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

Optimal. Leaf size=190 \[ -\frac{\left (d-\frac{2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{5/3}}-\frac{\left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{5/3}}-\frac{c+d x+e x^2}{3 b \left (a+b x^3\right )} \]

[Out]

-(c + d*x + e*x^2)/(3*b*(a + b*x^3)) - ((b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(a^(1/3
) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(5/3)) + ((b^(1/3)*d -
2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^(5/3)) - ((d - (2*a^(1/3)*e
)/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(4/3))

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Rubi [A]  time = 0.364204, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{\left (d-\frac{2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{5/3}}-\frac{\left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{5/3}}-\frac{c+d x+e x^2}{3 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c + d*x + e*x^2)/(3*b*(a + b*x^3)) - ((b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(a^(1/3
) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(5/3)) + ((b^(1/3)*d -
2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^(5/3)) - ((d - (2*a^(1/3)*e
)/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(4/3))

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Rubi in Sympy [A]  time = 49.2203, size = 175, normalized size = 0.92 \[ - \frac{c + d x + e x^{2}}{3 b \left (a + b x^{3}\right )} + \frac{\left (\sqrt [3]{a} e - \frac{\sqrt [3]{b} d}{2}\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{2}{3}} b^{\frac{5}{3}}} - \frac{\left (2 \sqrt [3]{a} e - \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (2 \sqrt [3]{a} e + \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.275483, size = 174, normalized size = 0.92 \[ \frac{\frac{\left (2 \sqrt [3]{a} e-\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{2 \left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac{2 \sqrt{3} \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{6 b^{2/3} (c+x (d+e x))}{a+b x^3}}{18 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-6*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3) - (2*Sqrt[3]*(b^(1/3)*d + 2*a^(1/3)*
e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(b^(1/3)*d - 2*a^(1
/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + ((-(b^(1/3)*d) + 2*a^(1/3)*e)*Log[a^(
2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(18*b^(5/3))

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Maple [A]  time = 0.012, size = 219, normalized size = 1.2 \[{\frac{1}{b{x}^{3}+a} \left ( -{\frac{e{x}^{2}}{3\,b}}-{\frac{dx}{3\,b}}-{\frac{c}{3\,b}} \right ) }+{\frac{d}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,e}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{9\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,e\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 4.31458, size = 109, normalized size = 0.57 \[ \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{5} + 54 t a b^{2} d e + 8 a e^{3} - b d^{3}, \left ( t \mapsto t \log{\left (x + \frac{162 t^{2} a^{2} b^{3} e + 9 t a b^{2} d^{2} + 8 a d e^{2}}{8 a e^{3} + b d^{3}} \right )} \right )\right )} - \frac{c + d x + e x^{2}}{3 a b + 3 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(729*_t**3*a**2*b**5 + 54*_t*a*b**2*d*e + 8*a*e**3 - b*d**3, Lambda(_t, _
t*log(x + (162*_t**2*a**2*b**3*e + 9*_t*a*b**2*d**2 + 8*a*d*e**2)/(8*a*e**3 + b*
d**3)))) - (c + d*x + e*x**2)/(3*a*b + 3*b**2*x**3)

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GIAC/XCAS [A]  time = 0.216969, size = 258, normalized size = 1.36 \[ -\frac{{\left (2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b} - \frac{x^{2} e + d x + c}{3 \,{\left (b x^{3} + a\right )} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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