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

Optimal. Leaf size=313 \[ \frac{b c-a d}{10 a^2 x^{10}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{16/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{16/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{16/3}}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5 x}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{4 a^4 x^4}-\frac{c}{13 a x^{13}} \]

[Out]

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

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Rubi [A]  time = 0.559978, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{b c-a d}{10 a^2 x^{10}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{16/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{16/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{16/3}}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5 x}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{4 a^4 x^4}-\frac{c}{13 a x^{13}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 101.679, size = 291, normalized size = 0.93 \[ - \frac{c}{13 a x^{13}} - \frac{a d - b c}{10 a^{2} x^{10}} - \frac{a^{2} e - a b d + b^{2} c}{7 a^{3} x^{7}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{4 a^{4} x^{4}} + \frac{b \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} x} - \frac{b^{\frac{4}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{16}{3}}} + \frac{b^{\frac{4}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{16}{3}}} - \frac{\sqrt{3} b^{\frac{4}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{16}{3}}} \]

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),x)

[Out]

-c/(13*a*x**13) - (a*d - b*c)/(10*a**2*x**10) - (a**2*e - a*b*d + b**2*c)/(7*a**
3*x**7) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(4*a**4*x**4) + b*(a**3*f - a*
*2*b*e + a*b**2*d - b**3*c)/(a**5*x) - b**(4/3)*(a**3*f - a**2*b*e + a*b**2*d -
b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(16/3)) + b**(4/3)*(a**3*f - a**2*b*e +
a*b**2*d - b**3*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(1
6/3)) - sqrt(3)*b**(4/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*atan(sqrt(3)*(a
**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(16/3))

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Mathematica [A]  time = 0.182712, size = 308, normalized size = 0.98 \[ \frac{b c-a d}{10 a^2 x^{10}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 a^{16/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{16/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{16/3}}+\frac{b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^5 x}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{4 a^4 x^4}-\frac{c}{13 a x^{13}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 546, 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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218643, size = 464, normalized size = 1.48 \[ \frac{\sqrt{3}{\left (910 \, \sqrt{3}{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{13} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 1820 \, \sqrt{3}{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{13} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 5460 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{13} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (1820 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 455 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 260 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 140 \, a^{4} c - 182 \,{\left (a^{3} b c - a^{4} d\right )} x^{3}\right )}\right )}}{16380 \, a^{5} x^{13}} \]

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)*x^14),x, algorithm="fricas")

[Out]

1/16380*sqrt(3)*(910*sqrt(3)*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*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)*(b^4*c - a*
b^3*d + a^2*b^2*e - a^3*b*f)*x^13*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) - 5460*
(b^4*c - a*b^3*d + a^2*b^2*e - a^3*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*(b^4*c - a*b^
3*d + a^2*b^2*e - a^3*b*f)*x^12 - 455*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^
9 + 260*(a^2*b^2*c - a^3*b*d + a^4*e)*x^6 + 140*a^4*c - 182*(a^3*b*c - a^4*d)*x^
3))/(a^5*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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220513, size = 566, normalized size = 1.81 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{6}} + \frac{{\left (b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{2} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{6}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{6}} - \frac{1820 \, b^{4} c x^{12} - 1820 \, a b^{3} d x^{12} - 1820 \, a^{3} b f x^{12} + 1820 \, a^{2} b^{2} x^{12} e - 455 \, a b^{3} c x^{9} + 455 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 455 \, a^{3} b x^{9} e + 260 \, a^{2} b^{2} c x^{6} - 260 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 182 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{5} x^{13}} \]

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)*x^14),x, algorithm="giac")

[Out]

1/3*sqrt(3)*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*
f + (-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^6 + 1/3*(b^5*c*(-a/b)^(1/3) - a*b^4*d*(-a/b)^(1/3) - a^3*b^2*f*(-a/b)^(1/3)
+ a^2*b^3*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^6 - 1/6*((-a*
b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3
)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^6 - 1/1820*(1820*b^4*c*x^12
- 1820*a*b^3*d*x^12 - 1820*a^3*b*f*x^12 + 1820*a^2*b^2*x^12*e - 455*a*b^3*c*x^9
+ 455*a^2*b^2*d*x^9 + 455*a^4*f*x^9 - 455*a^3*b*x^9*e + 260*a^2*b^2*c*x^6 - 260
*a^3*b*d*x^6 + 260*a^4*x^6*e - 182*a^3*b*c*x^3 + 182*a^4*d*x^3 + 140*a^4*c)/(a^5
*x^13)