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

Optimal. Leaf size=248 \[ -\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2} \]

[Out]

-(c + d*x + e*x^2)/(9*b*(a + b*x^3)^3) + (x*(d + 2*e*x))/(54*a*b*(a + b*x^3)^2)
+ (x*(5*d + 8*e*x))/(162*a^2*b*(a + b*x^3)) - ((5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(8/3)*b^(5/3)) + ((5
*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(8/3)*b^(5/3)) - ((5*
b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^
(8/3)*b^(5/3))

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Rubi [A]  time = 0.534477, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c + d*x + e*x^2)/(9*b*(a + b*x^3)^3) + (x*(d + 2*e*x))/(54*a*b*(a + b*x^3)^2)
+ (x*(5*d + 8*e*x))/(162*a^2*b*(a + b*x^3)) - ((5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(8/3)*b^(5/3)) + ((5
*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(8/3)*b^(5/3)) - ((5*
b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^
(8/3)*b^(5/3))

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Rubi in Sympy [A]  time = 75.8735, size = 228, normalized size = 0.92 \[ - \frac{c + d x + e x^{2}}{9 b \left (a + b x^{3}\right )^{3}} + \frac{x \left (d + 2 e x\right )}{54 a b \left (a + b x^{3}\right )^{2}} + \frac{x \left (5 d + 8 e x\right )}{162 a^{2} b \left (a + b x^{3}\right )} - \frac{\left (4 \sqrt [3]{a} e - 5 \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{243 a^{\frac{8}{3}} b^{\frac{5}{3}}} + \frac{\left (4 \sqrt [3]{a} e - 5 \sqrt [3]{b} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{486 a^{\frac{8}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (4 \sqrt [3]{a} e + 5 \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{243 a^{\frac{8}{3}} b^{\frac{5}{3}}} \]

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)**4,x)

[Out]

-(c + d*x + e*x**2)/(9*b*(a + b*x**3)**3) + x*(d + 2*e*x)/(54*a*b*(a + b*x**3)**
2) + x*(5*d + 8*e*x)/(162*a**2*b*(a + b*x**3)) - (4*a**(1/3)*e - 5*b**(1/3)*d)*l
og(a**(1/3) + b**(1/3)*x)/(243*a**(8/3)*b**(5/3)) + (4*a**(1/3)*e - 5*b**(1/3)*d
)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(486*a**(8/3)*b**(5/3)) -
sqrt(3)*(4*a**(1/3)*e + 5*b**(1/3)*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)
/a**(1/3))/(243*a**(8/3)*b**(5/3))

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Mathematica [A]  time = 0.478337, size = 230, normalized size = 0.93 \[ \frac{\frac{\left (4 \sqrt [3]{a} e-5 \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}+\frac{2 \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}-\frac{2 \sqrt{3} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{8/3}}+\frac{3 b^{2/3} x (5 d+8 e x)}{a^2 \left (a+b x^3\right )}-\frac{54 b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac{9 b^{2/3} x (d+2 e x)}{a \left (a+b x^3\right )^2}}{486 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((9*b^(2/3)*x*(d + 2*e*x))/(a*(a + b*x^3)^2) + (3*b^(2/3)*x*(5*d + 8*e*x))/(a^2*
(a + b*x^3)) - (54*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3)^3 - (2*Sqrt[3]*(5*b^(1
/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) + (2*(
5*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) + ((-5*b^(1/3)*d +
4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3))/(486*b^(5/
3))

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Maple [A]  time = 0.017, size = 275, normalized size = 1.1 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{3}} \left ({\frac{4\,be{x}^{8}}{81\,{a}^{2}}}+{\frac{5\,bd{x}^{7}}{162\,{a}^{2}}}+{\frac{11\,e{x}^{5}}{81\,a}}+{\frac{13\,d{x}^{4}}{162\,a}}-{\frac{2\,e{x}^{2}}{81\,b}}-{\frac{5\,dx}{81\,b}}-{\frac{c}{9\,b}} \right ) }+{\frac{5\,d}{243\,{a}^{2}{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,d}{486\,{a}^{2}{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,d\sqrt{3}}{243\,{a}^{2}{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,e}{243\,{a}^{2}{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,e}{243\,{a}^{2}{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{4\,e\sqrt{3}}{243\,{a}^{2}{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4/81/a^2*b*e*x^8+5/162/a^2*d*b*x^7+11/81/a*e*x^5+13/162*d/a*x^4-2/81*e*x^2/b-5/
81*d*x/b-1/9*c/b)/(b*x^3+a)^3+5/243*d/a^2/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-5/48
6*d/a^2/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/243*d/a^2/b^2/(a/b)^
(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-4/243*e/a^2/b^2/(a/b)^(1/3
)*ln(x+(a/b)^(1/3))+2/243*e/a^2/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)+4/243*e*3^(1/2)/a^2/b^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)^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^2/(b*x^3 + a)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 27.3954, size = 201, normalized size = 0.81 \[ \operatorname{RootSum}{\left (14348907 t^{3} a^{8} b^{5} + 14580 t a^{3} b^{2} d e + 64 a e^{3} - 125 b d^{3}, \left ( t \mapsto t \log{\left (x + \frac{236196 t^{2} a^{6} b^{3} e + 6075 t a^{3} b^{2} d^{2} + 160 a d e^{2}}{64 a e^{3} + 125 b d^{3}} \right )} \right )\right )} + \frac{- 18 a^{2} c - 10 a^{2} d x - 4 a^{2} e x^{2} + 13 a b d x^{4} + 22 a b e x^{5} + 5 b^{2} d x^{7} + 8 b^{2} e x^{8}}{162 a^{5} b + 486 a^{4} b^{2} x^{3} + 486 a^{3} b^{3} x^{6} + 162 a^{2} b^{4} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(14348907*_t**3*a**8*b**5 + 14580*_t*a**3*b**2*d*e + 64*a*e**3 - 125*b*d*
*3, Lambda(_t, _t*log(x + (236196*_t**2*a**6*b**3*e + 6075*_t*a**3*b**2*d**2 + 1
60*a*d*e**2)/(64*a*e**3 + 125*b*d**3)))) + (-18*a**2*c - 10*a**2*d*x - 4*a**2*e*
x**2 + 13*a*b*d*x**4 + 22*a*b*e*x**5 + 5*b**2*d*x**7 + 8*b**2*e*x**8)/(162*a**5*
b + 486*a**4*b**2*x**3 + 486*a**3*b**3*x**6 + 162*a**2*b**4*x**9)

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GIAC/XCAS [A]  time = 0.218112, size = 333, normalized size = 1.34 \[ -\frac{{\left (4 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{3} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{3} b^{3}} + \frac{8 \, b^{2} x^{8} e + 5 \, b^{2} d x^{7} + 22 \, a b x^{5} e + 13 \, a b d x^{4} - 4 \, a^{2} x^{2} e - 10 \, a^{2} d x - 18 \, a^{2} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{2} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{486 \, a^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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