3.25 \(\int \frac{b x+c x^2}{d+e x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.219905, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{b \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 \sqrt [3]{d} e^{2/3}}-\frac{b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac{b \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} \sqrt [3]{d} e^{2/3}}+\frac{c \log \left (d+e x^3\right )}{3 e} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)/(d + e*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.8705, size = 128, normalized size = 0.96 \[ - \frac{b \log{\left (\sqrt [3]{d} + \sqrt [3]{e} x \right )}}{3 \sqrt [3]{d} e^{\frac{2}{3}}} + \frac{b \log{\left (d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{d} e^{\frac{2}{3}}} - \frac{\sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} - \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 \sqrt [3]{d} e^{\frac{2}{3}}} + \frac{c \log{\left (d + e x^{3} \right )}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0678758, size = 122, normalized size = 0.91 \[ \frac{b \sqrt [3]{e} \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 b \sqrt [3]{e} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-2 \sqrt{3} b \sqrt [3]{e} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right )+2 c \sqrt [3]{d} \log \left (d+e x^3\right )}{6 \sqrt [3]{d} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 108, normalized size = 0.8 \[ -{\frac{b}{3\,e}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b}{6\,e}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{c\ln \left ( e{x}^{3}+d \right ) }{3\,e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)/(e*x^3+d),x)

[Out]

-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))
+1/3*c*ln(e*x^3+d)/e

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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 + b*x)/(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 + b*x)/(e*x^3 + d),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.428813, size = 75, normalized size = 0.56 \[ \operatorname{RootSum}{\left (27 t^{3} d e^{3} - 27 t^{2} c d e^{2} + 9 t c^{2} d e + b^{3} e - c^{3} d, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} d e^{2} - 6 t c d e + c^{2} d}{b^{2} e} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*d*e**3 - 27*_t**2*c*d*e**2 + 9*_t*c**2*d*e + b**3*e - c**3*d, L
ambda(_t, _t*log(x + (9*_t**2*d*e**2 - 6*_t*c*d*e + c**2*d)/(b**2*e))))

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GIAC/XCAS [A]  time = 0.213358, size = 162, normalized size = 1.21 \[ -\frac{\sqrt{3} \left (-d e^{2}\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, d} + \frac{1}{3} \, c e^{\left (-1\right )}{\rm ln}\left ({\left | x^{3} e + d \right |}\right ) + \frac{\left (-d e^{2}\right )^{\frac{2}{3}} b e^{\left (-2\right )}{\rm ln}\left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d} - \frac{\left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}} b{\rm ln}\left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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