3.176 \(\int \frac{a+\frac{b}{x^3}}{c+\frac{d}{x^3}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{(b c-a d) \log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{c} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} c^{4/3} d^{2/3}}+\frac{a x}{c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.257555, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -\frac{(b c-a d) \log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{c} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} c^{4/3} d^{2/3}}+\frac{a x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^3)/(c + d/x^3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.5772, size = 134, normalized size = 0.92 \[ \frac{a x}{c} - \frac{\left (a d - b c\right ) \log{\left (\sqrt [3]{c} x + \sqrt [3]{d} \right )}}{3 c^{\frac{4}{3}} d^{\frac{2}{3}}} + \frac{\left (a d - b c\right ) \log{\left (c^{\frac{2}{3}} x^{2} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} \right )}}{6 c^{\frac{4}{3}} d^{\frac{2}{3}}} + \frac{\sqrt{3} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{c} x}{3} + \frac{\sqrt [3]{d}}{3}\right )}{\sqrt [3]{d}} \right )}}{3 c^{\frac{4}{3}} d^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x/c - (a*d - b*c)*log(c**(1/3)*x + d**(1/3))/(3*c**(4/3)*d**(2/3)) + (a*d - b*
c)*log(c**(2/3)*x**2 - c**(1/3)*d**(1/3)*x + d**(2/3))/(6*c**(4/3)*d**(2/3)) + s
qrt(3)*(a*d - b*c)*atan(sqrt(3)*(-2*c**(1/3)*x/3 + d**(1/3)/3)/d**(1/3))/(3*c**(
4/3)*d**(2/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.165813, size = 129, normalized size = 0.89 \[ \frac{-(b c-a d) \log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )+2 (b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )-2 \sqrt{3} (b c-a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{c} x}{\sqrt [3]{d}}}{\sqrt{3}}\right )+6 a \sqrt [3]{c} d^{2/3} x}{6 c^{4/3} d^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^3)/(c + d/x^3),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 195, normalized size = 1.3 \[{\frac{ax}{c}}-{\frac{ad}{3\,{c}^{2}}\ln \left ( x+\sqrt [3]{{\frac{d}{c}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{3\,c}\ln \left ( x+\sqrt [3]{{\frac{d}{c}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ad}{6\,{c}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{c}}}+ \left ({\frac{d}{c}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{6\,c}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{c}}}+ \left ({\frac{d}{c}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}ad}{3\,{c}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{c}}}}}}-1 \right ) } \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{3\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{c}}}}}}-1 \right ) } \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)/(c+d/x^3),x)

[Out]

a*x/c-1/3/c^2/(d/c)^(2/3)*ln(x+(d/c)^(1/3))*a*d+1/3/c/(d/c)^(2/3)*ln(x+(d/c)^(1/
3))*b+1/6/c^2/(d/c)^(2/3)*ln(x^2-x*(d/c)^(1/3)+(d/c)^(2/3))*a*d-1/6/c/(d/c)^(2/3
)*ln(x^2-x*(d/c)^(1/3)+(d/c)^(2/3))*b-1/3/c^2/(d/c)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(d/c)^(1/3)*x-1))*a*d+1/3/c/(d/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d
/c)^(1/3)*x-1))*b

_______________________________________________________________________________________

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((a + b/x^3)/(c + d/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.239295, size = 184, normalized size = 1.27 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (-c d^{2}\right )^{\frac{1}{3}} a x + \sqrt{3}{\left (b c - a d\right )} \log \left (\left (-c d^{2}\right )^{\frac{2}{3}} x^{2} + \left (-c d^{2}\right )^{\frac{1}{3}} d x + d^{2}\right ) - 2 \, \sqrt{3}{\left (b c - a d\right )} \log \left (\left (-c d^{2}\right )^{\frac{1}{3}} x - d\right ) + 6 \,{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-c d^{2}\right )^{\frac{1}{3}} x + \sqrt{3} d}{3 \, d}\right )\right )}}{18 \, \left (-c d^{2}\right )^{\frac{1}{3}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(6*sqrt(3)*(-c*d^2)^(1/3)*a*x + sqrt(3)*(b*c - a*d)*log((-c*d^2)^(2
/3)*x^2 + (-c*d^2)^(1/3)*d*x + d^2) - 2*sqrt(3)*(b*c - a*d)*log((-c*d^2)^(1/3)*x
- d) + 6*(b*c - a*d)*arctan(1/3*(2*sqrt(3)*(-c*d^2)^(1/3)*x + sqrt(3)*d)/d))/((
-c*d^2)^(1/3)*c)

_______________________________________________________________________________________

Sympy [A]  time = 2.22337, size = 71, normalized size = 0.49 \[ \frac{a x}{c} + \operatorname{RootSum}{\left (27 t^{3} c^{4} d^{2} + a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t c d}{a d - b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x/c + RootSum(27*_t**3*c**4*d**2 + a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2
*d - b**3*c**3, Lambda(_t, _t*log(-3*_t*c*d/(a*d - b*c) + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219929, size = 217, normalized size = 1.5 \[ \frac{a x}{c} - \frac{{\left (b c - a d\right )} \left (-\frac{d}{c}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{d}{c}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d} + \frac{\sqrt{3}{\left (\left (-c^{2} d\right )^{\frac{1}{3}} b c - \left (-c^{2} d\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{d}{c}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{d}{c}\right )^{\frac{1}{3}}}\right )}{3 \, c^{2} d} + \frac{{\left (\left (-c^{2} d\right )^{\frac{1}{3}} b c - \left (-c^{2} d\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{d}{c}\right )^{\frac{1}{3}} + \left (-\frac{d}{c}\right )^{\frac{2}{3}}\right )}{6 \, c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x/c - 1/3*(b*c - a*d)*(-d/c)^(1/3)*ln(abs(x - (-d/c)^(1/3)))/(c*d) + 1/3*sqrt(
3)*((-c^2*d)^(1/3)*b*c - (-c^2*d)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-d/c)^(1
/3))/(-d/c)^(1/3))/(c^2*d) + 1/6*((-c^2*d)^(1/3)*b*c - (-c^2*d)^(1/3)*a*d)*ln(x^
2 + x*(-d/c)^(1/3) + (-d/c)^(2/3))/(c^2*d)