3.249 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{15} \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=315 \[ \frac{b c-a d}{11 a^2 x^{11}}-\frac{a^2 e-a b d+b^2 c}{8 a^3 x^8}+\frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{17/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{17/3}}+\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{17/3}}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^5 x^2}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac{c}{14 a x^{14}} \]

[Out]

-c/(14*a*x^14) + (b*c - a*d)/(11*a^2*x^11) - (b^2*c - a*b*d + a^2*e)/(8*a^3*x^8)
+ (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) - (b*(b^3*c - a*b^2*d + a^2*b
*e - a^3*f))/(2*a^5*x^2) + (b^(5/3)*(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^(17/3)) - (b^(5/3)*(b^3*c
- a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(17/3)) + (b^(5/3)*(
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^(17/3))

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Rubi [A]  time = 0.514963, antiderivative size = 315, 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 \[ \frac{b c-a d}{11 a^2 x^{11}}-\frac{a^2 e-a b d+b^2 c}{8 a^3 x^8}+\frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{17/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{17/3}}+\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{17/3}}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^5 x^2}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac{c}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^15*(a + b*x^3)),x]

[Out]

-c/(14*a*x^14) + (b*c - a*d)/(11*a^2*x^11) - (b^2*c - a*b*d + a^2*e)/(8*a^3*x^8)
+ (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) - (b*(b^3*c - a*b^2*d + a^2*b
*e - a^3*f))/(2*a^5*x^2) + (b^(5/3)*(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^(17/3)) - (b^(5/3)*(b^3*c
- a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(17/3)) + (b^(5/3)*(
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^(17/3))

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Rubi in Sympy [A]  time = 113.806, size = 294, normalized size = 0.93 \[ - \frac{c}{14 a x^{14}} - \frac{a d - b c}{11 a^{2} x^{11}} - \frac{a^{2} e - a b d + b^{2} c}{8 a^{3} x^{8}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{5 a^{4} x^{5}} + \frac{b \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 a^{5} x^{2}} + \frac{b^{\frac{5}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{17}{3}}} - \frac{b^{\frac{5}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{17}{3}}} - \frac{\sqrt{3} b^{\frac{5}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{17}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**15/(b*x**3+a),x)

[Out]

-c/(14*a*x**14) - (a*d - b*c)/(11*a**2*x**11) - (a**2*e - a*b*d + b**2*c)/(8*a**
3*x**8) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(5*a**4*x**5) + b*(a**3*f - a*
*2*b*e + a*b**2*d - b**3*c)/(2*a**5*x**2) + b**(5/3)*(a**3*f - a**2*b*e + a*b**2
*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(17/3)) - b**(5/3)*(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**(17/3)) - sqrt(3)*b**(5/3)*(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**(17/3))

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Mathematica [A]  time = 0.188541, size = 311, normalized size = 0.99 \[ \frac{b c-a d}{11 a^2 x^{11}}-\frac{a^2 e-a b d+b^2 c}{8 a^3 x^8}+\frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{17/3}}+\frac{b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 a^{17/3}}+\frac{b^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{17/3}}+\frac{b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^5 x^2}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac{c}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^15*(a + b*x^3)),x]

[Out]

-c/(14*a*x^14) + (b*c - a*d)/(11*a^2*x^11) - (b^2*c - a*b*d + a^2*e)/(8*a^3*x^8)
+ (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) + (b*(-(b^3*c) + a*b^2*d - a^
2*b*e + a^3*f))/(2*a^5*x^2) + (b^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTa
n[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(17/3)) + (b^(5/3)*(-(b^3*c)
+ a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(17/3)) + (b^(5/3)*(
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^(17/3))

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Maple [B]  time = 0.012, size = 548, 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^15/(b*x^3+a),x)

[Out]

1/3*b/a^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-1/3*b^2/
a^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+1/3*b^3/a^4/(a
/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-1/3*b^4/a^5/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c+1/3*b/a^2/(a/b)^(2/3)*ln(x
+(a/b)^(1/3))*f-1/3*b^2/a^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e+1/3*b^3/a^4/(a/b)^(2
/3)*ln(x+(a/b)^(1/3))*d-1/3*b^4/a^5/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/6*b/a^2/(a
/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/6*b^2/a^3/(a/b)^(2/3)*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))*e-1/6*b^3/a^4/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/
3))*d+1/5/a^2/x^5*b*e-1/5/a^3/x^5*b^2*d+1/5/a^4/x^5*b^3*c+1/2/a^2*b/x^2*f-1/2/a^
3*b^2/x^2*e+1/2/a^4*b^3/x^2*d-1/2/a^5*b^4/x^2*c+1/11/a^2/x^11*b*c+1/8/a^2/x^8*b*
d-1/8/a^3/x^8*b^2*c+1/6*b^4/a^5/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-
1/8/a/x^8*e-1/5/a/x^5*f-1/11/a/x^11*d-1/14*c/a/x^14

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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^15),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223861, size = 481, normalized size = 1.53 \[ \frac{\sqrt{3}{\left (1540 \, \sqrt{3}{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{14} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 3080 \, \sqrt{3}{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{14} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 9240 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{14} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (1540 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 616 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 385 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 220 \, a^{4} c - 280 \,{\left (a^{3} b c - a^{4} d\right )} x^{3}\right )}\right )}}{27720 \, a^{5} x^{14}} \]

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^15),x, algorithm="fricas")

[Out]

1/27720*sqrt(3)*(1540*sqrt(3)*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^14*(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)) - 3080*sqr
t(3)*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^14*(b^2/a^2)^(1/3)*log(b*x + a*(b
^2/a^2)^(1/3)) + 9240*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^14*(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)*(1540*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^12 - 616*(a*b^3*c -
a^2*b^2*d + a^3*b*e - a^4*f)*x^9 + 385*(a^2*b^2*c - a^3*b*d + a^4*e)*x^6 + 220*a
^4*c - 280*(a^3*b*c - a^4*d)*x^3))/(a^5*x^14)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217412, size = 531, normalized size = 1.69 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{6}} + \frac{{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{6}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{6}} - \frac{1540 \, b^{4} c x^{12} - 1540 \, a b^{3} d x^{12} - 1540 \, a^{3} b f x^{12} + 1540 \, a^{2} b^{2} x^{12} e - 616 \, a b^{3} c x^{9} + 616 \, a^{2} b^{2} d x^{9} + 616 \, a^{4} f x^{9} - 616 \, a^{3} b x^{9} e + 385 \, a^{2} b^{2} c x^{6} - 385 \, a^{3} b d x^{6} + 385 \, a^{4} x^{6} e - 280 \, a^{3} b c x^{3} + 280 \, a^{4} d x^{3} + 220 \, a^{4} c}{3080 \, a^{5} x^{14}} \]

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^15),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*b^4*c - (-a*b^2)^(1/3)*a*b^3*d - (-a*b^2)^(1/3)*a^3
*b*f + (-a*b^2)^(1/3)*a^2*b^2*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^4*d - a^3*b^2*f + a^2*b^3*e)*(-a/b)^(1/3)*ln(abs(x
- (-a/b)^(1/3)))/a^6 - 1/6*((-a*b^2)^(1/3)*b^4*c - (-a*b^2)^(1/3)*a*b^3*d - (-a
*b^2)^(1/3)*a^3*b*f + (-a*b^2)^(1/3)*a^2*b^2*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)
^(2/3))/a^6 - 1/3080*(1540*b^4*c*x^12 - 1540*a*b^3*d*x^12 - 1540*a^3*b*f*x^12 +
1540*a^2*b^2*x^12*e - 616*a*b^3*c*x^9 + 616*a^2*b^2*d*x^9 + 616*a^4*f*x^9 - 616*
a^3*b*x^9*e + 385*a^2*b^2*c*x^6 - 385*a^3*b*d*x^6 + 385*a^4*x^6*e - 280*a^3*b*c*
x^3 + 280*a^4*d*x^3 + 220*a^4*c)/(a^5*x^14)