3.694 \(\int \frac{1}{\sqrt [3]{x} (a+b x)^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac{\log (a+b x)}{9 a^{7/3} b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} b^{2/3}}+\frac{2 x^{2/3}}{3 a^2 (a+b x)}+\frac{x^{2/3}}{2 a (a+b x)^2} \]

[Out]

x^(2/3)/(2*a*(a + b*x)^2) + (2*x^(2/3))/(3*a^2*(a + b*x)) - (2*ArcTan[(a^(1/3) -
2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(2/3)) - Log[a^(1/3
) + b^(1/3)*x^(1/3)]/(3*a^(7/3)*b^(2/3)) + Log[a + b*x]/(9*a^(7/3)*b^(2/3))

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Rubi [A]  time = 0.113763, antiderivative size = 140, 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 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac{\log (a+b x)}{9 a^{7/3} b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} b^{2/3}}+\frac{2 x^{2/3}}{3 a^2 (a+b x)}+\frac{x^{2/3}}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(1/3)*(a + b*x)^3),x]

[Out]

x^(2/3)/(2*a*(a + b*x)^2) + (2*x^(2/3))/(3*a^2*(a + b*x)) - (2*ArcTan[(a^(1/3) -
2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(2/3)) - Log[a^(1/3
) + b^(1/3)*x^(1/3)]/(3*a^(7/3)*b^(2/3)) + Log[a + b*x]/(9*a^(7/3)*b^(2/3))

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Rubi in Sympy [A]  time = 15.7455, size = 129, normalized size = 0.92 \[ \frac{x^{\frac{2}{3}}}{2 a \left (a + b x\right )^{2}} + \frac{2 x^{\frac{2}{3}}}{3 a^{2} \left (a + b x\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{3 a^{\frac{7}{3}} b^{\frac{2}{3}}} + \frac{\log{\left (a + b x \right )}}{9 a^{\frac{7}{3}} b^{\frac{2}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(2/3)/(2*a*(a + b*x)**2) + 2*x**(2/3)/(3*a**2*(a + b*x)) - log(a**(1/3) + b**
(1/3)*x**(1/3))/(3*a**(7/3)*b**(2/3)) + log(a + b*x)/(9*a**(7/3)*b**(2/3)) - 2*s
qrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(1/3)/3)/a**(1/3))/(9*a**(7/3)*b
**(2/3))

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Mathematica [A]  time = 0.112105, size = 153, normalized size = 1.09 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{b^{2/3}}+\frac{9 a^{4/3} x^{2/3}}{(a+b x)^2}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{12 \sqrt [3]{a} x^{2/3}}{a+b x}}{18 a^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

((9*a^(4/3)*x^(2/3))/(a + b*x)^2 + (12*a^(1/3)*x^(2/3))/(a + b*x) - (4*Sqrt[3]*A
rcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/b^(2/3) - (4*Log[a^(1/3) + b^(
1/3)*x^(1/3)])/b^(2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2
/3)])/b^(2/3))/(18*a^(7/3))

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Maple [A]  time = 0.011, size = 136, normalized size = 1. \[{\frac{1}{2\,a \left ( bx+a \right ) ^{2}}{x}^{{\frac{2}{3}}}}+{\frac{2}{3\,{a}^{2} \left ( bx+a \right ) }{x}^{{\frac{2}{3}}}}-{\frac{2}{9\,{a}^{2}b}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{9\,{a}^{2}b}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}}{9\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/3)/(b*x+a)^3,x)

[Out]

1/2*x^(2/3)/a/(b*x+a)^2+2/3*x^(2/3)/a^2/(b*x+a)-2/9/a^2/b/(a/b)^(1/3)*ln(x^(1/3)
+(a/b)^(1/3))+1/9/a^2/b/(a/b)^(1/3)*ln(x^(2/3)-x^(1/3)*(a/b)^(1/3)+(a/b)^(2/3))+
2/9/a^2*3^(1/2)/b/(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(1/((b*x + a)^3*x^(1/3)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223345, size = 271, normalized size = 1.94 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}{\left (4 \, b x + 7 \, a\right )} x^{\frac{2}{3}} + 4 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) - 2 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (-a b + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{\frac{2}{3}} + \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) - 12 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}}{3 \, a b}\right )\right )}}{54 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(3*sqrt(3)*(-a*b^2)^(1/3)*(4*b*x + 7*a)*x^(2/3) + 4*sqrt(3)*(b^2*x^
2 + 2*a*b*x + a^2)*log(a*b + (-a*b^2)^(2/3)*x^(1/3)) - 2*sqrt(3)*(b^2*x^2 + 2*a*
b*x + a^2)*log(-a*b + (-a*b^2)^(1/3)*b*x^(2/3) + (-a*b^2)^(2/3)*x^(1/3)) - 12*(b
^2*x^2 + 2*a*b*x + a^2)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x^(1
/3))/(a*b)))/((a^2*b^2*x^2 + 2*a^3*b*x + a^4)*(-a*b^2)^(1/3))

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Sympy [A]  time = 5.3247, size = 1693, normalized size = 12.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(14/3)*b**(4/3)*x**2*exp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*
pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gam
ma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 4*a**(1
4/3)*b**(4/3)*x**2*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3
)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x*
*4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 4*a**(14/3)*b**(4/3)*x**2*exp(2*
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
**7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamm
a(5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 12*a**(11/3)*b**(7/3)*x**3*exp(10*I*pi/
3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**
2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3)
+ 27*a**4*b**5*x**5*gamma(5/3)) - 12*a**(11/3)*b**(7/3)*x**3*log(1 - b**(1/3)*x*
*(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a
**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamm
a(5/3)) - 12*a**(11/3)*b**(7/3)*x**3*exp(2*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**7*b**2*x**2*gamma(5/3) + 81*a**6*b*
*3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/3)
) - 12*a**(8/3)*b**(10/3)*x**4*exp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_pola
r(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3
*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 12*
a**(8/3)*b**(10/3)*x**4*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamm
a(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b*
*4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 12*a**(8/3)*b**(10/3)*x**4*
exp(2*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**7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**
4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 4*a**(5/3)*b**(13/3)*x**5*exp(10*
I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**
7*b**2*x**2*gamma(5/3) + 81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(
5/3) + 27*a**4*b**5*x**5*gamma(5/3)) - 4*a**(5/3)*b**(13/3)*x**5*log(1 - b**(1/3
)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) +
81*a**6*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*
gamma(5/3)) - 4*a**(5/3)*b**(13/3)*x**5*exp(2*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**7*b**2*x**2*gamma(5/3) + 81*a**6
*b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5
/3)) + 21*a**4*b**2*x**(8/3)*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*
b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/
3)) + 33*a**3*b**3*x**(11/3)*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*
b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/
3)) + 12*a**2*b**4*x**(14/3)*gamma(2/3)/(27*a**7*b**2*x**2*gamma(5/3) + 81*a**6*
b**3*x**3*gamma(5/3) + 81*a**5*b**4*x**4*gamma(5/3) + 27*a**4*b**5*x**5*gamma(5/
3))

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GIAC/XCAS [A]  time = 0.224405, size = 193, normalized size = 1.38 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} + \frac{4 \, b x^{\frac{5}{3}} + 7 \, a x^{\frac{2}{3}}}{6 \,{\left (b x + a\right )}^{2} a^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(-a/b)^(2/3)*ln(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 - 2/9*sqrt(3)*(-a*b^2)^(2/
3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^2) + 1/6*(
4*b*x^(5/3) + 7*a*x^(2/3))/((b*x + a)^2*a^2) + 1/9*(-a*b^2)^(2/3)*ln(x^(2/3) + x
^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2)