3.275 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=375 \[ \frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt{3} a^{19/3}}-\frac{b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}-\frac{b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]

[Out]

-c/(13*a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^2*e)/(7*
a^4*x^7) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) - (b*(5*b^3*c -
4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^6*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e -
7*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(19/3))
+ (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)
*x])/(9*a^(19/3)) - (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(19/3))

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Rubi [A]  time = 1.13424, antiderivative size = 375, 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 \[ \frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt{3} a^{19/3}}-\frac{b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}-\frac{b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^2),x]

[Out]

-c/(13*a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^2*e)/(7*
a^4*x^7) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) - (b*(5*b^3*c -
4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^6*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e -
7*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(19/3))
+ (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)
*x])/(9*a^(19/3)) - (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*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**14/(b*x**3+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.793809, size = 370, normalized size = 0.99 \[ \frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-10 a^2 b e+13 a b^2 d-16 b^3 c\right )}{18 a^{19/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt{3} a^{19/3}}+\frac{b^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac{b \left (2 a^3 f-3 a^2 b e+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^2),x]

[Out]

-c/(13*a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^2*e)/(7*
a^4*x^7) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) + (b*(-5*b^3*c
+ 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e
+ a^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*
e - 7*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(3*Sqrt[3]*a^(19/3)) +
(b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x
])/(9*a^(19/3)) + (b^(4/3)*(-16*b^3*c + 13*a*b^2*d - 10*a^2*b*e + 7*a^3*f)*Log[a
^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(19/3))

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Maple [A]  time = 0.024, size = 631, 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)^2,x)

[Out]

7/9*b/a^3*f*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*b^2
/a^4*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*b^3/a^5*
d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-16/9*b^4/a^6*c*3^(
1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*b^4/a^5*x^2/(b*x^3+
a)*d-3/7/a^4/x^7*b^2*c-1/4/a^2/x^4*f-1/10/a^2/x^10*d-1/7/a^2/x^7*e+1/2/a^3/x^4*b
*e-3/4/a^4/x^4*b^2*d+1/a^5/x^4*b^3*c+2/a^3*b/x*f-3/a^4*b^2/x*e+4/a^5*b^3/x*d-5/a
^6*b^4/x*c+1/5/a^3/x^10*b*c+2/7/a^3/x^7*b*d-1/3*b^5/a^6*x^2/(b*x^3+a)*c+13/18*b^
3/a^5*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+16/9*b^4/a^6*c/(a/b)^(1/3)
*ln(x+(a/b)^(1/3))-8/9*b^4/a^6*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-7
/9*b/a^3*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/18*b/a^3*f/(a/b)^(1/3)*ln(x^2-x*(a/b)
^(1/3)+(a/b)^(2/3))+1/3*b^2/a^3*x^2/(b*x^3+a)*f-1/3*b^3/a^4*x^2/(b*x^3+a)*e+10/9
*b^2/a^4*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-5/9*b^2/a^4*e/(a/b)^(1/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))-13/9*b^3/a^5*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/13*c/a^2/x^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)^2*x^14),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229478, size = 721, normalized size = 1.92 \[ \frac{\sqrt{3}{\left (910 \, \sqrt{3}{\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} +{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 1820 \, \sqrt{3}{\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} +{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 5460 \,{\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} +{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (1820 \,{\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 1365 \,{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 195 \,{\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 78 \,{\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 420 \, a^{5} c - 42 \,{\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3}\right )}\right )}}{49140 \,{\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} \]

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)^2*x^14),x, algorithm="fricas")

[Out]

1/49140*sqrt(3)*(910*sqrt(3)*((16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*
f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*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)*((16*b^5*c - 1
3*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a
^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) - 5460*((16*b
^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*
d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sq
rt(3)*a*(-b/a)^(2/3))/(a*(-b/a)^(2/3))) - 3*sqrt(3)*(1820*(16*b^5*c - 13*a*b^4*d
+ 10*a^2*b^3*e - 7*a^3*b^2*f)*x^15 + 1365*(16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b
^2*e - 7*a^4*b*f)*x^12 - 195*(16*a^2*b^3*c - 13*a^3*b^2*d + 10*a^4*b*e - 7*a^5*f
)*x^9 + 78*(16*a^3*b^2*c - 13*a^4*b*d + 10*a^5*e)*x^6 + 420*a^5*c - 42*(16*a^4*b
*c - 13*a^5*d)*x^3))/(a^6*b*x^16 + a^7*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218039, size = 651, normalized size = 1.74 \[ \frac{\sqrt{3}{\left (16 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{7}} + \frac{{\left (16 \, b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 13 \, a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, a^{3} b^{2} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 10 \, a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{7}} - \frac{{\left (16 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{7}} - \frac{b^{5} c x^{2} - a b^{4} d x^{2} - a^{3} b^{2} f x^{2} + a^{2} b^{3} x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{6}} - \frac{9100 \, b^{4} c x^{12} - 7280 \, a b^{3} d x^{12} - 3640 \, a^{3} b f x^{12} + 5460 \, a^{2} b^{2} x^{12} e - 1820 \, a b^{3} c x^{9} + 1365 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 910 \, a^{3} b x^{9} e + 780 \, a^{2} b^{2} c x^{6} - 520 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 364 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{6} x^{13}} \]

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)^2*x^14),x, algorithm="giac")

[Out]

1/9*sqrt(3)*(16*(-a*b^2)^(2/3)*b^3*c - 13*(-a*b^2)^(2/3)*a*b^2*d - 7*(-a*b^2)^(2
/3)*a^3*f + 10*(-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 + 1/9*(16*b^5*c*(-a/b)^(1/3) - 13*a*b^4*d*(-a/b)^(1/3) - 7*a^3*
b^2*f*(-a/b)^(1/3) + 10*a^2*b^3*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(
1/3)))/a^7 - 1/18*(16*(-a*b^2)^(2/3)*b^3*c - 13*(-a*b^2)^(2/3)*a*b^2*d - 7*(-a*b
^2)^(2/3)*a^3*f + 10*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/a^7 - 1/3*(b^5*c*x^2 - a*b^4*d*x^2 - a^3*b^2*f*x^2 + a^2*b^3*x^2*e)/((b*x^3
+ a)*a^6) - 1/1820*(9100*b^4*c*x^12 - 7280*a*b^3*d*x^12 - 3640*a^3*b*f*x^12 + 5
460*a^2*b^2*x^12*e - 1820*a*b^3*c*x^9 + 1365*a^2*b^2*d*x^9 + 455*a^4*f*x^9 - 910
*a^3*b*x^9*e + 780*a^2*b^2*c*x^6 - 520*a^3*b*d*x^6 + 260*a^4*x^6*e - 364*a^3*b*c
*x^3 + 182*a^4*d*x^3 + 140*a^4*c)/(a^6*x^13)