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

Optimal. Leaf size=239 \[ \frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}}-\frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}-\frac{\left (a^{2/3} e+2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{4/3}}-\frac{3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac{x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2} \]

[Out]

-(x*(a*e - b*c*x - b*d*x^2))/(6*a*b*(a + b*x^3)^2) - (3*a*d - x*(a*e + 4*b*c*x))
/(18*a^2*b*(a + b*x^3)) - ((2*b^(2/3)*c + a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)
*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(4/3)) - ((2*b^(2/3)*c - a^(2/3)*e)
*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(4/3)) + ((2*b^(2/3)*c - a^(2/3)*e)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(4/3))

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Rubi [A]  time = 0.460489, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}}-\frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}-\frac{\left (a^{2/3} e+2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{4/3}}-\frac{3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac{x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(a*e - b*c*x - b*d*x^2))/(6*a*b*(a + b*x^3)^2) - (3*a*d - x*(a*e + 4*b*c*x))
/(18*a^2*b*(a + b*x^3)) - ((2*b^(2/3)*c + a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)
*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(4/3)) - ((2*b^(2/3)*c - a^(2/3)*e)
*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(4/3)) + ((2*b^(2/3)*c - a^(2/3)*e)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(4/3))

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Rubi in Sympy [A]  time = 73.7936, size = 216, normalized size = 0.9 \[ - \frac{x \left (a e - b c x - b d x^{2}\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{3 a d - x \left (a e + 4 b c x\right )}{18 a^{2} b \left (a + b x^{3}\right )} + \frac{\left (a^{\frac{2}{3}} e - 2 b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{4}{3}}} - \frac{\left (a^{\frac{2}{3}} e - 2 b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{7}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a^{\frac{2}{3}} e + 2 b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a*e - b*c*x - b*d*x**2)/(6*a*b*(a + b*x**3)**2) - (3*a*d - x*(a*e + 4*b*c*x)
)/(18*a**2*b*(a + b*x**3)) + (a**(2/3)*e - 2*b**(2/3)*c)*log(a**(1/3) + b**(1/3)
*x)/(27*a**(7/3)*b**(4/3)) - (a**(2/3)*e - 2*b**(2/3)*c)*log(a**(2/3) - a**(1/3)
*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(7/3)*b**(4/3)) - sqrt(3)*(a**(2/3)*e + 2*b*
*(2/3)*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(7/3)*b**(
4/3))

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Mathematica [A]  time = 0.556766, size = 214, normalized size = 0.9 \[ \frac{\left (2 a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt{3} a^{2/3} \sqrt [3]{b} \left (a^{2/3} e+2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+2 \left (a^{4/3} \sqrt [3]{b} e-2 a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a b^{2/3} \left (-a^2 (3 d+2 e x)+a b x^2 \left (7 c+e x^2\right )+4 b^2 c x^5\right )}{\left (a+b x^3\right )^2}}{54 a^3 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((3*a*b^(2/3)*(4*b^2*c*x^5 - a^2*(3*d + 2*e*x) + a*b*x^2*(7*c + e*x^2)))/(a + b*
x^3)^2 - 2*Sqrt[3]*a^(2/3)*b^(1/3)*(2*b^(2/3)*c + a^(2/3)*e)*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-2*a^(2/3)*b*c + a^(4/3)*b^(1/3)*e)*Log[a^(1/3) +
b^(1/3)*x] + (2*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])/(54*a^3*b^(5/3))

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Maple [A]  time = 0.014, size = 256, normalized size = 1.1 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{2\,bc{x}^{5}}{9\,{a}^{2}}}+{\frac{e{x}^{4}}{18\,a}}+{\frac{7\,c{x}^{2}}{18\,a}}-{\frac{ex}{9\,b}}-{\frac{d}{6\,b}} \right ) }+{\frac{e}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}e}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,c}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,c\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2/9/a^2*c*b*x^5+1/18/a*e*x^4+7/18/a*c*x^2-1/9*e*x/b-1/6/b*d)/(b*x^3+a)^2+1/27/a
/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-1/54/a/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)
+(a/b)^(2/3))*e+1/27/a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)
*x-1))*e-2/27/b/a^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+1/27/b/a^2/(a/b)^(1/3)*ln(x^
2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+2/27/b/a^2*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)^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/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 7.1414, size = 170, normalized size = 0.71 \[ \operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{4} + 162 t a^{3} b^{2} c e - a^{2} e^{3} + 8 b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{1458 t^{2} a^{5} b^{3} c + 27 t a^{4} b e^{2} + 8 a b c^{2} e}{a^{2} e^{3} + 8 b^{2} c^{3}} \right )} \right )\right )} + \frac{- 3 a^{2} d - 2 a^{2} e x + 7 a b c x^{2} + a b e x^{4} + 4 b^{2} c x^{5}}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(19683*_t**3*a**7*b**4 + 162*_t*a**3*b**2*c*e - a**2*e**3 + 8*b**2*c**3,
Lambda(_t, _t*log(x + (1458*_t**2*a**5*b**3*c + 27*_t*a**4*b*e**2 + 8*a*b*c**2*e
)/(a**2*e**3 + 8*b**2*c**3)))) + (-3*a**2*d - 2*a**2*e*x + 7*a*b*c*x**2 + a*b*e*
x**4 + 4*b**2*c*x**5)/(18*a**4*b + 36*a**3*b**2*x**3 + 18*a**2*b**3*x**6)

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GIAC/XCAS [A]  time = 0.217957, size = 306, normalized size = 1.28 \[ -\frac{{\left (2 \, b c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{4 \, b^{2} c x^{5} + a b x^{4} e + 7 \, a b c x^{2} - 2 \, a^{2} x e - 3 \, a^{2} d}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(2*b*c*(-a/b)^(1/3) + a*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^3*b)
+ 1/27*sqrt(3)*((-a*b^2)^(1/3)*a*e - 2*(-a*b^2)^(2/3)*c)*arctan(1/3*sqrt(3)*(2*x
+ (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^2) + 1/18*(4*b^2*c*x^5 + a*b*x^4*e + 7*a*b
*c*x^2 - 2*a^2*x*e - 3*a^2*d)/((b*x^3 + a)^2*a^2*b) + 1/54*((-a*b^2)^(1/3)*a*b^2
*e + 2*(-a*b^2)^(2/3)*b^2*c)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^4)