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

Optimal. Leaf size=215 \[ -\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{5/3}}-\frac{\left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{5/3}}-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )} \]

[Out]

-(c + d*x + e*x^2)/(6*b*(a + b*x^3)^2) + (x*(d + 2*e*x))/(18*a*b*(a + b*x^3)) -
((b^(1/3)*d + a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*S
qrt[3]*a^(5/3)*b^(5/3)) + ((b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27
*a^(5/3)*b^(5/3)) - ((d - (a^(1/3)*e)/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(54*a^(5/3)*b^(4/3))

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Rubi [A]  time = 0.441702, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{5/3}}-\frac{\left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{5/3}}-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c + d*x + e*x^2)/(6*b*(a + b*x^3)^2) + (x*(d + 2*e*x))/(18*a*b*(a + b*x^3)) -
((b^(1/3)*d + a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*S
qrt[3]*a^(5/3)*b^(5/3)) + ((b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27
*a^(5/3)*b^(5/3)) - ((d - (a^(1/3)*e)/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(54*a^(5/3)*b^(4/3))

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Rubi in Sympy [A]  time = 61.8711, size = 192, normalized size = 0.89 \[ - \frac{c + d x + e x^{2}}{6 b \left (a + b x^{3}\right )^{2}} + \frac{x \left (d + 2 e x\right )}{18 a b \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} e - \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{5}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} e - \sqrt [3]{b} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{5}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (\sqrt [3]{a} e + \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{5}{3}} b^{\frac{5}{3}}} \]

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)**3,x)

[Out]

-(c + d*x + e*x**2)/(6*b*(a + b*x**3)**2) + x*(d + 2*e*x)/(18*a*b*(a + b*x**3))
- (a**(1/3)*e - b**(1/3)*d)*log(a**(1/3) + b**(1/3)*x)/(27*a**(5/3)*b**(5/3)) +
(a**(1/3)*e - b**(1/3)*d)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(5
4*a**(5/3)*b**(5/3)) - sqrt(3)*(a**(1/3)*e + b**(1/3)*d)*atan(sqrt(3)*(a**(1/3)/
3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(5/3)*b**(5/3))

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Mathematica [A]  time = 0.4436, size = 198, normalized size = 0.92 \[ \frac{\frac{\left (\sqrt [3]{a} e-\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{2 \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac{2 \sqrt{3} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}-\frac{9 b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^2}+\frac{3 b^{2/3} x (d+2 e x)}{a \left (a+b x^3\right )}}{54 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((3*b^(2/3)*x*(d + 2*e*x))/(a*(a + b*x^3)) - (9*b^(2/3)*(c + x*(d + e*x)))/(a +
b*x^3)^2 - (2*Sqrt[3]*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))
/Sqrt[3]])/a^(5/3) + (2*(b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3
) + ((-(b^(1/3)*d) + a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
a^(5/3))/(54*b^(5/3))

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Maple [A]  time = 0.013, size = 255, normalized size = 1.2 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{e{x}^{5}}{9\,a}}+{\frac{d{x}^{4}}{18\,a}}-{\frac{e{x}^{2}}{18\,b}}-{\frac{dx}{9\,b}}-{\frac{c}{6\,b}} \right ) }+{\frac{d}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d\sqrt{3}}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}e}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/9/a*e*x^5+1/18*d/a*x^4-1/18*e*x^2/b-1/9*d*x/b-1/6*c/b)/(b*x^3+a)^2+1/27/b^2/a
/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-1/54/b^2/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/
b)^(2/3))*d+1/27/b^2/a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))*d-1/27/a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e+1/54/a/b^2/(a/b)^(1/3)*ln(x^2-x*
(a/b)^(1/3)+(a/b)^(2/3))*e+1/27/a/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/
(a/b)^(1/3)*x-1))*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)*x^2/(b*x^3 + a)^3,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)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 10.6732, size = 148, normalized size = 0.69 \[ \operatorname{RootSum}{\left (19683 t^{3} a^{5} b^{5} + 81 t a^{2} b^{2} d e + a e^{3} - b d^{3}, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{4} b^{3} e + 27 t a^{2} b^{2} d^{2} + 2 a d e^{2}}{a e^{3} + b d^{3}} \right )} \right )\right )} + \frac{- 3 a c - 2 a d x - a e x^{2} + b d x^{4} + 2 b e x^{5}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(19683*_t**3*a**5*b**5 + 81*_t*a**2*b**2*d*e + a*e**3 - b*d**3, Lambda(_t
, _t*log(x + (729*_t**2*a**4*b**3*e + 27*_t*a**2*b**2*d**2 + 2*a*d*e**2)/(a*e**3
+ b*d**3)))) + (-3*a*c - 2*a*d*x - a*e*x**2 + b*d*x**4 + 2*b*e*x**5)/(18*a**3*b
+ 36*a**2*b**2*x**3 + 18*a*b**3*x**6)

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GIAC/XCAS [A]  time = 0.217986, size = 288, normalized size = 1.34 \[ -\frac{{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} e + d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac{2 \, b x^{5} e + b d x^{4} - a x^{2} e - 2 \, a d x - 3 \, a c}{18 \,{\left (b x^{3} + a\right )}^{2} a b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{2}{3}} a b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*((-a/b)^(1/3)*e + d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b) + 1/27
*sqrt(3)*((-a*b^2)^(1/3)*b*d - (-a*b^2)^(2/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b
)^(1/3))/(-a/b)^(1/3))/(a^2*b^3) + 1/18*(2*b*x^5*e + b*d*x^4 - a*x^2*e - 2*a*d*x
- 3*a*c)/((b*x^3 + a)^2*a*b) + 1/54*((-a*b^2)^(1/3)*a*b^2*d + (-a*b^2)^(2/3)*a*
b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^4)