3.250 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=351 \[ \frac{b c-a d}{13 a^2 x^{13}}-\frac{a^2 e-a b d+b^2 c}{10 a^3 x^{10}}+\frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{19/3}}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{19/3}}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{19/3}}+\frac{b^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6 x}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 x^4}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{7 a^4 x^7}-\frac{c}{16 a x^{16}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.599039, antiderivative size = 351, 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 \[ \frac{b c-a d}{13 a^2 x^{13}}-\frac{a^2 e-a b d+b^2 c}{10 a^3 x^{10}}+\frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{19/3}}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{19/3}}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{19/3}}+\frac{b^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6 x}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 x^4}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{7 a^4 x^7}-\frac{c}{16 a x^{16}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^17*(a + b*x^3)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 155.236, size = 326, normalized size = 0.93 \[ - \frac{c}{16 a x^{16}} - \frac{a d - b c}{13 a^{2} x^{13}} - \frac{a^{2} e - a b d + b^{2} c}{10 a^{3} x^{10}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{7 a^{4} x^{7}} + \frac{b \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{4 a^{5} x^{4}} - \frac{b^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{6} x} + \frac{b^{\frac{7}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{19}{3}}} - \frac{b^{\frac{7}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{19}{3}}} + \frac{\sqrt{3} b^{\frac{7}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{19}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**17/(b*x**3+a),x)

[Out]

-c/(16*a*x**16) - (a*d - b*c)/(13*a**2*x**13) - (a**2*e - a*b*d + b**2*c)/(10*a*
*3*x**10) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(7*a**4*x**7) + b*(a**3*f -
a**2*b*e + a*b**2*d - b**3*c)/(4*a**5*x**4) - b**2*(a**3*f - a**2*b*e + a*b**2*d
- b**3*c)/(a**6*x) + b**(7/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1
/3) + b**(1/3)*x)/(3*a**(19/3)) - b**(7/3)*(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**(19/3)) + sqrt(3)*b
**(7/3)*(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**(19/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.203014, size = 346, normalized size = 0.99 \[ \frac{b c-a d}{13 a^2 x^{13}}-\frac{a^2 e-a b d+b^2 c}{10 a^3 x^{10}}+\frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{19/3}}+\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 a^{19/3}}+\frac{b^{7/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt{3} a^{19/3}}+\frac{b^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 a^5 x^4}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{7 a^4 x^7}-\frac{c}{16 a x^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^17*(a + b*x^3)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 600, 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^17/(b*x^3+a),x)

[Out]

1/3*b^2/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-1/3*b^4/a^5*3^(1/2)/(a/b)^(1/3)*arct
an(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+1/3*b^5/a^6*3^(1/2)/(a/b)^(1/3)*arctan(1/3
*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c-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))*f+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))*e-1/7/a/x^7*f-1/13/a/x^13*d-1/10/a/x^10*e+1/3*b^4/a^5/(a/b)^(1
/3)*ln(x+(a/b)^(1/3))*d-1/3*b^5/a^6/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-1/6*b^2/a^3/
(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/6*b^3/a^4/(a/b)^(1/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))*e-1/6*b^4/a^5/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))*d+1/6*b^5/a^6/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+1/a^6*b^5/x*
c+1/4/a^2*b/x^4*f-1/4/a^3*b^2/x^4*e+1/4/a^4*b^3/x^4*d-1/4/a^5*b^4/x^4*c+1/13/a^2
/x^13*b*c+1/10/a^2/x^10*b*d-1/10/a^3/x^10*b^2*c+1/7/a^2/x^7*b*e-1/7/a^3/x^7*b^2*
d+1/7/a^4/x^7*b^3*c-1/a^3*b^2/x*f+1/a^4*b^3/x*e-1/a^5*b^4/x*d-1/3*b^3/a^4/(a/b)^
(1/3)*ln(x+(a/b)^(1/3))*e-1/16*c/a/x^16

_______________________________________________________________________________________

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^17),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.219368, size = 514, normalized size = 1.46 \[ \frac{\sqrt{3}{\left (3640 \, \sqrt{3}{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{16} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 7280 \, \sqrt{3}{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{16} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 21840 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{16} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (7280 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} - 1820 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 1040 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{9} - 728 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{6} - 455 \, a^{5} c + 560 \,{\left (a^{4} b c - a^{5} d\right )} x^{3}\right )}\right )}}{65520 \, a^{6} x^{16}} \]

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^17),x, algorithm="fricas")

[Out]

1/65520*sqrt(3)*(3640*sqrt(3)*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^16*(b/
a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 7280*sqrt(3)*(b^5*c - a*
b^4*d + a^2*b^3*e - a^3*b^2*f)*x^16*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) - 21840
*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^16*(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)*(7280*(b^5*c - a*b^
4*d + a^2*b^3*e - a^3*b^2*f)*x^15 - 1820*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*
b*f)*x^12 + 1040*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^9 - 728*(a^3*b^2*c
- a^4*b*d + a^5*e)*x^6 - 455*a^5*c + 560*(a^4*b*c - a^5*d)*x^3))/(a^6*x^16)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218915, size = 640, normalized size = 1.82 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{3} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} b f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b^{2} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{7}} - \frac{{\left (b^{6} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{5} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{4} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{7}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{3} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} b f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b^{2} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{7}} + \frac{7280 \, b^{5} c x^{15} - 7280 \, a b^{4} d x^{15} - 7280 \, a^{3} b^{2} f x^{15} + 7280 \, a^{2} b^{3} x^{15} e - 1820 \, a b^{4} c x^{12} + 1820 \, a^{2} b^{3} d x^{12} + 1820 \, a^{4} b f x^{12} - 1820 \, a^{3} b^{2} x^{12} e + 1040 \, a^{2} b^{3} c x^{9} - 1040 \, a^{3} b^{2} d x^{9} - 1040 \, a^{5} f x^{9} + 1040 \, a^{4} b x^{9} e - 728 \, a^{3} b^{2} c x^{6} + 728 \, a^{4} b d x^{6} - 728 \, a^{5} x^{6} e + 560 \, a^{4} b c x^{3} - 560 \, a^{5} d x^{3} - 455 \, a^{5} c}{7280 \, a^{6} x^{16}} \]

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^17),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(2/3)*b^4*c - (-a*b^2)^(2/3)*a*b^3*d - (-a*b^2)^(2/3)*a^3
*b*f + (-a*b^2)^(2/3)*a^2*b^2*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^
(1/3))/a^7 - 1/3*(b^6*c*(-a/b)^(1/3) - a*b^5*d*(-a/b)^(1/3) - a^3*b^3*f*(-a/b)^(
1/3) + a^2*b^4*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^7 + 1/6*
((-a*b^2)^(2/3)*b^4*c - (-a*b^2)^(2/3)*a*b^3*d - (-a*b^2)^(2/3)*a^3*b*f + (-a*b^
2)^(2/3)*a^2*b^2*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^7 + 1/7280*(7280*b
^5*c*x^15 - 7280*a*b^4*d*x^15 - 7280*a^3*b^2*f*x^15 + 7280*a^2*b^3*x^15*e - 1820
*a*b^4*c*x^12 + 1820*a^2*b^3*d*x^12 + 1820*a^4*b*f*x^12 - 1820*a^3*b^2*x^12*e +
1040*a^2*b^3*c*x^9 - 1040*a^3*b^2*d*x^9 - 1040*a^5*f*x^9 + 1040*a^4*b*x^9*e - 72
8*a^3*b^2*c*x^6 + 728*a^4*b*d*x^6 - 728*a^5*x^6*e + 560*a^4*b*c*x^3 - 560*a^5*d*
x^3 - 455*a^5*c)/(a^6*x^16)