3.302 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=380 \[ \frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{54 a^{20/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{9 \sqrt{3} a^{20/3}}+\frac{b x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

[Out]

-c/(11*a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^
5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b^3*c - 23*a*b^2*d +
17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) - (b^(2/3)*(119*b^3*c - 77*a*b^2*
d + 44*a^2*b*e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9
*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) - (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a
^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))

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Rubi [A]  time = 1.39485, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{54 a^{20/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{9 \sqrt{3} a^{20/3}}+\frac{b x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]

[Out]

-c/(11*a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^
5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b^3*c - 23*a*b^2*d +
17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) - (b^(2/3)*(119*b^3*c - 77*a*b^2*
d + 44*a^2*b*e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9
*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) - (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a
^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(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((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.869465, size = 376, normalized size = 0.99 \[ \frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (20 a^3 f-44 a^2 b e+77 a b^2 d-119 b^3 c\right )}{54 a^{20/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (20 a^3 f-44 a^2 b e+77 a b^2 d-119 b^3 c\right )}{9 \sqrt{3} a^{20/3}}+\frac{b x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]

[Out]

-c/(11*a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^
5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b^3*c - 23*a*b^2*d +
17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2
*d - 44*a^2*b*e + 20*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt
[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^
(1/3) + b^(1/3)*x])/(27*a^(20/3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2*d - 44*a^2*b
*e + 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))

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Maple [A]  time = 0.026, size = 651, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x)

[Out]

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

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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)/((b*x^3 + a)^3*x^12),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221767, size = 911, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^12),x, algorithm="fricas")

[Out]

1/35640*sqrt(3)*(220*sqrt(3)*((119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^
2*f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14 + (1
19*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*log(
b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 440*sqrt(3)*((119*b^5
*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^
3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14 + (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b
*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)) + 1320*((119
*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a*b^4*c - 77*a^
2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14 + (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a
^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x + sqrt(3)*a*
(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 3*sqrt(3)*(220*(119*b^5*c - 77*a*b^4*d
+ 44*a^2*b^3*e - 20*a^3*b^2*f)*x^15 + 352*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*
b^2*e - 20*a^4*b*f)*x^12 + 99*(119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^
5*f)*x^9 - 18*(119*a^3*b^2*c - 77*a^4*b*d + 44*a^5*e)*x^6 - 360*a^5*c + 45*(17*a
^4*b*c - 11*a^5*d)*x^3))/(a^6*b^2*x^17 + 2*a^7*b*x^14 + a^8*x^11)

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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((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21837, size = 594, normalized size = 1.56 \[ \frac{\sqrt{3}{\left (119 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{7}} - \frac{{\left (119 \, b^{4} c - 77 \, a b^{3} d - 20 \, a^{3} b f + 44 \, a^{2} b^{2} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7}} + \frac{{\left (119 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{7}} + \frac{29 \, b^{5} c x^{4} - 23 \, a b^{4} d x^{4} - 11 \, a^{3} b^{2} f x^{4} + 17 \, a^{2} b^{3} x^{4} e + 32 \, a b^{4} c x - 26 \, a^{2} b^{3} d x - 14 \, a^{4} b f x + 20 \, a^{3} b^{2} x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{6}} + \frac{2200 \, b^{3} c x^{9} - 1320 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 660 \, a^{2} b x^{9} e - 528 \, a b^{2} c x^{6} + 264 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 165 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{6} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^12),x, algorithm="giac")

[Out]

1/27*sqrt(3)*(119*(-a*b^2)^(1/3)*b^3*c - 77*(-a*b^2)^(1/3)*a*b^2*d - 20*(-a*b^2)
^(1/3)*a^3*f + 44*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3)
)/(-a/b)^(1/3))/a^7 - 1/27*(119*b^4*c - 77*a*b^3*d - 20*a^3*b*f + 44*a^2*b^2*e)*
(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^7 + 1/54*(119*(-a*b^2)^(1/3)*b^3*c - 77
*(-a*b^2)^(1/3)*a*b^2*d - 20*(-a*b^2)^(1/3)*a^3*f + 44*(-a*b^2)^(1/3)*a^2*b*e)*l
n(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^7 + 1/18*(29*b^5*c*x^4 - 23*a*b^4*d*x^4
- 11*a^3*b^2*f*x^4 + 17*a^2*b^3*x^4*e + 32*a*b^4*c*x - 26*a^2*b^3*d*x - 14*a^4*
b*f*x + 20*a^3*b^2*x*e)/((b*x^3 + a)^2*a^6) + 1/440*(2200*b^3*c*x^9 - 1320*a*b^2
*d*x^9 - 220*a^3*f*x^9 + 660*a^2*b*x^9*e - 528*a*b^2*c*x^6 + 264*a^2*b*d*x^6 - 8
8*a^3*x^6*e + 165*a^2*b*c*x^3 - 55*a^3*d*x^3 - 40*a^3*c)/(a^6*x^11)