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

Optimal. Leaf size=200 \[ \frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{4/3}}-\frac{x \left (a e-b c x-b d x^2\right )}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.36624, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{4/3}}-\frac{x \left (a e-b c x-b d x^2\right )}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.9996, size = 178, normalized size = 0.89 \[ - \frac{x \left (a e - b c x - b d x^{2}\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (a^{\frac{2}{3}} e - b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{4}{3}}} - \frac{\left (a^{\frac{2}{3}} e - b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a^{\frac{2}{3}} e + b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a*e - b*c*x - b*d*x**2)/(3*a*b*(a + b*x**3)) + (a**(2/3)*e - b**(2/3)*c)*log
(a**(1/3) + b**(1/3)*x)/(9*a**(4/3)*b**(4/3)) - (a**(2/3)*e - b**(2/3)*c)*log(a*
*(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(4/3)*b**(4/3)) - sqrt(3)*(
a**(2/3)*e + b**(2/3)*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9
*a**(4/3)*b**(4/3))

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Mathematica [A]  time = 0.45468, size = 186, normalized size = 0.93 \[ \frac{-\left (a^{4/3} \sqrt [3]{b} e-a^{2/3} b c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \left (a^{4/3} \sqrt [3]{b} e-a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \left (a^{2/3} b c+a^{4/3} \sqrt [3]{b} e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{6 a b^{2/3} \left (a (d+e x)-b c x^2\right )}{a+b x^3}}{18 a^2 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 228, normalized size = 1.1 \[{\frac{1}{b{x}^{3}+a} \left ({\frac{c{x}^{2}}{3\,a}}-{\frac{ex}{3\,b}}-{\frac{d}{3\,b}} \right ) }+{\frac{e}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{e\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.5051, size = 124, normalized size = 0.62 \[ \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{4} + 27 t a^{2} b^{2} c e - a^{2} e^{3} + b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{3} b^{3} c + 9 t a^{3} b e^{2} + 2 a b c^{2} e}{a^{2} e^{3} + b^{2} c^{3}} \right )} \right )\right )} + \frac{- a d - a e x + b c x^{2}}{3 a^{2} b + 3 a b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(729*_t**3*a**4*b**4 + 27*_t*a**2*b**2*c*e - a**2*e**3 + b**2*c**3, Lambd
a(_t, _t*log(x + (81*_t**2*a**3*b**3*c + 9*_t*a**3*b*e**2 + 2*a*b*c**2*e)/(a**2*
e**3 + b**2*c**3)))) + (-a*d - a*e*x + b*c*x**2)/(3*a**2*b + 3*a*b**2*x**3)

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GIAC/XCAS [A]  time = 0.215455, size = 273, normalized size = 1.36 \[ -\frac{{\left (b c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{b c x^{2} - a x e - a d}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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