3]{x} \Gamma \left (\frac{7}{3}\right )}{b^{2} \Gamma \left (\frac{10}{3}\right )} + \frac{7 x^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )}{4 b \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(4/3)/(b*x+a),x)

[Out]

-7*a**(4/3)*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*
gamma(7/3)/(3*b**(7/3)*gamma(10/3)) + 7*a**(4/3)*log(1 - b**(1/3)*x**(1/3)*exp_p
olar(I*pi)/a**(1/3))*gamma(7/3)/(3*b**(7/3)*gamma(10/3)) - 7*a**(4/3)*exp(I*pi/3
)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(7/3)/(3*b**(7/3)
*gamma(10/3)) - 7*a*x**(1/3)*gamma(7/3)/(b**2*gamma(10/3)) + 7*x**(4/3)*gamma(7/
3)/(4*b*gamma(10/3))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229183, size = 184, normalized size = 1.5 \[ -\frac{a \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{b^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} a \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{b^{3}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} a{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, b^{3}} + \frac{3 \,{\left (b^{3} x^{\frac{4}{3}} - 4 \, a b^{2} x^{\frac{1}{3}}\right )}}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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