3.12 \(\int \frac{a+b x}{d-e x^3} \, dx\)

Optimal. Leaf size=161 \[ \frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.215671, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(d - e*x^3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.7272, size = 150, normalized size = 0.93 \[ \frac{\sqrt{3} \left (a \sqrt [3]{e} - b \sqrt [3]{d}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} + \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 d^{\frac{2}{3}} e^{\frac{2}{3}}} - \frac{\left (a \sqrt [3]{e} + b \sqrt [3]{d}\right ) \log{\left (\sqrt [3]{d} - \sqrt [3]{e} x \right )}}{3 d^{\frac{2}{3}} e^{\frac{2}{3}}} + \frac{\left (a \sqrt [3]{e} + b \sqrt [3]{d}\right ) \log{\left (d^{\frac{2}{3}} + \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} e^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*(a*e**(1/3) - b*d**(1/3))*atan(sqrt(3)*(d**(1/3)/3 + 2*e**(1/3)*x/3)/d**
(1/3))/(3*d**(2/3)*e**(2/3)) - (a*e**(1/3) + b*d**(1/3))*log(d**(1/3) - e**(1/3)
*x)/(3*d**(2/3)*e**(2/3)) + (a*e**(1/3) + b*d**(1/3))*log(d**(2/3) + d**(1/3)*e*
*(1/3)*x + e**(2/3)*x**2)/(6*d**(2/3)*e**(2/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0878635, size = 125, normalized size = 0.78 \[ \frac{-\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \left (2 \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )-\log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )\right )-2 \sqrt{3} \left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )}{6 d^{2/3} e^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(d - e*x^3),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 188, normalized size = 1.2 \[ -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b}{6\,e}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-{\frac{b\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-e*x^3+d),x)

[Out]

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

_______________________________________________________________________________________

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)/(e*x^3 - d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 1.06266, size = 78, normalized size = 0.48 \[ - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{2} - 9 t a b d e - a^{3} e - b^{3} d, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b d^{2} e - 3 t a^{2} d e - 2 a b^{2} d}{a^{3} e - b^{3} d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(27*_t**3*d**2*e**2 - 9*_t*a*b*d*e - a**3*e - b**3*d, Lambda(_t, _t*log(
x + (9*_t**2*b*d**2*e - 3*_t*a**2*d*e - 2*a*b**2*d)/(a**3*e - b**3*d))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213404, size = 155, normalized size = 0.96 \[ -\frac{\sqrt{3}{\left (b d^{\frac{2}{3}} e^{\frac{4}{3}} - a d^{\frac{1}{3}} e^{\frac{5}{3}}\right )} \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, d} - \frac{{\left (b d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + a\right )} e^{\left (-\frac{1}{3}\right )}{\rm ln}\left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} + \frac{{\left (b d^{\frac{2}{3}} e^{\frac{4}{3}} + a d^{\frac{1}{3}} e^{\frac{5}{3}}\right )} e^{\left (-2\right )}{\rm ln}\left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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