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

Optimal. Leaf size=199 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}-\frac{a e-b x (c+d x)}{3 a b \left (a+b x^3\right )} \]

[Out]

-(a*e - b*x*(c + d*x))/(3*a*b*(a + b*x^3)) - ((2*b^(1/3)*c + a^(1/3)*d)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(2/3)) + ((2*b^(
1/3)*c - a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(2/3)) - ((2*b^(1/3)*
c - a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(2/
3))

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Rubi [A]  time = 0.314682, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}-\frac{a e-b x (c+d x)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a*e - b*x*(c + d*x))/(3*a*b*(a + b*x^3)) - ((2*b^(1/3)*c + a^(1/3)*d)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(2/3)) + ((2*b^(
1/3)*c - a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(2/3)) - ((2*b^(1/3)*
c - a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(2/
3))

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Rubi in Sympy [A]  time = 46.5138, size = 178, normalized size = 0.89 \[ - \frac{a e - b x \left (c + d x\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (\frac{\sqrt [3]{a} d}{2} - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\left (\sqrt [3]{a} d - 2 \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (\sqrt [3]{a} d + 2 \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*e - b*x*(c + d*x))/(3*a*b*(a + b*x**3)) + (a**(1/3)*d/2 - b**(1/3)*c)*log(a*
*(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(9*a**(5/3)*b**(2/3)) - (a**(1/3)*
d - 2*b**(1/3)*c)*log(a**(1/3) + b**(1/3)*x)/(9*a**(5/3)*b**(2/3)) - sqrt(3)*(a*
*(1/3)*d + 2*b**(1/3)*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9
*a**(5/3)*b**(2/3))

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Mathematica [A]  time = 0.382753, size = 189, normalized size = 0.95 \[ \frac{\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d-2 \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\left (4 \sqrt [3]{a} b^{2/3} c-2 a^{2/3} \sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{6 a (b x (c+d x)-a e)}{a+b x^3}-2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^2 b} \]

Antiderivative was successfully verified.

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

[Out]

((6*a*(-(a*e) + b*x*(c + d*x)))/(a + b*x^3) - 2*Sqrt[3]*a^(1/3)*b^(1/3)*(2*b^(1/
3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (4*a^(1/3)*b^(2/
3)*c - 2*a^(2/3)*b^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x] + a^(1/3)*b^(1/3)*(-2*b^(1/
3)*c + a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^2*b)

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Maple [A]  time = 0.006, size = 253, normalized size = 1.3 \[{\frac{cx}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2\,c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,c\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{d}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e}{3\,b \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x/a/(b*x^3+a)+2/9*c/a/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/9*c/a/b/(a/b)^(2/3
)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/9*c/a/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))+1/3*d*x^2/a/(b*x^3+a)-1/9*d/a/b/(a/b)^(1/3)*ln(x+(a/b)
^(1/3))+1/18*d/a/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/9*d/a*3^(1/2)
/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3/b/(b*x^3+a)*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((e*x^2 + d*x + c)/(b*x^3 + a)^2,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)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.61076, size = 116, normalized size = 0.58 \[ \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{2} + 54 t a^{2} b c d + a d^{3} - 8 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{4} b d + 36 t a^{2} b c^{2} + 4 a c d^{2}}{a d^{3} + 8 b c^{3}} \right )} \right )\right )} + \frac{- a e + b c x + b d x^{2}}{3 a^{2} b + 3 a b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(729*_t**3*a**5*b**2 + 54*_t*a**2*b*c*d + a*d**3 - 8*b*c**3, Lambda(_t, _
t*log(x + (81*_t**2*a**4*b*d + 36*_t*a**2*b*c**2 + 4*a*c*d**2)/(a*d**3 + 8*b*c**
3)))) + (-a*e + b*c*x + b*d*x**2)/(3*a**2*b + 3*a*b**2*x**3)

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GIAC/XCAS [A]  time = 0.216552, size = 266, normalized size = 1.34 \[ -\frac{{\left (d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{b d x^{2} + b c x - a e}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(d*(-a/b)^(1/3) + 2*c)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^2 + 1/9*sqr
t(3)*(2*(-a*b^2)^(1/3)*b*c - (-a*b^2)^(2/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^
(1/3))/(-a/b)^(1/3))/(a^2*b^2) + 1/3*(b*d*x^2 + b*c*x - a*e)/((b*x^3 + a)*a*b) +
1/18*(2*(-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^3*b^4)