3.412 \(\int \frac{1}{x^3 (a+b x)^{2/3}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{8/3}}-\frac{5 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac{\sqrt [3]{a+b x}}{2 a x^2} \]

[Out]

-(a + b*x)^(1/3)/(2*a*x^2) + (5*b*(a + b*x)^(1/3))/(6*a^2*x) - (5*b^2*ArcTan[(a^
(1/3) + 2*(a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)) - (5*b^2*Log[
x])/(18*a^(8/3)) + (5*b^2*Log[a^(1/3) - (a + b*x)^(1/3)])/(6*a^(8/3))

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Rubi [A]  time = 0.116346, antiderivative size = 130, 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 \[ -\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{8/3}}-\frac{5 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac{\sqrt [3]{a+b x}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^(2/3)),x]

[Out]

-(a + b*x)^(1/3)/(2*a*x^2) + (5*b*(a + b*x)^(1/3))/(6*a^2*x) - (5*b^2*ArcTan[(a^
(1/3) + 2*(a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)) - (5*b^2*Log[
x])/(18*a^(8/3)) + (5*b^2*Log[a^(1/3) - (a + b*x)^(1/3)])/(6*a^(8/3))

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Rubi in Sympy [A]  time = 11.3346, size = 122, normalized size = 0.94 \[ - \frac{\sqrt [3]{a + b x}}{2 a x^{2}} + \frac{5 b \sqrt [3]{a + b x}}{6 a^{2} x} - \frac{5 b^{2} \log{\left (x \right )}}{18 a^{\frac{8}{3}}} + \frac{5 b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{6 a^{\frac{8}{3}}} - \frac{5 \sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x+a)**(2/3),x)

[Out]

-(a + b*x)**(1/3)/(2*a*x**2) + 5*b*(a + b*x)**(1/3)/(6*a**2*x) - 5*b**2*log(x)/(
18*a**(8/3)) + 5*b**2*log(a**(1/3) - (a + b*x)**(1/3))/(6*a**(8/3)) - 5*sqrt(3)*
b**2*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x)**(1/3)/3)/a**(1/3))/(9*a**(8/3))

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Mathematica [C]  time = 0.0396424, size = 79, normalized size = 0.61 \[ \frac{-3 a^2-5 b^2 x^2 \left (\frac{a}{b x}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )+2 a b x+5 b^2 x^2}{6 a^2 x^2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 117, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}}\sqrt [3]{bx+a}}+{\frac{5\,b}{6\,{a}^{2}x}\sqrt [3]{bx+a}}+{\frac{5\,{b}^{2}}{9}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{8}{3}}}}-{\frac{5\,{b}^{2}}{18}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{8}{3}}}}-{\frac{5\,{b}^{2}\sqrt{3}}{9}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{8}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^(2/3),x)

[Out]

-1/2*(b*x+a)^(1/3)/a/x^2+5/6*b*(b*x+a)^(1/3)/a^2/x+5/9*b^2/a^(8/3)*ln((b*x+a)^(1
/3)-a^(1/3))-5/18*b^2/a^(8/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a^(1/3)+a^(2/3))-5/
9*b^2/a^(8/3)*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(1/((b*x + a)^(2/3)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220784, size = 204, normalized size = 1.57 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3} b^{2} x^{2} \log \left (a^{2} +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 10 \, \sqrt{3} b^{2} x^{2} \log \left (-a +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) + 30 \, b^{2} x^{2} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (5 \, b x - 3 \, a\right )}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2}\right )}^{\frac{1}{3}} a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(2/3)*x^3),x, algorithm="fricas")

[Out]

-1/54*sqrt(3)*(5*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)) - 10*sqrt(3)*b^2*x^2*log(-a + (a^2)^(1/3)*(b*x + a)^(1/3
)) + 30*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)*(5*b*x - 3*a)*(b*x + a)^(1/3))/((a^2)^(1/3)*a^2*x^2)

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Sympy [A]  time = 8.21233, size = 1904, normalized size = 14.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x+a)**(2/3),x)

[Out]

-10*a**(13/3)*b**3*(a/b + x)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(1
/3)/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 1
62*a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) - 1
0*a**(13/3)*b**3*(a/b + x)*exp(4*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_p
olar(2*I*pi/3)/a**(1/3))*gamma(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*
b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b
**4*(a/b + x)**4*gamma(4/3)) - 10*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(1/3)/(-54*a**7*b*
(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/
b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) + 30*a**(10/3)*b**4
*(a/b + x)**2*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(1/3)/(-54*a**7*b
*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a
/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) + 30*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(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b +
x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x
)**4*gamma(4/3)) + 30*a**(10/3)*b**4*(a/b + x)**2*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(1/3)/(-54*a**7*b*(a/b + x)
*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3
*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) - 30*a**(7/3)*b**5*(a/b + x)
**3*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(1/3)/(-54*a**7*b*(a/b + x)
*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3
*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) - 30*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(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma
(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(
4/3)) - 30*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(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3)
+ 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3)
+ 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) + 10*a**(4/3)*b**6*(a/b + x)**4*log(1 -
b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3)
+ 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3)
+ 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) + 10*a**(4/3)*b**6*(a/b + x)**4*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(1/3)
/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*
a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) + 10*a
**(4/3)*b**6*(a/b + x)**4*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_po
lar(4*I*pi/3)/a**(1/3))*gamma(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b
**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b*
*4*(a/b + x)**4*gamma(4/3)) + 24*a**4*b**(10/3)*(a/b + x)**(4/3)*gamma(1/3)/(-54
*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*
b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3)) - 39*a**3*b
**(13/3)*(a/b + x)**(7/3)*gamma(1/3)/(-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6
*b**2*(a/b + x)**2*gamma(4/3) - 162*a**5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*
b**4*(a/b + x)**4*gamma(4/3)) + 15*a**2*b**(16/3)*(a/b + x)**(10/3)*gamma(1/3)/(
-54*a**7*b*(a/b + x)*gamma(4/3) + 162*a**6*b**2*(a/b + x)**2*gamma(4/3) - 162*a*
*5*b**3*(a/b + x)**3*gamma(4/3) + 54*a**4*b**4*(a/b + x)**4*gamma(4/3))

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GIAC/XCAS [A]  time = 0.558353, size = 176, normalized size = 1.35 \[ -\frac{\frac{10 \, \sqrt{3} b^{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{8}{3}}} + \frac{5 \, b^{3}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{8}{3}}} - \frac{10 \, b^{3}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{8}{3}}} - \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{4}{3}} b^{3} - 8 \,{\left (b x + a\right )}^{\frac{1}{3}} a b^{3}\right )}}{a^{2} b^{2} x^{2}}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(2/3)*x^3),x, algorithm="giac")

[Out]

-1/18*(10*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/
a^(8/3) + 5*b^3*ln((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(8/3)
- 10*b^3*ln(abs((b*x + a)^(1/3) - a^(1/3)))/a^(8/3) - 3*(5*(b*x + a)^(4/3)*b^3 -
8*(b*x + a)^(1/3)*a*b^3)/(a^2*b^2*x^2))/b