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

Optimal. Leaf size=127 \[ \frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{5/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{2 x^2}-\frac{b \sqrt [3]{a+b x}}{6 a x} \]

[Out]

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

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Rubi [A]  time = 0.121913, antiderivative size = 127, 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 \[ \frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{5/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{2 x^2}-\frac{b \sqrt [3]{a+b x}}{6 a x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/3)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.1332, size = 112, normalized size = 0.88 \[ - \frac{\sqrt [3]{a + b x}}{2 x^{2}} - \frac{b \sqrt [3]{a + b x}}{6 a x} + \frac{b^{2} \log{\left (x \right )}}{18 a^{\frac{5}{3}}} - \frac{b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{6 a^{\frac{5}{3}}} + \frac{\sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0381378, size = 78, normalized size = 0.61 \[ \frac{-3 a^2+b^2 x^2 \left (\frac{a}{b x}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )-4 a b x-b^2 x^2}{6 a x^2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^2 - 4*a*b*x - b^2*x^2 + b^2*(1 + a/(b*x))^(2/3)*x^2*Hypergeometric2F1[2/3,
2/3, 5/3, -(a/(b*x))])/(6*a*x^2*(a + b*x)^(2/3))

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Maple [A]  time = 0.016, size = 113, normalized size = 0.9 \[ -{\frac{1}{6\,a{x}^{2}} \left ( bx+a \right ) ^{{\frac{4}{3}}}}-{\frac{1}{3\,{x}^{2}}\sqrt [3]{bx+a}}-{\frac{{b}^{2}}{9}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{5}{3}}}}+{\frac{{b}^{2}}{18}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{5}{3}}}}+{\frac{{b}^{2}\sqrt{3}}{9}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/3)/x^3,x)

[Out]

-1/6/x^2/a*(b*x+a)^(4/3)-1/3*(b*x+a)^(1/3)/x^2-1/9*b^2/a^(5/3)*ln((b*x+a)^(1/3)-
a^(1/3))+1/18*b^2/a^(5/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a^(1/3)+a^(2/3))+1/9*b^
2/a^(5/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(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((b*x + a)^(1/3)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219531, size = 216, normalized size = 1.7 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b^{2} x^{2} \log \left (a^{2} - \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a + \left (-a^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b^{2} x^{2} \log \left (a + \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) - 6 \, b^{2} x^{2} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 3 \, \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + 3 \, a\right )}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{54 \, \left (-a^{2}\right )^{\frac{1}{3}} a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.00036, size = 1731, normalized size = 13.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(16/3)*b**2*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(27*a**
7*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(
7/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) - 4*a**(16/3)*b**2*exp(4*I*pi/3)*lo
g(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**
7*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(
7/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) - 4*a**(16/3)*b**2*exp(2*I*pi/3)*lo
g(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**
7*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(
7/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) + 12*a**(13/3)*b**3*(a/b + x)*log(1
- b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(27*a**7*gamma(7/3) - 81*a**6*
b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a**4*b**3*(a/
b + x)**3*gamma(7/3)) + 12*a**(13/3)*b**3*(a/b + x)*exp(4*I*pi/3)*log(1 - b**(1/
3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*gamma(7/3)
- 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a*
*4*b**3*(a/b + x)**3*gamma(7/3)) + 12*a**(13/3)*b**3*(a/b + x)*exp(2*I*pi/3)*log
(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7
*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7
/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) - 12*a**(10/3)*b**4*(a/b + x)**2*log
(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(27*a**7*gamma(7/3) - 81*a**
6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a**4*b**3*(
a/b + x)**3*gamma(7/3)) - 12*a**(10/3)*b**4*(a/b + x)**2*exp(4*I*pi/3)*log(1 - b
**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*gamma
(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) -
27*a**4*b**3*(a/b + x)**3*gamma(7/3)) - 12*a**(10/3)*b**4*(a/b + x)**2*exp(2*I*p
i/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(4/3)/
(27*a**7*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2
*gamma(7/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) + 4*a**(7/3)*b**5*(a/b + x)*
*3*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(27*a**7*gamma(7/3) -
81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a**4*
b**3*(a/b + x)**3*gamma(7/3)) + 4*a**(7/3)*b**5*(a/b + x)**3*exp(4*I*pi/3)*log(1
- b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*g
amma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3
) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) + 4*a**(7/3)*b**5*(a/b + x)**3*exp(2*I
*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(4/3
)/(27*a**7*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)*
*2*gamma(7/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) - 12*a**5*b**(7/3)*(a/b +
x)**(1/3)*gamma(4/3)/(27*a**7*gamma(7/3) - 81*a**6*b*(a/b + x)*gamma(7/3) + 81*a
**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a**4*b**3*(a/b + x)**3*gamma(7/3)) + 6*a**
4*b**(10/3)*(a/b + x)**(4/3)*gamma(4/3)/(27*a**7*gamma(7/3) - 81*a**6*b*(a/b + x
)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a**4*b**3*(a/b + x)**3*
gamma(7/3)) + 6*a**3*b**(13/3)*(a/b + x)**(7/3)*gamma(4/3)/(27*a**7*gamma(7/3) -
81*a**6*b*(a/b + x)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*gamma(7/3) - 27*a**4
*b**3*(a/b + x)**3*gamma(7/3))

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GIAC/XCAS [A]  time = 0.535527, size = 173, normalized size = 1.36 \[ \frac{\frac{2 \, \sqrt{3} b^{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{5}{3}}} + \frac{b^{3}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{5}{3}}} - \frac{2 \, b^{3}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}}} - \frac{3 \,{\left ({\left (b x + a\right )}^{\frac{4}{3}} b^{3} + 2 \,{\left (b x + a\right )}^{\frac{1}{3}} a b^{3}\right )}}{a b^{2} x^{2}}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(2*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^
(5/3) + b^3*ln((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(5/3) - 2*
b^3*ln(abs((b*x + a)^(1/3) - a^(1/3)))/a^(5/3) - 3*((b*x + a)^(4/3)*b^3 + 2*(b*x
+ a)^(1/3)*a*b^3)/(a*b^2*x^2))/b