3.26 \(\int \frac{a+c x^2}{d-e x^3} \, dx\)

Optimal. Leaf size=134 \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.181389, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)/(d - e*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 31.2776, size = 128, normalized size = 0.96 \[ - \frac{a \log{\left (\sqrt [3]{d} - \sqrt [3]{e} x \right )}}{3 d^{\frac{2}{3}} \sqrt [3]{e}} + \frac{a \log{\left (d^{\frac{2}{3}} + \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} \sqrt [3]{e}} + \frac{\sqrt{3} a \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} + \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 d^{\frac{2}{3}} \sqrt [3]{e}} - \frac{c \log{\left (d - e x^{3} \right )}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)/(-e*x**3+d),x)

[Out]

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

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Mathematica [A]  time = 0.0612387, size = 123, normalized size = 0.92 \[ \frac{a e^{2/3} \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 a e^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \sqrt{3} a e^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )-2 c d^{2/3} \log \left (d-e x^3\right )}{6 d^{2/3} e} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)/(d - e*x^3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 111, normalized size = 0.8 \[ -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\ln \left ( e{x}^{3}-d \right ) }{3\,e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+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*c/e*ln(e*x^3-d)

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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(-(c*x^2 + a)/(e*x^3 - d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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(-(c*x^2 + a)/(e*x^3 - d),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.643426, size = 70, normalized size = 0.52 \[ - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{3} - 27 t^{2} c d^{2} e^{2} + 9 t c^{2} d^{2} e - a^{3} e^{2} - c^{3} d^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t d e + c d}{a e} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)/(-e*x**3+d),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212121, size = 128, normalized size = 0.96 \[ -\frac{1}{3} \, c e^{\left (-1\right )}{\rm ln}\left ({\left | x^{3} e - d \right |}\right ) + \frac{\sqrt{3} a \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-\frac{1}{3}\right )}}{3 \, d^{\frac{2}{3}}} + \frac{a e^{\left (-\frac{1}{3}\right )}{\rm ln}\left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d^{\frac{2}{3}}} - \frac{a e^{\left (-\frac{1}{3}\right )}{\rm ln}\left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*x^2 + a)/(e*x^3 - d),x, algorithm="giac")

[Out]

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