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

Optimal. Leaf size=152 \[ -\frac{10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{11/3}}+\frac{10 b^{2/3} \log (a+b x)}{9 a^{11/3}}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}-\frac{10}{3 a^3 x^{2/3}}+\frac{4}{3 a^2 x^{2/3} (a+b x)}+\frac{1}{2 a x^{2/3} (a+b x)^2} \]

[Out]

-10/(3*a^3*x^(2/3)) + 1/(2*a*x^(2/3)*(a + b*x)^2) + 4/(3*a^2*x^(2/3)*(a + b*x))
+ (20*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3
]*a^(11/3)) - (10*b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(3*a^(11/3)) + (10*b^(
2/3)*Log[a + b*x])/(9*a^(11/3))

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Rubi [A]  time = 0.148813, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{11/3}}+\frac{10 b^{2/3} \log (a+b x)}{9 a^{11/3}}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}-\frac{10}{3 a^3 x^{2/3}}+\frac{4}{3 a^2 x^{2/3} (a+b x)}+\frac{1}{2 a x^{2/3} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/3)*(a + b*x)^3),x]

[Out]

-10/(3*a^3*x^(2/3)) + 1/(2*a*x^(2/3)*(a + b*x)^2) + 4/(3*a^2*x^(2/3)*(a + b*x))
+ (20*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3
]*a^(11/3)) - (10*b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(3*a^(11/3)) + (10*b^(
2/3)*Log[a + b*x])/(9*a^(11/3))

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Rubi in Sympy [A]  time = 20.142, size = 146, normalized size = 0.96 \[ \frac{1}{2 a x^{\frac{2}{3}} \left (a + b x\right )^{2}} + \frac{4}{3 a^{2} x^{\frac{2}{3}} \left (a + b x\right )} - \frac{10}{3 a^{3} x^{\frac{2}{3}}} - \frac{10 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{3 a^{\frac{11}{3}}} + \frac{10 b^{\frac{2}{3}} \log{\left (a + b x \right )}}{9 a^{\frac{11}{3}}} + \frac{20 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*a*x**(2/3)*(a + b*x)**2) + 4/(3*a**2*x**(2/3)*(a + b*x)) - 10/(3*a**3*x**(2
/3)) - 10*b**(2/3)*log(a**(1/3) + b**(1/3)*x**(1/3))/(3*a**(11/3)) + 10*b**(2/3)
*log(a + b*x)/(9*a**(11/3)) + 20*sqrt(3)*b**(2/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b
**(1/3)*x**(1/3)/3)/a**(1/3))/(9*a**(11/3))

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Mathematica [A]  time = 0.1354, size = 167, normalized size = 1.1 \[ \frac{20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-\frac{9 a^{5/3} b \sqrt [3]{x}}{(a+b x)^2}-\frac{33 a^{2/3} b \sqrt [3]{x}}{a+b x}-\frac{27 a^{2/3}}{x^{2/3}}-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+40 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-27*a^(2/3))/x^(2/3) - (9*a^(5/3)*b*x^(1/3))/(a + b*x)^2 - (33*a^(2/3)*b*x^(1/
3))/(a + b*x) + 40*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt
[3]] - 40*b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] + 20*b^(2/3)*Log[a^(2/3) - a^(1
/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(18*a^(11/3))

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Maple [A]  time = 0.023, size = 139, normalized size = 0.9 \[ -{\frac{3}{2\,{a}^{3}}{x}^{-{\frac{2}{3}}}}-{\frac{11\,{b}^{2}}{6\,{a}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{4}{3}}}}-{\frac{7\,b}{3\,{a}^{2} \left ( bx+a \right ) ^{2}}\sqrt [3]{x}}-{\frac{20}{9\,{a}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10}{9\,{a}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,\sqrt{3}}{9\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/3)/(b*x+a)^3,x)

[Out]

-3/2/a^3/x^(2/3)-11/6/a^3*b^2/(b*x+a)^2*x^(4/3)-7/3/a^2*b/(b*x+a)^2*x^(1/3)-20/9
/a^3/(a/b)^(2/3)*ln(x^(1/3)+(a/b)^(1/3))+10/9/a^3/(a/b)^(2/3)*ln(x^(2/3)-x^(1/3)
*(a/b)^(1/3)+(a/b)^(2/3))-20/9/a^3/(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)^3*x^(5/3)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229832, size = 344, normalized size = 2.26 \[ -\frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 40 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{\frac{1}{3}} - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 120 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x^{\frac{1}{3}} + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (20 \, b^{2} x^{2} + 32 \, a b x + 9 \, a^{2}\right )}\right )}}{54 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(20*sqrt(3)*(b^2*x^2 + 2*a*b*x + a^2)*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)) - 40*sqrt(3)
*(b^2*x^2 + 2*a*b*x + a^2)*x^(2/3)*(-b^2/a^2)^(1/3)*log(b*x^(1/3) - a*(-b^2/a^2)
^(1/3)) + 120*(b^2*x^2 + 2*a*b*x + a^2)*x^(2/3)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*s
qrt(3)*b*x^(1/3) + sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 3*sqrt(3)
*(20*b^2*x^2 + 32*a*b*x + 9*a^2))/((a^3*b^2*x^2 + 2*a^4*b*x + a^5)*x^(2/3))

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Sympy [A]  time = 9.05024, size = 1921, normalized size = 12.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*a**(14/3)*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(
5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(
11/3)*gamma(1/3)) + 123*a**(11/3)*b*x*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1
/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3)
+ 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) + 156*a**(8/3)*b**2*x**2*gamma(-2/3)/
(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(
17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) + 60*a*
*(5/3)*b**3*x**3*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*
x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*
x**(11/3)*gamma(1/3)) - 40*a**4*b**(2/3)*x**(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)/(27*a**(23/3)*x**(2/3)*gamma(1
/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3)
+ 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) + 40*a**4*b**(2/3)*x**(2/3)*log(1 - b
**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*ga
mma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma
(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) - 40*a**4*b**(2/3)*x**(2/3)*exp(
I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-2/3)/(27*
a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3
)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) - 120*a**3*
b**(5/3)*x**(5/3)*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(
1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*ga
mma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*g
amma(1/3)) + 120*a**3*b**(5/3)*x**(5/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi
)/a**(1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5
/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(1
1/3)*gamma(1/3)) - 120*a**3*b**(5/3)*x**(5/3)*exp(I*pi/3)*log(1 - b**(1/3)*x**(1
/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3)
+ 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 2
7*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) - 120*a**2*b**(8/3)*x**(8/3)*exp(5*I*pi/3
)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-2/3)/(27*a**(23/3
)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x
**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) + 120*a**2*b**(8/3)
*x**(8/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-2/3)/(27*a*
*(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*
b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3)) - 120*a**2*b*
*(8/3)*x**(8/3)*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/
3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamm
a(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gam
ma(1/3)) - 40*a*b**(11/3)*x**(11/3)*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_
polar(I*pi/3)/a**(1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a**(2
0/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(14/3
)*b**3*x**(11/3)*gamma(1/3)) + 40*a*b**(11/3)*x**(11/3)*log(1 - b**(1/3)*x**(1/3
)*exp_polar(I*pi)/a**(1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/3)*gamma(1/3) + 81*a
**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*gamma(1/3) + 27*a**(
14/3)*b**3*x**(11/3)*gamma(1/3)) - 40*a*b**(11/3)*x**(11/3)*exp(I*pi/3)*log(1 -
b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-2/3)/(27*a**(23/3)*x**(2/
3)*gamma(1/3) + 81*a**(20/3)*b*x**(5/3)*gamma(1/3) + 81*a**(17/3)*b**2*x**(8/3)*
gamma(1/3) + 27*a**(14/3)*b**3*x**(11/3)*gamma(1/3))

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GIAC/XCAS [A]  time = 0.222017, size = 203, normalized size = 1.34 \[ \frac{20 \, b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4}} - \frac{20 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4}} - \frac{10 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{4}} - \frac{20 \, b^{2} x^{2} + 32 \, a b x + 9 \, a^{2}}{6 \,{\left (b x^{\frac{4}{3}} + a x^{\frac{1}{3}}\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20/9*b*(-a/b)^(1/3)*ln(abs(x^(1/3) - (-a/b)^(1/3)))/a^4 - 20/9*sqrt(3)*(-a*b^2)^
(1/3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/a^4 - 10/9*(-a
*b^2)^(1/3)*ln(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/a^4 - 1/6*(20*b^2*
x^2 + 32*a*b*x + 9*a^2)/((b*x^(4/3) + a*x^(1/3))^2*a^3)