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

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

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

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Rubi [A]  time = 0.254427, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a-\frac{b \sqrt [3]{d}}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]

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

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Rubi in Sympy [A]  time = 31.5348, size = 150, normalized size = 0.93 \[ \frac{\left (a \sqrt [3]{e} - b \sqrt [3]{d}\right ) \log{\left (\sqrt [3]{d} + \sqrt [3]{e} x \right )}}{3 d^{\frac{2}{3}} e^{\frac{2}{3}}} - \frac{\left (a \sqrt [3]{e} - b \sqrt [3]{d}\right ) \log{\left (d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} e^{\frac{2}{3}}} - \frac{\sqrt{3} \left (a \sqrt [3]{e} + b \sqrt [3]{d}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} - \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 d^{\frac{2}{3}} e^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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)) - 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))

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Mathematica [A]  time = 0.104646, size = 125, normalized size = 0.78 \[ \frac{-\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \left (2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )\right )-2 \sqrt{3} \left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right )}{6 d^{2/3} e^{2/3}} \]

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))

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Maple [A]  time = 0.007, size = 186, 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))

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 1.04167, size = 76, normalized size = 0.47 \[ \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{2} + 9 t a b d e - a^{3} e + b^{3} d, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b d^{2} e + 3 t a^{2} d e + 2 a b^{2} d}{a^{3} e + b^{3} d} \right )} \right )\right )} \]

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))))

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GIAC/XCAS [A]  time = 0.21255, size = 197, normalized size = 1.22 \[ \frac{\sqrt{3}{\left (\left (-d e^{2}\right )^{\frac{1}{3}} a e - \left (-d e^{2}\right )^{\frac{2}{3}} b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, d} - \frac{\left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}{\left (\left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} b + a\right )}{\rm ln}\left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} + \frac{{\left (\left (-d e^{2}\right )^{\frac{2}{3}} b d e^{2} + \left (-d e^{2}\right )^{\frac{1}{3}} a d e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d^{2}} \]

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