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

Optimal. Leaf size=143 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}-\frac{\log (a+b x)}{18 a^{5/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{4/3}}+\frac{\sqrt [3]{x}}{6 a b (a+b x)}-\frac{\sqrt [3]{x}}{2 b (a+b x)^2} \]

[Out]

-x^(1/3)/(2*b*(a + b*x)^2) + x^(1/3)/(6*a*b*(a + b*x)) - ArcTan[(a^(1/3) - 2*b^(
1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))]/(3*Sqrt[3]*a^(5/3)*b^(4/3)) + Log[a^(1/3) + b^(
1/3)*x^(1/3)]/(6*a^(5/3)*b^(4/3)) - Log[a + b*x]/(18*a^(5/3)*b^(4/3))

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Rubi [A]  time = 0.117544, antiderivative size = 143, 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 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}-\frac{\log (a+b x)}{18 a^{5/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{4/3}}+\frac{\sqrt [3]{x}}{6 a b (a+b x)}-\frac{\sqrt [3]{x}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(1/3)/(a + b*x)^3,x]

[Out]

-x^(1/3)/(2*b*(a + b*x)^2) + x^(1/3)/(6*a*b*(a + b*x)) - ArcTan[(a^(1/3) - 2*b^(
1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))]/(3*Sqrt[3]*a^(5/3)*b^(4/3)) + Log[a^(1/3) + b^(
1/3)*x^(1/3)]/(6*a^(5/3)*b^(4/3)) - Log[a + b*x]/(18*a^(5/3)*b^(4/3))

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Rubi in Sympy [A]  time = 16.3284, size = 126, normalized size = 0.88 \[ - \frac{\sqrt [3]{x}}{2 b \left (a + b x\right )^{2}} + \frac{\sqrt [3]{x}}{6 a b \left (a + b x\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{6 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\log{\left (a + b x \right )}}{18 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(1/3)/(2*b*(a + b*x)**2) + x**(1/3)/(6*a*b*(a + b*x)) + log(a**(1/3) + b**(1
/3)*x**(1/3))/(6*a**(5/3)*b**(4/3)) - log(a + b*x)/(18*a**(5/3)*b**(4/3)) - sqrt
(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(1/3)/3)/a**(1/3))/(9*a**(5/3)*b**(
4/3))

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Mathematica [A]  time = 0.125547, size = 156, normalized size = 1.09 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{a^{5/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}+\frac{3 \sqrt [3]{b} \sqrt [3]{x}}{a^2+a b x}-\frac{9 \sqrt [3]{b} \sqrt [3]{x}}{(a+b x)^2}}{18 b^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-9*b^(1/3)*x^(1/3))/(a + b*x)^2 + (3*b^(1/3)*x^(1/3))/(a^2 + a*b*x) - (2*Sqrt[
3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*Log[a^(1/3) +
b^(1/3)*x^(1/3)])/a^(5/3) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(
2/3)]/a^(5/3))/(18*b^(4/3))

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Maple [A]  time = 0.017, size = 132, normalized size = 0.9 \[ 3\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( 1/18\,{\frac{{x}^{4/3}}{a}}-1/9\,{\frac{\sqrt [3]{x}}{b}} \right ) }+{\frac{1}{9\,{b}^{2}a}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{18\,{b}^{2}a}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{9\,{b}^{2}a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/3)/(b*x+a)^3,x)

[Out]

3*(1/18/a*x^(4/3)-1/9*x^(1/3)/b)/(b*x+a)^2+1/9/b^2/a/(a/b)^(2/3)*ln(x^(1/3)+(a/b
)^(1/3))-1/18/b^2/a/(a/b)^(2/3)*ln(x^(2/3)-x^(1/3)*(a/b)^(1/3)+(a/b)^(2/3))+1/9/
b^2/a/(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(x^(1/3)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218344, size = 255, normalized size = 1.78 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (a^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right ) - 2 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (a + \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right ) - 3 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (b x - 2 \, a\right )} x^{\frac{1}{3}} - 6 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}}{3 \, a}\right )\right )}}{54 \,{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(b^2*x^2 + 2*a*b*x + a^2)*log(a^2 - (a^2*b)^(1/3)*a*x^(1/
3) + (a^2*b)^(2/3)*x^(2/3)) - 2*sqrt(3)*(b^2*x^2 + 2*a*b*x + a^2)*log(a + (a^2*b
)^(1/3)*x^(1/3)) - 3*sqrt(3)*(a^2*b)^(1/3)*(b*x - 2*a)*x^(1/3) - 6*(b^2*x^2 + 2*
a*b*x + a^2)*arctan(-1/3*(sqrt(3)*a - 2*sqrt(3)*(a^2*b)^(1/3)*x^(1/3))/a))/((a*b
^3*x^2 + 2*a^2*b^2*x + a^3*b)*(a^2*b)^(1/3))

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Sympy [A]  time = 5.32307, size = 1676, normalized size = 11.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(16/3)*b*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3
))*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81
*a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) + 4*a**(16/
3)*b*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(4/3)/(27*a**7*b**
(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma
(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) - 4*a**(16/3)*b*exp(I*pi/3)*log(1 - b
**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*b**(7/3)*gamm
a(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7/3) + 27
*a**4*b**(16/3)*x**3*gamma(7/3)) - 12*a**(13/3)*b**2*x*exp(5*I*pi/3)*log(1 - b**
(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/
3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**
4*b**(16/3)*x**3*gamma(7/3)) + 12*a**(13/3)*b**2*x*log(1 - b**(1/3)*x**(1/3)*exp
_polar(I*pi)/a**(1/3))*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/
3)*x*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gam
ma(7/3)) - 12*a**(13/3)*b**2*x*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5
*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x
*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7
/3)) - 12*a**(10/3)*b**3*x**2*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(
I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*
gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/
3)) + 12*a**(10/3)*b**3*x**2*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))
*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a
**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) - 12*a**(10/3
)*b**3*x**2*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*
gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a*
*5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) - 4*a**(7/3)*b
**4*x**3*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gam
ma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*
b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) + 4*a**(7/3)*b**4
*x**3*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(4/3)/(27*a**7*b*
*(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamm
a(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) - 4*a**(7/3)*b**4*x**3*exp(I*pi/3)*l
og(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*b**(7
/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7
/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) - 12*a**5*b**(4/3)*x**(1/3)*gamma(4/3)/
(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*a**5*b**(13/3
)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) - 6*a**4*b**(7/3)*x**(4/3
)*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*gamma(7/3) + 81*
a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)) + 6*a**3*b**
(10/3)*x**(7/3)*gamma(4/3)/(27*a**7*b**(7/3)*gamma(7/3) + 81*a**6*b**(10/3)*x*ga
mma(7/3) + 81*a**5*b**(13/3)*x**2*gamma(7/3) + 27*a**4*b**(16/3)*x**3*gamma(7/3)
)

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GIAC/XCAS [A]  time = 0.220085, size = 200, normalized size = 1.4 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{2}} + \frac{b x^{\frac{4}{3}} - 2 \, a x^{\frac{1}{3}}}{6 \,{\left (b x + a\right )}^{2} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(-a/b)^(1/3)*ln(abs(x^(1/3) - (-a/b)^(1/3)))/(a^2*b) + 1/9*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^2*b^2) + 1
/18*(-a*b^2)^(1/3)*ln(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^2) +
1/6*(b*x^(4/3) - 2*a*x^(1/3))/((b*x + a)^2*a*b)