3.303 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=424 \[ \frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{9 \sqrt{3} a^{22/3}}-\frac{b^2 x^2 \left (-8 a^3 f+11 a^2 b e-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac{b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac{b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2} \]

[Out]

-c/(13*a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*
a^5*x^7) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) - (b*(15*b^3*c
- 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) - (b^2*(17*b^3*c - 14*a*b^2*d + 11*a^2*b*e -
8*a^3*f)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*
b*e - 35*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^
(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(22/3)) - (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e -
35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(22/3))

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Rubi [A]  time = 1.70244, antiderivative size = 424, 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 \[ \frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{9 \sqrt{3} a^{22/3}}-\frac{b^2 x^2 \left (-8 a^3 f+11 a^2 b e-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac{b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac{b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]

[Out]

-c/(13*a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*
a^5*x^7) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) - (b*(15*b^3*c
- 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) - (b^2*(17*b^3*c - 14*a*b^2*d + 11*a^2*b*e -
8*a^3*f)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*
b*e - 35*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^
(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(22/3)) - (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e -
35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(22/3))

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

[Out]

Timed out

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Mathematica [A]  time = 1.08831, size = 419, normalized size = 0.99 \[ \frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^3 f-65 a^2 b e+104 a b^2 d-152 b^3 c\right )}{54 a^{22/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{9 \sqrt{3} a^{22/3}}+\frac{b^2 x^2 \left (8 a^3 f-11 a^2 b e+14 a b^2 d-17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}+\frac{b \left (3 a^3 f-6 a^2 b e+10 a b^2 d-15 b^3 c\right )}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}+\frac{b^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]

[Out]

-c/(13*a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*
a^5*x^7) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) + (b*(-15*b^3*
c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*
b*e + a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) + (b^2*(-17*b^3*c + 14*a*b^2*d - 11*a^2*
b*e + 8*a^3*f)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65
*a^2*b*e - 35*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(
22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(22/3)) + (b^(4/3)*(-152*b^3*c + 104*a*b^2*d - 65*a^2*b*e +
35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(22/3))

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Maple [A]  time = 0.03, size = 716, 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^14/(b*x^3+a)^3,x)

[Out]

35/27*b/a^4*f*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*e*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*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-152/27*b^4/a^
7*c*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*d
/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+14/9*b^5/a^6/(b*x^3+a)^2*x^5*d+31/18*b^4/a^5/(b*x
^3+a)^2*x^2*d-37/18*b^5/a^6/(b*x^3+a)^2*x^2*c-35/27*b/a^4*f/(a/b)^(1/3)*ln(x+(a/
b)^(1/3))+52/27*b^3/a^6*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+152/27*b
^4/a^7*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-76/27*b^4/a^7*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*f-25/18*b^3/a^4/(b*x^3+a)^2*x
^2*e-17/9*b^6/a^7/(b*x^3+a)^2*x^5*c+3*b/a^4/x*f-6*b^2/a^5/x*e+10*b^3/a^6/x*d-15*
b^4/a^7/x*c+3/10/a^4/x^10*b*c+3/7/a^4/x^7*b*d-6/7/a^5/x^7*b^2*c+3/4/a^4/x^4*b*e+
35/54*b/a^4*f/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+65/27*b^2/a^5*e/(a/b
)^(1/3)*ln(x+(a/b)^(1/3))-65/54*b^2/a^5*e/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)
^(2/3))-3/2/a^5/x^4*b^2*d+5/2/a^6/x^4*b^3*c-1/13*c/a^3/x^13-1/10/a^3/x^10*d-1/7/
a^3/x^7*e-1/4/a^3/x^4*f+8/9*b^3/a^4/(b*x^3+a)^2*x^5*f-11/9*b^4/a^5/(b*x^3+a)^2*x
^5*e

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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^14),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243658, size = 963, normalized size = 2.27 \[ \text{result too large to display} \]

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^14),x, algorithm="fricas")

[Out]

1/147420*sqrt(3)*(910*sqrt(3)*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*
b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^16
+ (152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3
)*log(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) - 1820*sqrt(3)*((152*b^6*c - 10
4*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d +
65*a^3*b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2
*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) - 5460*((152*b^6*c
- 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^
4*d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^
4*b^2*e - 35*a^5*b*f)*x^13)*(-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*sqrt(3)*(1820*(152*b^6*c - 104*a*b^5*d + 65*
a^2*b^4*e - 35*a^3*b^3*f)*x^18 + 3185*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*b^3*
e - 35*a^4*b^2*f)*x^15 + 1170*(152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35
*a^5*b*f)*x^12 - 117*(152*a^3*b^3*c - 104*a^4*b^2*d + 65*a^5*b*e - 35*a^6*f)*x^9
+ 1260*a^6*c + 36*(152*a^4*b^2*c - 104*a^5*b*d + 65*a^6*e)*x^6 - 126*(19*a^5*b*
c - 13*a^6*d)*x^3))/(a^7*b^2*x^19 + 2*a^8*b*x^16 + a^9*x^13)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221781, size = 717, normalized size = 1.69 \[ \frac{\sqrt{3}{\left (152 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{8}} + \frac{{\left (152 \, b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 104 \, a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 35 \, a^{3} b^{2} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 65 \, a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{8}} - \frac{{\left (152 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{8}} - \frac{34 \, b^{6} c x^{5} - 28 \, a b^{5} d x^{5} - 16 \, a^{3} b^{3} f x^{5} + 22 \, a^{2} b^{4} x^{5} e + 37 \, a b^{5} c x^{2} - 31 \, a^{2} b^{4} d x^{2} - 19 \, a^{4} b^{2} f x^{2} + 25 \, a^{3} b^{3} x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{7}} - \frac{27300 \, b^{4} c x^{12} - 18200 \, a b^{3} d x^{12} - 5460 \, a^{3} b f x^{12} + 10920 \, a^{2} b^{2} x^{12} e - 4550 \, a b^{3} c x^{9} + 2730 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 1365 \, a^{3} b x^{9} e + 1560 \, a^{2} b^{2} c x^{6} - 780 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 546 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{7} x^{13}} \]

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^14),x, algorithm="giac")

[Out]

1/27*sqrt(3)*(152*(-a*b^2)^(2/3)*b^3*c - 104*(-a*b^2)^(2/3)*a*b^2*d - 35*(-a*b^2
)^(2/3)*a^3*f + 65*(-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^8 + 1/27*(152*b^5*c*(-a/b)^(1/3) - 104*a*b^4*d*(-a/b)^(1/3) -
35*a^3*b^2*f*(-a/b)^(1/3) + 65*a^2*b^3*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x -
(-a/b)^(1/3)))/a^8 - 1/54*(152*(-a*b^2)^(2/3)*b^3*c - 104*(-a*b^2)^(2/3)*a*b^2*d
- 35*(-a*b^2)^(2/3)*a^3*f + 65*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/a^8 - 1/18*(34*b^6*c*x^5 - 28*a*b^5*d*x^5 - 16*a^3*b^3*f*x^5 + 2
2*a^2*b^4*x^5*e + 37*a*b^5*c*x^2 - 31*a^2*b^4*d*x^2 - 19*a^4*b^2*f*x^2 + 25*a^3*
b^3*x^2*e)/((b*x^3 + a)^2*a^7) - 1/1820*(27300*b^4*c*x^12 - 18200*a*b^3*d*x^12 -
5460*a^3*b*f*x^12 + 10920*a^2*b^2*x^12*e - 4550*a*b^3*c*x^9 + 2730*a^2*b^2*d*x^
9 + 455*a^4*f*x^9 - 1365*a^3*b*x^9*e + 1560*a^2*b^2*c*x^6 - 780*a^3*b*d*x^6 + 26
0*a^4*x^6*e - 546*a^3*b*c*x^3 + 182*a^4*d*x^3 + 140*a^4*c)/(a^7*x^13)