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

Optimal. Leaf size=328 \[ \frac{x^4 \left (3 a^2 f-2 a b e+b^2 d\right )}{4 b^4}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{9 b^{16/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \sqrt{3} b^{16/3}}+\frac{a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{b^5}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{18 b^{16/3}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^{10}}{10 b^2} \]

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 + ((b^2*d - 2*a*b*e + 3*a^2*f)
*x^4)/(4*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^10)/(10*b^2) + (a*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(3*b^5*(a + b*x^3)) + (a^(1/3)*(4*b^3*c - 7*a*b^2*d
+ 10*a^2*b*e - 13*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*S
qrt[3]*b^(16/3)) - (a^(1/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*Log[a^
(1/3) + b^(1/3)*x])/(9*b^(16/3)) + (a^(1/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e -
13*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(16/3))

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Rubi [A]  time = 0.803358, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^4 \left (3 a^2 f-2 a b e+b^2 d\right )}{4 b^4}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{9 b^{16/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \sqrt{3} b^{16/3}}+\frac{a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{b^5}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{18 b^{16/3}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^{10}}{10 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 + ((b^2*d - 2*a*b*e + 3*a^2*f)
*x^4)/(4*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^10)/(10*b^2) + (a*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(3*b^5*(a + b*x^3)) + (a^(1/3)*(4*b^3*c - 7*a*b^2*d
+ 10*a^2*b*e - 13*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*S
qrt[3]*b^(16/3)) - (a^(1/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*Log[a^
(1/3) + b^(1/3)*x])/(9*b^(16/3)) + (a^(1/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e -
13*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(16/3))

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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**6*(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 = 0.516766, size = 315, normalized size = 0.96 \[ \frac{315 b^{4/3} x^4 \left (3 a^2 f-2 a b e+b^2 d\right )+\frac{420 a \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+1260 \sqrt [3]{b} x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right )-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right )+180 b^{7/3} x^7 (b e-2 a f)+126 b^{10/3} f x^{10}}{1260 b^{16/3}} \]

Antiderivative was successfully verified.

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

[Out]

(1260*b^(1/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x + 315*b^(4/3)*(b^2*d -
2*a*b*e + 3*a^2*f)*x^4 + 180*b^(7/3)*(b*e - 2*a*f)*x^7 + 126*b^(10/3)*f*x^10 +
(420*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3) - 140*Sqrt[3]*
a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)
/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*
Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^
3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1260*b^(16/3))

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Maple [B]  time = 0.016, size = 567, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9*a/b^3*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*a^
4/b^6*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*a^3/b^5
*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+7/9*a^2/b^4*d/(a/
b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3*a^4/b^5*x/(b*x^3+a)
*f+1/3*a^3/b^4*x/(b*x^3+a)*e+1/7/b^2*x^7*e+1/4/b^2*x^4*d+1/b^2*c*x+1/10*f*x^10/b
^2+1/3*a/b^2*x/(b*x^3+a)*c-10/9*a^3/b^5*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+5/9*a^3/
b^5*e/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/9*a^2/b^4*d/(a/b)^(2/3)*ln
(x+(a/b)^(1/3))-7/18*a^2/b^4*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9
*a/b^3*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+2/9*a/b^3*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1
/3)+(a/b)^(2/3))+13/9*a^4/b^6*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-13/18*a^4/b^6*f/(a
/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3*a^2/b^3*x/(b*x^3+a)*d-2/7/b^3*x^
7*a*f+3/4/b^4*x^4*a^2*f-1/2/b^3*x^4*a*e-4/b^5*a^3*f*x+3/b^4*a^2*e*x-2/b^3*a*d*x

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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^6/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240745, size = 590, normalized size = 1.8 \[ \frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f +{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 140 \, \sqrt{3}{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f +{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f +{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (42 \, b^{4} f x^{13} + 6 \,{\left (10 \, b^{4} e - 13 \, a b^{3} f\right )} x^{10} + 15 \,{\left (7 \, b^{4} d - 10 \, a b^{3} e + 13 \, a^{2} b^{2} f\right )} x^{7} + 105 \,{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{4} + 140 \,{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} x\right )}\right )}}{3780 \,{\left (b^{6} x^{3} + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3780*sqrt(3)*(70*sqrt(3)*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f + (4
*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/
b)^(1/3) + (a/b)^(2/3)) - 140*sqrt(3)*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13
*a^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*log(
x + (a/b)^(1/3)) + 420*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f + (4*b^4
*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*arctan(-1/3*(2*sqrt
(3)*x - sqrt(3)*(a/b)^(1/3))/(a/b)^(1/3)) + 3*sqrt(3)*(42*b^4*f*x^13 + 6*(10*b^4
*e - 13*a*b^3*f)*x^10 + 15*(7*b^4*d - 10*a*b^3*e + 13*a^2*b^2*f)*x^7 + 105*(4*b^
4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^4 + 140*(4*a*b^3*c - 7*a^2*b^2*d
+ 10*a^3*b*e - 13*a^4*f)*x))/(b^6*x^3 + a*b^5)

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Sympy [A]  time = 18.5485, size = 440, normalized size = 1.34 \[ - \frac{x \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{16} - 2197 a^{10} f^{3} + 5070 a^{9} b e f^{2} - 3549 a^{8} b^{2} d f^{2} - 3900 a^{8} b^{2} e^{2} f + 2028 a^{7} b^{3} c f^{2} + 5460 a^{7} b^{3} d e f + 1000 a^{7} b^{3} e^{3} - 3120 a^{6} b^{4} c e f - 1911 a^{6} b^{4} d^{2} f - 2100 a^{6} b^{4} d e^{2} + 2184 a^{5} b^{5} c d f + 1200 a^{5} b^{5} c e^{2} + 1470 a^{5} b^{5} d^{2} e - 624 a^{4} b^{6} c^{2} f - 1680 a^{4} b^{6} c d e - 343 a^{4} b^{6} d^{3} + 480 a^{3} b^{7} c^{2} e + 588 a^{3} b^{7} c d^{2} - 336 a^{2} b^{8} c^{2} d + 64 a b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t b^{5}}{13 a^{3} f - 10 a^{2} b e + 7 a b^{2} d - 4 b^{3} c} + x \right )} \right )\right )} + \frac{f x^{10}}{10 b^{2}} - \frac{x^{7} \left (2 a f - b e\right )}{7 b^{3}} + \frac{x^{4} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{4 b^{4}} - \frac{x \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c)/(3*a*b**5 + 3*b**6*x**3) + RootS
um(729*_t**3*b**16 - 2197*a**10*f**3 + 5070*a**9*b*e*f**2 - 3549*a**8*b**2*d*f**
2 - 3900*a**8*b**2*e**2*f + 2028*a**7*b**3*c*f**2 + 5460*a**7*b**3*d*e*f + 1000*
a**7*b**3*e**3 - 3120*a**6*b**4*c*e*f - 1911*a**6*b**4*d**2*f - 2100*a**6*b**4*d
*e**2 + 2184*a**5*b**5*c*d*f + 1200*a**5*b**5*c*e**2 + 1470*a**5*b**5*d**2*e - 6
24*a**4*b**6*c**2*f - 1680*a**4*b**6*c*d*e - 343*a**4*b**6*d**3 + 480*a**3*b**7*
c**2*e + 588*a**3*b**7*c*d**2 - 336*a**2*b**8*c**2*d + 64*a*b**9*c**3, Lambda(_t
, _t*log(9*_t*b**5/(13*a**3*f - 10*a**2*b*e + 7*a*b**2*d - 4*b**3*c) + x))) + f*
x**10/(10*b**2) - x**7*(2*a*f - b*e)/(7*b**3) + x**4*(3*a**2*f - 2*a*b*e + b**2*
d)/(4*b**4) - x*(4*a**3*f - 3*a**2*b*e + 2*a*b**2*d - b**3*c)/b**5

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GIAC/XCAS [A]  time = 0.215678, size = 532, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{6}} + \frac{{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d - 13 \, a^{4} f + 10 \, a^{3} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{5}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{6}} + \frac{a b^{3} c x - a^{2} b^{2} d x - a^{4} f x + a^{3} b x e}{3 \,{\left (b x^{3} + a\right )} b^{5}} + \frac{14 \, b^{18} f x^{10} - 40 \, a b^{17} f x^{7} + 20 \, b^{18} x^{7} e + 35 \, b^{18} d x^{4} + 105 \, a^{2} b^{16} f x^{4} - 70 \, a b^{17} x^{4} e + 140 \, b^{18} c x - 280 \, a b^{17} d x - 560 \, a^{3} b^{15} f x + 420 \, a^{2} b^{16} x e}{140 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*(4*(-a*b^2)^(1/3)*b^3*c - 7*(-a*b^2)^(1/3)*a*b^2*d - 13*(-a*b^2)^(1
/3)*a^3*f + 10*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(
-a/b)^(1/3))/b^6 + 1/9*(4*a*b^3*c - 7*a^2*b^2*d - 13*a^4*f + 10*a^3*b*e)*(-a/b)^
(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/18*(4*(-a*b^2)^(1/3)*b^3*c - 7*(-a*b
^2)^(1/3)*a*b^2*d - 13*(-a*b^2)^(1/3)*a^3*f + 10*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6 + 1/3*(a*b^3*c*x - a^2*b^2*d*x - a^4*f*x +
a^3*b*x*e)/((b*x^3 + a)*b^5) + 1/140*(14*b^18*f*x^10 - 40*a*b^17*f*x^7 + 20*b^18
*x^7*e + 35*b^18*d*x^4 + 105*a^2*b^16*f*x^4 - 70*a*b^17*x^4*e + 140*b^18*c*x - 2
80*a*b^17*d*x - 560*a^3*b^15*f*x + 420*a^2*b^16*x*e)/b^20