3.347 \(\int \frac{x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx\)

Optimal. Leaf size=270 \[ \frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}-\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3} \]

[Out]

-(x*(a*e - b*c*x - b*d*x^2))/(9*a*b*(a + b*x^3)^3) + (x*(5*a*e + 28*b*c*x))/(162
*a^3*b*(a + b*x^3)) - (6*a*d - x*(a*e + 7*b*c*x))/(54*a^2*b*(a + b*x^3)^2) - ((1
4*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(8
1*Sqrt[3]*a^(10/3)*b^(4/3)) - ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3
)*x])/(243*a^(10/3)*b^(4/3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^(10/3)*b^(4/3))

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Rubi [A]  time = 0.548044, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}-\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(x*(c + d*x + e*x^2))/(a + b*x^3)^4,x]

[Out]

-(x*(a*e - b*c*x - b*d*x^2))/(9*a*b*(a + b*x^3)^3) + (x*(5*a*e + 28*b*c*x))/(162
*a^3*b*(a + b*x^3)) - (6*a*d - x*(a*e + 7*b*c*x))/(54*a^2*b*(a + b*x^3)^2) - ((1
4*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(8
1*Sqrt[3]*a^(10/3)*b^(4/3)) - ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3
)*x])/(243*a^(10/3)*b^(4/3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^(10/3)*b^(4/3))

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Rubi in Sympy [A]  time = 91.2819, size = 250, normalized size = 0.93 \[ - \frac{x \left (a e - b c x - b d x^{2}\right )}{9 a b \left (a + b x^{3}\right )^{3}} - \frac{6 a d - x \left (a e + 7 b c x\right )}{54 a^{2} b \left (a + b x^{3}\right )^{2}} + \frac{x \left (5 a e + 28 b c x\right )}{162 a^{3} b \left (a + b x^{3}\right )} + \frac{\left (5 a^{\frac{2}{3}} e - 14 b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{243 a^{\frac{10}{3}} b^{\frac{4}{3}}} - \frac{\left (5 a^{\frac{2}{3}} e - 14 b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{486 a^{\frac{10}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (5 a^{\frac{2}{3}} e + 14 b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{243 a^{\frac{10}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**4,x)

[Out]

-x*(a*e - b*c*x - b*d*x**2)/(9*a*b*(a + b*x**3)**3) - (6*a*d - x*(a*e + 7*b*c*x)
)/(54*a**2*b*(a + b*x**3)**2) + x*(5*a*e + 28*b*c*x)/(162*a**3*b*(a + b*x**3)) +
(5*a**(2/3)*e - 14*b**(2/3)*c)*log(a**(1/3) + b**(1/3)*x)/(243*a**(10/3)*b**(4/
3)) - (5*a**(2/3)*e - 14*b**(2/3)*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/
3)*x**2)/(486*a**(10/3)*b**(4/3)) - sqrt(3)*(5*a**(2/3)*e + 14*b**(2/3)*c)*atan(
sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(243*a**(10/3)*b**(4/3))

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Mathematica [A]  time = 0.713206, size = 241, normalized size = 0.89 \[ \frac{a^{2/3} \sqrt [3]{b} \left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt{3} a^{2/3} \sqrt [3]{b} \left (5 a^{2/3} e+14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+2 \left (5 a^{4/3} \sqrt [3]{b} e-14 a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a b^{2/3} \left (-2 a^3 (9 d+5 e x)+a^2 b x^2 \left (67 c+13 e x^2\right )+a b^2 x^5 \left (77 c+5 e x^2\right )+28 b^3 c x^8\right )}{\left (a+b x^3\right )^3}}{486 a^4 b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^4,x]

[Out]

((3*a*b^(2/3)*(28*b^3*c*x^8 - 2*a^3*(9*d + 5*e*x) + a*b^2*x^5*(77*c + 5*e*x^2) +
a^2*b*x^2*(67*c + 13*e*x^2)))/(a + b*x^3)^3 - 2*Sqrt[3]*a^(2/3)*b^(1/3)*(14*b^(
2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-14*a^(2/
3)*b*c + 5*a^(4/3)*b^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x] + a^(2/3)*b^(1/3)*(14*b^(
2/3)*c - 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^4*b
^(5/3))

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Maple [A]  time = 0.016, size = 278, normalized size = 1. \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{3}} \left ({\frac{14\,{b}^{2}c{x}^{8}}{81\,{a}^{3}}}+{\frac{5\,be{x}^{7}}{162\,{a}^{2}}}+{\frac{77\,bc{x}^{5}}{162\,{a}^{2}}}+{\frac{13\,e{x}^{4}}{162\,a}}+{\frac{67\,c{x}^{2}}{162\,a}}-{\frac{5\,ex}{81\,b}}-{\frac{d}{9\,b}} \right ) }+{\frac{5\,e}{243\,{a}^{2}{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,e}{486\,{a}^{2}{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,e\sqrt{3}}{243\,{a}^{2}{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,c}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,c}{243\,{a}^{3}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,c\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x^2+d*x+c)/(b*x^3+a)^4,x)

[Out]

(14/81*c/a^3*b^2*x^8+5/162/a^2*b*e*x^7+77/162/a^2*c*b*x^5+13/162/a*e*x^4+67/162/
a*c*x^2-5/81*e*x/b-1/9/b*d)/(b*x^3+a)^3+5/243/a^2*e/b^2/(a/b)^(2/3)*ln(x+(a/b)^(
1/3))-5/486/a^2*e/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/243/a^2*e/
b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/243/b*c/a^3/(
a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/243/b*c/a^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b
)^(2/3))+14/243/b*c*3^(1/2)/a^3/(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/(b*x^3 + a)^4,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)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 15.0171, size = 214, normalized size = 0.79 \[ \operatorname{RootSum}{\left (14348907 t^{3} a^{10} b^{4} + 51030 t a^{4} b^{2} c e - 125 a^{2} e^{3} + 2744 b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{826686 t^{2} a^{7} b^{3} c + 6075 t a^{5} b e^{2} + 1960 a b c^{2} e}{125 a^{2} e^{3} + 2744 b^{2} c^{3}} \right )} \right )\right )} + \frac{- 18 a^{3} d - 10 a^{3} e x + 67 a^{2} b c x^{2} + 13 a^{2} b e x^{4} + 77 a b^{2} c x^{5} + 5 a b^{2} e x^{7} + 28 b^{3} c x^{8}}{162 a^{6} b + 486 a^{5} b^{2} x^{3} + 486 a^{4} b^{3} x^{6} + 162 a^{3} b^{4} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**4,x)

[Out]

RootSum(14348907*_t**3*a**10*b**4 + 51030*_t*a**4*b**2*c*e - 125*a**2*e**3 + 274
4*b**2*c**3, Lambda(_t, _t*log(x + (826686*_t**2*a**7*b**3*c + 6075*_t*a**5*b*e*
*2 + 1960*a*b*c**2*e)/(125*a**2*e**3 + 2744*b**2*c**3)))) + (-18*a**3*d - 10*a**
3*e*x + 67*a**2*b*c*x**2 + 13*a**2*b*e*x**4 + 77*a*b**2*c*x**5 + 5*a*b**2*e*x**7
+ 28*b**3*c*x**8)/(162*a**6*b + 486*a**5*b**2*x**3 + 486*a**4*b**3*x**6 + 162*a
**3*b**4*x**9)

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GIAC/XCAS [A]  time = 0.217852, size = 346, normalized size = 1.28 \[ -\frac{{\left (14 \, b c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{4} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{4} b^{2}} + \frac{28 \, b^{3} c x^{8} + 5 \, a b^{2} x^{7} e + 77 \, a b^{2} c x^{5} + 13 \, a^{2} b x^{4} e + 67 \, a^{2} b c x^{2} - 10 \, a^{3} x e - 18 \, a^{3} d}{162 \,{\left (b x^{3} + a\right )}^{3} a^{3} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{486 \, a^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)*x/(b*x^3 + a)^4,x, algorithm="giac")

[Out]

-1/243*(14*b*c*(-a/b)^(1/3) + 5*a*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^4
*b) + 1/243*sqrt(3)*(5*(-a*b^2)^(1/3)*a*e - 14*(-a*b^2)^(2/3)*c)*arctan(1/3*sqrt
(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/162*(28*b^3*c*x^8 + 5*a*b^2
*x^7*e + 77*a*b^2*c*x^5 + 13*a^2*b*x^4*e + 67*a^2*b*c*x^2 - 10*a^3*x*e - 18*a^3*
d)/((b*x^3 + a)^3*a^3*b) + 1/486*(5*(-a*b^2)^(1/3)*a*b^2*e + 14*(-a*b^2)^(2/3)*b
^2*c)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^4)