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

Optimal. Leaf size=177 \[ -\frac{\left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3}}+\frac{e \log \left (a+b x^3\right )}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.290702, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3}}+\frac{e \log \left (a+b x^3\right )}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.4738, size = 163, normalized size = 0.92 \[ \frac{e \log{\left (a + b x^{3} \right )}}{3 b} - \frac{\left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}}} + \frac{\left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.18897, size = 176, normalized size = 0.99 \[ \frac{-\sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+2 a e \log \left (a+b x^3\right )}{6 a b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 200, normalized size = 1.1 \[{\frac{c}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d*x+c)/(b*x^3+a),x)

[Out]

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

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.25022, size = 160, normalized size = 0.9 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{3} - 27 t^{2} a^{2} b^{2} e + t \left (9 a^{2} b e^{2} + 9 a b^{2} c d\right ) - a^{2} e^{3} - 3 a b c d e + a b d^{3} - b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a^{2} b^{2} d - 6 t a^{2} b d e + 3 t a b^{2} c^{2} + a^{2} d e^{2} - a b c^{2} e + 2 a b c d^{2}}{a b d^{3} + b^{2} c^{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216631, size = 246, normalized size = 1.39 \[ \frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac{{\left (b d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + b c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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