3.263 \(\int \frac{x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=298 \[ \frac{x^2 \left (3 a^2 f-2 a b e+b^2 d\right )}{2 b^4}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{9 \sqrt [3]{a} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{14/3}}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{18 \sqrt [3]{a} b^{14/3}}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^8}{8 b^2} \]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x^2)/(2*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^8
)/(8*b^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*b^4*(a + b*x^3)) - ((2*
b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3
]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(14/3)) - ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e -
11*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(14/3)) + ((2*b^3*c - 5*a*b^2*d
+ 8*a^2*b*e - 11*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(
1/3)*b^(14/3))

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Rubi [A]  time = 0.968636, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^2 \left (3 a^2 f-2 a b e+b^2 d\right )}{2 b^4}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{9 \sqrt [3]{a} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{14/3}}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{18 \sqrt [3]{a} b^{14/3}}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^8}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x^2)/(2*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^8
)/(8*b^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*b^4*(a + b*x^3)) - ((2*
b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3
]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(14/3)) - ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e -
11*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(14/3)) + ((2*b^3*c - 5*a*b^2*d
+ 8*a^2*b*e - 11*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(
1/3)*b^(14/3))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.302505, size = 282, normalized size = 0.95 \[ \frac{180 b^{2/3} x^2 \left (3 a^2 f-2 a b e+b^2 d\right )+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^3 f-8 a^2 b e+5 a b^2 d-2 b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (11 a^3 f-8 a^2 b e+5 a b^2 d-2 b^3 c\right )}{\sqrt [3]{a}}-\frac{120 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{\sqrt [3]{a}}+72 b^{5/3} x^5 (b e-2 a f)+45 b^{8/3} f x^8}{360 b^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(180*b^(2/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^2 + 72*b^(5/3)*(b*e - 2*a*f)*x^5 + 45
*b^(8/3)*f*x^8 - (120*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^
3) + (40*Sqrt[3]*(-2*b^3*c + 5*a*b^2*d - 8*a^2*b*e + 11*a^3*f)*ArcTan[(1 - (2*b^
(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-2*b^3*c + 5*a*b^2*d - 8*a^2*b*e + 11
*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(2*b^3*c - 5*a*b^2*d + 8*a^2*b*e
- 11*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(360*b^(14
/3))

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Maple [B]  time = 0.015, size = 529, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

1/8*f*x^8/b^2-2/5/b^3*x^5*a*f+1/5/b^2*x^5*e+3/2/b^4*x^2*a^2*f-1/b^3*x^2*a*e+1/2*
x^2*d/b^2+1/3/b^4*x^2/(b*x^3+a)*a^3*f-1/3/b^3*x^2/(b*x^3+a)*a^2*e+1/3/b^2*x^2/(b
*x^3+a)*a*d-1/3/b*x^2/(b*x^3+a)*c+11/9/b^5*a^3*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1
1/18/b^5*a^3*f/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-11/9/b^5*a^3*f*3^(1
/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-8/9/b^4*a^2*e/(a/b)^(1/3
)*ln(x+(a/b)^(1/3))+4/9/b^4*a^2*e/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+
8/9/b^4*a^2*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+5/9/b^
3*a*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-5/18/b^3*a*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3
)+(a/b)^(2/3))-5/9/b^3*a*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)
*x-1))-2/9/b^2*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/9/b^2*c/(a/b)^(1/3)*ln(x^2-x*(a
/b)^(1/3)+(a/b)^(2/3))+2/9/b^2*c*3^(1/2)/(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((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239034, size = 541, normalized size = 1.82 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f +{\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 40 \, \sqrt{3}{\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f +{\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 120 \,{\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f +{\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (15 \, b^{3} f x^{11} + 3 \,{\left (8 \, b^{3} e - 11 \, a b^{2} f\right )} x^{8} + 12 \,{\left (5 \, b^{3} d - 8 \, a b^{2} e + 11 \, a^{2} b f\right )} x^{5} - 20 \,{\left (2 \, b^{3} c - 5 \, a b^{2} d + 8 \, a^{2} b e - 11 \, a^{3} f\right )} x^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{1080 \,{\left (b^{5} x^{3} + a b^{4}\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/1080*sqrt(3)*(20*sqrt(3)*(2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f + (2*
b^4*c - 5*a*b^3*d + 8*a^2*b^2*e - 11*a^3*b*f)*x^3)*log((a*b^2)^(1/3)*b*x^2 + a*b
- (a*b^2)^(2/3)*x) - 40*sqrt(3)*(2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f
+ (2*b^4*c - 5*a*b^3*d + 8*a^2*b^2*e - 11*a^3*b*f)*x^3)*log(a*b + (a*b^2)^(2/3)
*x) + 120*(2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f + (2*b^4*c - 5*a*b^3*d
+ 8*a^2*b^2*e - 11*a^3*b*f)*x^3)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(a*b^2)^(
2/3)*x)/(a*b)) + 3*sqrt(3)*(15*b^3*f*x^11 + 3*(8*b^3*e - 11*a*b^2*f)*x^8 + 12*(5
*b^3*d - 8*a*b^2*e + 11*a^2*b*f)*x^5 - 20*(2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*
a^3*f)*x^2)*(a*b^2)^(1/3))/((b^5*x^3 + a*b^4)*(a*b^2)^(1/3))

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Sympy [A]  time = 51.1795, size = 484, normalized size = 1.62 \[ \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a b^{14} - 1331 a^{9} f^{3} + 2904 a^{8} b e f^{2} - 1815 a^{7} b^{2} d f^{2} - 2112 a^{7} b^{2} e^{2} f + 726 a^{6} b^{3} c f^{2} + 2640 a^{6} b^{3} d e f + 512 a^{6} b^{3} e^{3} - 1056 a^{5} b^{4} c e f - 825 a^{5} b^{4} d^{2} f - 960 a^{5} b^{4} d e^{2} + 660 a^{4} b^{5} c d f + 384 a^{4} b^{5} c e^{2} + 600 a^{4} b^{5} d^{2} e - 132 a^{3} b^{6} c^{2} f - 480 a^{3} b^{6} c d e - 125 a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e + 150 a^{2} b^{7} c d^{2} - 60 a b^{8} c^{2} d + 8 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a b^{9}}{121 a^{6} f^{2} - 176 a^{5} b e f + 110 a^{4} b^{2} d f + 64 a^{4} b^{2} e^{2} - 44 a^{3} b^{3} c f - 80 a^{3} b^{3} d e + 32 a^{2} b^{4} c e + 25 a^{2} b^{4} d^{2} - 20 a b^{5} c d + 4 b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{8}}{8 b^{2}} - \frac{x^{5} \left (2 a f - b e\right )}{5 b^{3}} + \frac{x^{2} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

x**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*a*b**4 + 3*b**5*x**3) + RootSum(
729*_t**3*a*b**14 - 1331*a**9*f**3 + 2904*a**8*b*e*f**2 - 1815*a**7*b**2*d*f**2
- 2112*a**7*b**2*e**2*f + 726*a**6*b**3*c*f**2 + 2640*a**6*b**3*d*e*f + 512*a**6
*b**3*e**3 - 1056*a**5*b**4*c*e*f - 825*a**5*b**4*d**2*f - 960*a**5*b**4*d*e**2
+ 660*a**4*b**5*c*d*f + 384*a**4*b**5*c*e**2 + 600*a**4*b**5*d**2*e - 132*a**3*b
**6*c**2*f - 480*a**3*b**6*c*d*e - 125*a**3*b**6*d**3 + 96*a**2*b**7*c**2*e + 15
0*a**2*b**7*c*d**2 - 60*a*b**8*c**2*d + 8*b**9*c**3, Lambda(_t, _t*log(81*_t**2*
a*b**9/(121*a**6*f**2 - 176*a**5*b*e*f + 110*a**4*b**2*d*f + 64*a**4*b**2*e**2 -
44*a**3*b**3*c*f - 80*a**3*b**3*d*e + 32*a**2*b**4*c*e + 25*a**2*b**4*d**2 - 20
*a*b**5*c*d + 4*b**6*c**2) + x))) + f*x**8/(8*b**2) - x**5*(2*a*f - b*e)/(5*b**3
) + x**2*(3*a**2*f - 2*a*b*e + b**2*d)/(2*b**4)

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GIAC/XCAS [A]  time = 0.219401, size = 537, normalized size = 1.8 \[ -\frac{{\left (2 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 11 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 8 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{4}} - \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} b^{4}} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 11 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{6}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 11 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{6}} + \frac{5 \, b^{14} f x^{8} - 16 \, a b^{13} f x^{5} + 8 \, b^{14} x^{5} e + 20 \, b^{14} d x^{2} + 60 \, a^{2} b^{12} f x^{2} - 40 \, a b^{13} x^{2} e}{40 \, b^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-1/9*(2*b^3*c*(-a/b)^(1/3) - 5*a*b^2*d*(-a/b)^(1/3) - 11*a^3*f*(-a/b)^(1/3) + 8*
a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^4) - 1/3*(b^3*
c*x^2 - a*b^2*d*x^2 - a^3*f*x^2 + a^2*b*x^2*e)/((b*x^3 + a)*b^4) - 1/9*sqrt(3)*(
2*(-a*b^2)^(2/3)*b^3*c - 5*(-a*b^2)^(2/3)*a*b^2*d - 11*(-a*b^2)^(2/3)*a^3*f + 8*
(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a
*b^6) + 1/18*(2*(-a*b^2)^(2/3)*b^3*c - 5*(-a*b^2)^(2/3)*a*b^2*d - 11*(-a*b^2)^(2
/3)*a^3*f + 8*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a
*b^6) + 1/40*(5*b^14*f*x^8 - 16*a*b^13*f*x^5 + 8*b^14*x^5*e + 20*b^14*d*x^2 + 60
*a^2*b^12*f*x^2 - 40*a*b^13*x^2*e)/b^16