3.301 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=381 \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{9 \sqrt{3} a^{19/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac{b x^2 \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

[Out]

-c/(10*a^3*x^10) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(4*a^
5*x^4) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*x^2)/(6*a^5*(a + b*x^3)^2) + (b*(14*b^3*c - 11*a*b^2*d + 8*
a^2*b*e - 5*a^3*f)*x^2)/(9*a^6*(a + b*x^3)) - (b^(1/3)*(104*b^3*c - 65*a*b^2*d +
35*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sq
rt[3]*a^(19/3)) - (b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[
a^(1/3) + b^(1/3)*x])/(27*a^(19/3)) + (b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*
b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(19/3))

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Rubi [A]  time = 1.40591, antiderivative size = 381, 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 \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{9 \sqrt{3} a^{19/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac{b x^2 \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]

[Out]

-c/(10*a^3*x^10) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(4*a^
5*x^4) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*x^2)/(6*a^5*(a + b*x^3)^2) + (b*(14*b^3*c - 11*a*b^2*d + 8*
a^2*b*e - 5*a^3*f)*x^2)/(9*a^6*(a + b*x^3)) - (b^(1/3)*(104*b^3*c - 65*a*b^2*d +
35*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sq
rt[3]*a^(19/3)) - (b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[
a^(1/3) + b^(1/3)*x])/(27*a^(19/3)) + (b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*
b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(19/3))

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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**11/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.850349, size = 366, normalized size = 0.96 \[ \frac{-\frac{540 a^{7/3} (a d-3 b c)}{x^7}-\frac{378 a^{10/3} c}{x^{10}}-\frac{945 a^{4/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^4}-\frac{420 \sqrt [3]{a} b x^2 \left (5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right )}{a+b x^3}-\frac{3780 \sqrt [3]{a} \left (a^3 f-3 a^2 b e+6 a b^2 d-10 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^3 f-35 a^2 b e+65 a b^2 d-104 b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )-\frac{630 a^{4/3} b x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{3780 a^{19/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]

[Out]

((-378*a^(10/3)*c)/x^10 - (540*a^(7/3)*(-3*b*c + a*d))/x^7 - (945*a^(4/3)*(6*b^2
*c - 3*a*b*d + a^2*e))/x^4 - (3780*a^(1/3)*(-10*b^3*c + 6*a*b^2*d - 3*a^2*b*e +
a^3*f))/x - (630*a^(4/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^
3)^2 - (420*a^(1/3)*b*(-14*b^3*c + 11*a*b^2*d - 8*a^2*b*e + 5*a^3*f)*x^2)/(a + b
*x^3) - 140*Sqrt[3]*b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Arc
Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-104*b^3*c + 65*a*b^2*d
- 35*a^2*b*e + 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 70*b^(1/3)*(104*b^3*c - 65*a
*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
(3780*a^(19/3))

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Maple [A]  time = 0.025, size = 659, 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^11/(b*x^3+a)^3,x)

[Out]

8/9*b^3/a^4/(b*x^3+a)^2*x^5*e-11/9*b^4/a^5/(b*x^3+a)^2*x^5*d+14/9*b^5/a^6/(b*x^3
+a)^2*x^5*c-13/18*b/a^2/(b*x^3+a)^2*x^2*f-65/54*b^2/a^5*d/(a/b)^(1/3)*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))-104/27*b^3/a^6*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+35/27*b/a
^4*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-65/27*b^2/a^5*d
*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+104/27*b^3/a^6*c*3^
(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+65/27*b^2/a^5*d/(a/b)^
(1/3)*ln(x+(a/b)^(1/3))-5/9*b^2/a^3/(b*x^3+a)^2*x^5*f-1/4/a^3/x^4*e-1/a^3/x*f-1/
7/a^3/x^7*d-1/10*c/a^3/x^10+14/27/a^3*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-7/27/a^3*f
/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+3/7/a^4/x^7*b*c+3/4/a^4/x^4*b*d-3
/2/a^5/x^4*b^2*c+3/a^4/x*b*e-6/a^5/x*b^2*d+10/a^6/x*b^3*c+52/27*b^3/a^6*c/(a/b)^
(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+19/18*b^2/a^3/(b*x^3+a)^2*x^2*e-25/18*b^
3/a^4/(b*x^3+a)^2*x^2*d+35/54*b/a^4*e/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/
3))+31/18*b^4/a^5/(b*x^3+a)^2*x^2*c-35/27*b/a^4*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-
14/27/a^3*f*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)/((b*x^3 + a)^3*x^11),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230684, size = 873, normalized size = 2.29 \[ \text{result too large to display} \]

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^11),x, algorithm="fricas")

[Out]

1/11340*sqrt(3)*(70*sqrt(3)*((104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2
*f)*x^16 + 2*(104*a*b^4*c - 65*a^2*b^3*d + 35*a^3*b^2*e - 14*a^4*b*f)*x^13 + (10
4*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^10)*(b/a)^(1/3)*log(b*x^2
- a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 140*sqrt(3)*((104*b^5*c - 65*a*b^4*d + 35*a
^2*b^3*e - 14*a^3*b^2*f)*x^16 + 2*(104*a*b^4*c - 65*a^2*b^3*d + 35*a^3*b^2*e - 1
4*a^4*b*f)*x^13 + (104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^10)*(
b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) - 420*((104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*
e - 14*a^3*b^2*f)*x^16 + 2*(104*a*b^4*c - 65*a^2*b^3*d + 35*a^3*b^2*e - 14*a^4*b
*f)*x^13 + (104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^10)*(b/a)^(1
/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(b/a)^(2/3))/(a*(b/a)^(2/3))) + 3*sqr
t(3)*(140*(104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2*f)*x^15 + 245*(104
*a*b^4*c - 65*a^2*b^3*d + 35*a^3*b^2*e - 14*a^4*b*f)*x^12 + 90*(104*a^2*b^3*c -
65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^9 - 9*(104*a^3*b^2*c - 65*a^4*b*d + 35*a
^5*e)*x^6 - 126*a^5*c + 36*(8*a^4*b*c - 5*a^5*d)*x^3))/(a^6*b^2*x^16 + 2*a^7*b*x
^13 + a^8*x^10)

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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**11/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219558, size = 656, normalized size = 1.72 \[ -\frac{{\left (104 \, b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 65 \, a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 35 \, a^{2} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7}} - \frac{\sqrt{3}{\left (104 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{7} b} + \frac{{\left (104 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{7} b} + \frac{28 \, b^{5} c x^{5} - 22 \, a b^{4} d x^{5} - 10 \, a^{3} b^{2} f x^{5} + 16 \, a^{2} b^{3} x^{5} e + 31 \, a b^{4} c x^{2} - 25 \, a^{2} b^{3} d x^{2} - 13 \, a^{4} b f x^{2} + 19 \, a^{3} b^{2} x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{6}} + \frac{1400 \, b^{3} c x^{9} - 840 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 420 \, a^{2} b x^{9} e - 210 \, a b^{2} c x^{6} + 105 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 60 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{6} x^{10}} \]

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^11),x, algorithm="giac")

[Out]

-1/27*(104*b^4*c*(-a/b)^(1/3) - 65*a*b^3*d*(-a/b)^(1/3) - 14*a^3*b*f*(-a/b)^(1/3
) + 35*a^2*b^2*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^7 - 1/27
*sqrt(3)*(104*(-a*b^2)^(2/3)*b^3*c - 65*(-a*b^2)^(2/3)*a*b^2*d - 14*(-a*b^2)^(2/
3)*a^3*f + 35*(-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^7*b) + 1/54*(104*(-a*b^2)^(2/3)*b^3*c - 65*(-a*b^2)^(2/3)*a*b^2*d
- 14*(-a*b^2)^(2/3)*a^3*f + 35*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a^7*b) + 1/18*(28*b^5*c*x^5 - 22*a*b^4*d*x^5 - 10*a^3*b^2*f*x^5
+ 16*a^2*b^3*x^5*e + 31*a*b^4*c*x^2 - 25*a^2*b^3*d*x^2 - 13*a^4*b*f*x^2 + 19*a^
3*b^2*x^2*e)/((b*x^3 + a)^2*a^6) + 1/140*(1400*b^3*c*x^9 - 840*a*b^2*d*x^9 - 140
*a^3*f*x^9 + 420*a^2*b*x^9*e - 210*a*b^2*c*x^6 + 105*a^2*b*d*x^6 - 35*a^3*x^6*e
+ 60*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^6*x^10)