3.682 \(\int \frac{x^{5/3}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=129 \[ \frac{5 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}-\frac{5 a^{2/3} \log (a+b x)}{6 b^{8/3}}+\frac{5 a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{8/3}}-\frac{x^{5/3}}{b (a+b x)}+\frac{5 x^{2/3}}{2 b^2} \]

[Out]

(5*x^(2/3))/(2*b^2) - x^(5/3)/(b*(a + b*x)) + (5*a^(2/3)*ArcTan[(a^(1/3) - 2*b^(
1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(8/3)) + (5*a^(2/3)*Log[a^(1/3) + b
^(1/3)*x^(1/3)])/(2*b^(8/3)) - (5*a^(2/3)*Log[a + b*x])/(6*b^(8/3))

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Rubi [A]  time = 0.113116, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{5 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}-\frac{5 a^{2/3} \log (a+b x)}{6 b^{8/3}}+\frac{5 a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{8/3}}-\frac{x^{5/3}}{b (a+b x)}+\frac{5 x^{2/3}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/3)/(a + b*x)^2,x]

[Out]

(5*x^(2/3))/(2*b^2) - x^(5/3)/(b*(a + b*x)) + (5*a^(2/3)*ArcTan[(a^(1/3) - 2*b^(
1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(8/3)) + (5*a^(2/3)*Log[a^(1/3) + b
^(1/3)*x^(1/3)])/(2*b^(8/3)) - (5*a^(2/3)*Log[a + b*x])/(6*b^(8/3))

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Rubi in Sympy [A]  time = 16.506, size = 124, normalized size = 0.96 \[ \frac{5 a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 b^{\frac{8}{3}}} - \frac{5 a^{\frac{2}{3}} \log{\left (a + b x \right )}}{6 b^{\frac{8}{3}}} + \frac{5 \sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{8}{3}}} - \frac{x^{\frac{5}{3}}}{b \left (a + b x\right )} + \frac{5 x^{\frac{2}{3}}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/3)/(b*x+a)**2,x)

[Out]

5*a**(2/3)*log(a**(1/3) + b**(1/3)*x**(1/3))/(2*b**(8/3)) - 5*a**(2/3)*log(a + b
*x)/(6*b**(8/3)) + 5*sqrt(3)*a**(2/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(
1/3)/3)/a**(1/3))/(3*b**(8/3)) - x**(5/3)/(b*(a + b*x)) + 5*x**(2/3)/(2*b**2)

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Mathematica [A]  time = 0.13883, size = 147, normalized size = 1.14 \[ \frac{-5 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )+10 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+10 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{6 a b^{2/3} x^{2/3}}{a+b x}+9 b^{2/3} x^{2/3}}{6 b^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/3)/(a + b*x)^2,x]

[Out]

(9*b^(2/3)*x^(2/3) + (6*a*b^(2/3)*x^(2/3))/(a + b*x) + 10*Sqrt[3]*a^(2/3)*ArcTan
[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] + 10*a^(2/3)*Log[a^(1/3) + b^(1/3)*x
^(1/3)] - 5*a^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(6
*b^(8/3))

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Maple [A]  time = 0.017, size = 123, normalized size = 1. \[{\frac{3}{2\,{b}^{2}}{x}^{{\frac{2}{3}}}}+{\frac{a}{{b}^{2} \left ( bx+a \right ) }{x}^{{\frac{2}{3}}}}+{\frac{5\,a}{3\,{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,a}{6\,{b}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,a\sqrt{3}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/3)/(b*x+a)^2,x)

[Out]

3/2*x^(2/3)/b^2+a/b^2*x^(2/3)/(b*x+a)+5/3*a/b^3/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/
3))-5/6*a/b^3/(a/b)^(1/3)*ln(x^(2/3)-x^(1/3)*(a/b)^(1/3)+(a/b)^(2/3))-5/3*a/b^3*
3^(1/2)/(a/b)^(1/3)*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^(5/3)/(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226783, size = 242, normalized size = 1.88 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b x + a\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (-b x^{\frac{1}{3}} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a x^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3}{\left (b x + a\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a x^{\frac{1}{3}}\right ) + 30 \,{\left (b x + a\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - 2 \, \sqrt{3} a x^{\frac{1}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (3 \, b x + 5 \, a\right )} x^{\frac{2}{3}}\right )}}{18 \,{\left (b^{3} x + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/3)/(b*x + a)^2,x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(5*sqrt(3)*(b*x + a)*(a^2/b^2)^(1/3)*log(-b*x^(1/3)*(a^2/b^2)^(2/3
) + a*x^(2/3) + a*(a^2/b^2)^(1/3)) - 10*sqrt(3)*(b*x + a)*(a^2/b^2)^(1/3)*log(b*
(a^2/b^2)^(2/3) + a*x^(1/3)) + 30*(b*x + a)*(a^2/b^2)^(1/3)*arctan(-1/3*(sqrt(3)
*b*(a^2/b^2)^(2/3) - 2*sqrt(3)*a*x^(1/3))/(b*(a^2/b^2)^(2/3))) - 3*sqrt(3)*(3*b*
x + 5*a)*x^(2/3))/(b^3*x + a*b^2)

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Sympy [A]  time = 8.44365, size = 581, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/3)/(b*x+a)**2,x)

[Out]

40*a**(11/3)*b**3*exp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**
(1/3))*gamma(8/3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3)
) + 40*a**(11/3)*b**3*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(
8/3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3)) + 40*a**(11
/3)*b**3*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*g
amma(8/3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3)) + 40*a
**(8/3)*b**4*x*exp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/
3))*gamma(8/3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3)) +
40*a**(8/3)*b**4*x*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(8/
3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3)) + 40*a**(8/3)
*b**4*x*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*ga
mma(8/3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3)) + 60*a*
*3*b**(11/3)*x**(2/3)*gamma(8/3)/(9*a**3*b**(17/3)*gamma(11/3) + 9*a**2*b**(20/3
)*x*gamma(11/3)) + 36*a**2*b**(14/3)*x**(5/3)*gamma(8/3)/(9*a**3*b**(17/3)*gamma
(11/3) + 9*a**2*b**(20/3)*x*gamma(11/3))

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GIAC/XCAS [A]  time = 0.218803, size = 182, normalized size = 1.41 \[ \frac{5 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b^{2}} + \frac{a x^{\frac{2}{3}}}{{\left (b x + a\right )} b^{2}} + \frac{3 \, x^{\frac{2}{3}}}{2 \, b^{2}} + \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{5 \, \left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/3)/(b*x + a)^2,x, algorithm="giac")

[Out]

5/3*(-a/b)^(2/3)*ln(abs(x^(1/3) - (-a/b)^(1/3)))/b^2 + a*x^(2/3)/((b*x + a)*b^2)
+ 3/2*x^(2/3)/b^2 + 5/3*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^(1/3) +
(-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 5/6*(-a*b^2)^(2/3)*ln(x^(2/3) + x^(1/3)*(-a/b)
^(1/3) + (-a/b)^(2/3))/b^4