3.287 \(\int \frac{x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=384 \[ \frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac{a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6
*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) - (a^2*(b
^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2) + (a*(7*b^3*c - 10*
a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) + (a^(2/3)*(20*b^3*c -
44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^
(1/3))])/(9*Sqrt[3]*b^(20/3)) + (a^(2/3)*(20*b^3*c - 44*a*b^2*d + 77*a^2*b*e - 1
19*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) - (a^(2/3)*(20*b^3*c - 44*a*b^
2*d + 77*a^2*b*e - 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(5
4*b^(20/3))

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Rubi [A]  time = 2.05217, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac{a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6
*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) - (a^2*(b
^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2) + (a*(7*b^3*c - 10*
a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) + (a^(2/3)*(20*b^3*c -
44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^
(1/3))])/(9*Sqrt[3]*b^(20/3)) + (a^(2/3)*(20*b^3*c - 44*a*b^2*d + 77*a^2*b*e - 1
19*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) - (a^(2/3)*(20*b^3*c - 44*a*b^
2*d + 77*a^2*b*e - 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(5
4*b^(20/3))

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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**10*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.977661, size = 380, normalized size = 0.99 \[ \frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}+\frac{a^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{54 b^{20/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{27 b^{20/3}}-\frac{a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6
*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) + (a^2*(-
(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2) + (a*(7*b^3*c -
10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) - (a^(2/3)*(-20*b^3
*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqr
t[3]])/(9*Sqrt[3]*b^(20/3)) - (a^(2/3)*(-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 11
9*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) + (a^(2/3)*(-20*b^3*c + 44*a*b^
2*d - 77*a^2*b*e + 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(5
4*b^(20/3))

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Maple [B]  time = 0.021, size = 668, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

-3/8/b^4*x^8*a*f+6/5/b^5*x^5*a^2*f-3/5/b^4*x^5*a*e-5/b^6*x^2*a^3*f+3/b^5*x^2*a^2
*e-3/2/b^4*x^2*a*d+1/11*f*x^11/b^3+20/27*a/b^4*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1
19/27*a^4/b^7*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+119/54*a^4/b^7*f/(a/b)^(1/3)*ln(x^
2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/8/b^3*x^8*e+1/5/b^3*x^5*d+1/2/b^3*x^2*c+7/9*a/b^2
/(b*x^3+a)^2*x^5*c-29/18*a^5/b^6/(b*x^3+a)^2*x^2*f+23/18*a^4/b^5/(b*x^3+a)^2*x^2
*e+77/27*a^3/b^6*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-77/54*a^3/b^6*e/(a/b)^(1/3)*ln(
x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-10/27*a/b^4*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a
/b)^(2/3))-16/9*a^4/b^5/(b*x^3+a)^2*x^5*f+13/9*a^3/b^4/(b*x^3+a)^2*x^5*e-10/9*a^
2/b^3/(b*x^3+a)^2*x^5*d-44/27*a^2/b^5*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+22/27*a^2/
b^5*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+119/27*a^4/b^7*f*3^(1/2)/(a/
b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-77/27*a^3/b^6*e*3^(1/2)/(a/b)^(
1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+44/27*a^2/b^5*d*3^(1/2)/(a/b)^(1/3)
*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-20/27*a/b^4*c*3^(1/2)/(a/b)^(1/3)*arcta
n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-17/18*a^3/b^4/(b*x^3+a)^2*x^2*d+11/18*a^2/b^3
/(b*x^3+a)^2*x^2*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((f*x^9 + e*x^6 + d*x^3 + c)*x^10/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240825, size = 887, normalized size = 2.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^10/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/35640*sqrt(3)*(220*sqrt(3)*((20*b^5*c - 44*a*b^4*d + 77*a^2*b^3*e - 119*a^3*b^
2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2*(20*a*b^4*c
- 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^2)^(1/3)*log(a*x^2 - b
*x*(-a^2/b^2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 440*sqrt(3)*((20*b^5*c - 44*a*b^4*d
+ 77*a^2*b^3*e - 119*a^3*b^2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e -
119*a^5*f + 2*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a
^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/3)) - 1320*((20*b^5*c - 44*a*b^4*d + 77*
a^2*b^3*e - 119*a^3*b^2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*
a^5*f + 2*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^
2)^(1/3)*arctan(-1/3*(2*sqrt(3)*a*x - sqrt(3)*b*(-a^2/b^2)^(2/3))/(b*(-a^2/b^2)^
(2/3))) + 3*sqrt(3)*(360*b^5*f*x^17 + 45*(11*b^5*e - 17*a*b^4*f)*x^14 + 18*(44*b
^5*d - 77*a*b^4*e + 119*a^2*b^3*f)*x^11 + 99*(20*b^5*c - 44*a*b^4*d + 77*a^2*b^3
*e - 119*a^3*b^2*f)*x^8 + 352*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^
4*b*f)*x^5 + 220*(20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f)*x^2))/(b
^8*x^6 + 2*a*b^7*x^3 + a^2*b^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223775, size = 663, normalized size = 1.73 \[ \frac{{\left (20 \, a b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 44 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 119 \, a^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 77 \, a^{3} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{6}} + \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, b^{8}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, b^{8}} + \frac{14 \, a b^{4} c x^{5} - 20 \, a^{2} b^{3} d x^{5} - 32 \, a^{4} b f x^{5} + 26 \, a^{3} b^{2} x^{5} e + 11 \, a^{2} b^{3} c x^{2} - 17 \, a^{3} b^{2} d x^{2} - 29 \, a^{5} f x^{2} + 23 \, a^{4} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{40 \, b^{30} f x^{11} - 165 \, a b^{29} f x^{8} + 55 \, b^{30} x^{8} e + 88 \, b^{30} d x^{5} + 528 \, a^{2} b^{28} f x^{5} - 264 \, a b^{29} x^{5} e + 220 \, b^{30} c x^{2} - 660 \, a b^{29} d x^{2} - 2200 \, a^{3} b^{27} f x^{2} + 1320 \, a^{2} b^{28} x^{2} e}{440 \, b^{33}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^10/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

1/27*(20*a*b^3*c*(-a/b)^(1/3) - 44*a^2*b^2*d*(-a/b)^(1/3) - 119*a^4*f*(-a/b)^(1/
3) + 77*a^3*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^6) + 1
/27*sqrt(3)*(20*(-a*b^2)^(2/3)*b^3*c - 44*(-a*b^2)^(2/3)*a*b^2*d - 119*(-a*b^2)^
(2/3)*a^3*f + 77*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))
/(-a/b)^(1/3))/b^8 - 1/54*(20*(-a*b^2)^(2/3)*b^3*c - 44*(-a*b^2)^(2/3)*a*b^2*d -
119*(-a*b^2)^(2/3)*a^3*f + 77*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/b^8 + 1/18*(14*a*b^4*c*x^5 - 20*a^2*b^3*d*x^5 - 32*a^4*b*f*x^5 +
26*a^3*b^2*x^5*e + 11*a^2*b^3*c*x^2 - 17*a^3*b^2*d*x^2 - 29*a^5*f*x^2 + 23*a^4*b
*x^2*e)/((b*x^3 + a)^2*b^6) + 1/440*(40*b^30*f*x^11 - 165*a*b^29*f*x^8 + 55*b^30
*x^8*e + 88*b^30*d*x^5 + 528*a^2*b^28*f*x^5 - 264*a*b^29*x^5*e + 220*b^30*c*x^2
- 660*a*b^29*d*x^2 - 2200*a^3*b^27*f*x^2 + 1320*a^2*b^28*x^2*e)/b^33