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

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]

-1/14*c/a/x^14+1/11*(-a*d+b*c)/a^2/x^11+1/8*(-a^2*e+a*b*d-b^2*c)/a^3/x^8+1/5*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^
4/x^5-1/2*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^5/x^2-1/3*b^(5/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/
3)*x)/a^(17/3)+1/6*b^(5/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)+1
/3*b^(5/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(17/3)*3^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1834, 200, 31, 634, 617, 204, 628} \[ -\frac {b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^5 x^2}+\frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{17/3}}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{17/3}}+\frac {b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} a^{17/3}}-\frac {a^2 e-a b d+b^2 c}{8 a^3 x^8}+\frac {b c-a d}{11 a^2 x^{11}}-\frac {c}{14 a x^{14}} \]

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))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^{15} \left (a+b x^3\right )} \, dx &=\int \left (\frac {c}{a x^{15}}+\frac {-b c+a d}{a^2 x^{12}}+\frac {b^2 c-a b d+a^2 e}{a^3 x^9}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^6}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^3}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac {c}{14 a x^{14}}+\frac {b c-a d}{11 a^2 x^{11}}-\frac {b^2 c-a b d+a^2 e}{8 a^3 x^8}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{2 a^5 x^2}-\frac {\left (b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{a^5}\\ &=-\frac {c}{14 a x^{14}}+\frac {b c-a d}{11 a^2 x^{11}}-\frac {b^2 c-a b d+a^2 e}{8 a^3 x^8}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{2 a^5 x^2}-\frac {\left (b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{17/3}}-\frac {\left (b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{17/3}}\\ &=-\frac {c}{14 a x^{14}}+\frac {b c-a d}{11 a^2 x^{11}}-\frac {b^2 c-a b d+a^2 e}{8 a^3 x^8}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{2 a^5 x^2}-\frac {b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{17/3}}+\frac {\left (b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{17/3}}-\frac {\left (b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{16/3}}\\ &=-\frac {c}{14 a x^{14}}+\frac {b c-a d}{11 a^2 x^{11}}-\frac {b^2 c-a b d+a^2 e}{8 a^3 x^8}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{2 a^5 x^2}-\frac {b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{17/3}}+\frac {b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{17/3}}-\frac {\left (b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{17/3}}\\ &=-\frac {c}{14 a x^{14}}+\frac {b c-a d}{11 a^2 x^{11}}-\frac {b^2 c-a b d+a^2 e}{8 a^3 x^8}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{2 a^5 x^2}+\frac {b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{17/3}}-\frac {b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{17/3}}+\frac {b^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{17/3}}\\ \end {align*}

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Mathematica [A]  time = 0.15, 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]

-1/14*c/(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[(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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fricas [A]  time = 0.59, size = 335, normalized size = 1.06 \[ -\frac {3080 \, \sqrt {3} {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{14} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 1540 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{14} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 3080 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{14} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 4620 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 1848 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 1155 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 660 \, a^{4} c - 840 \, {\left (a^{3} b c - a^{4} d\right )} x^{3}}{9240 \, a^{5} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^15/(b*x^3+a),x, algorithm="fricas")

[Out]

-1/9240*(3080*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)*arctan(1/3*(2*sqrt(3)*a*x*(
b^2/a^2)^(2/3) - sqrt(3)*b)/b) - 1540*(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*(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)) + 4620*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^12 - 1848*(a*b^3*c - a^2*b
^2*d + a^3*b*e - a^4*f)*x^9 + 1155*(a^2*b^2*c - a^3*b*d + a^4*e)*x^6 + 660*a^4*c - 840*(a^3*b*c - a^4*d)*x^3)/
(a^5*x^14)

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giac [A]  time = 0.19, size = 393, normalized size = 1.25 \[ -\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{6}} + \frac {{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{6}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{6}} - \frac {1540 \, b^{4} c x^{12} - 1540 \, a b^{3} d x^{12} - 1540 \, a^{3} b f x^{12} + 1540 \, a^{2} b^{2} x^{12} e - 616 \, a b^{3} c x^{9} + 616 \, a^{2} b^{2} d x^{9} + 616 \, a^{4} f x^{9} - 616 \, a^{3} b x^{9} e + 385 \, a^{2} b^{2} c x^{6} - 385 \, a^{3} b d x^{6} + 385 \, a^{4} x^{6} e - 280 \, a^{3} b c x^{3} + 280 \, a^{4} d x^{3} + 220 \, a^{4} c}{3080 \, a^{5} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^15/(b*x^3+a),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)*log(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)*log(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)

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maple [B]  time = 0.06, size = 548, normalized size = 1.74 \[ \frac {\sqrt {3}\, b f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {b f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {b f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {\sqrt {3}\, b^{2} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {b^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {b^{2} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {\sqrt {3}\, b^{3} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {b^{3} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {b^{3} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {\sqrt {3}\, b^{4} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}-\frac {b^{4} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}+\frac {b^{4} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}+\frac {b f}{2 a^{2} x^{2}}-\frac {b^{2} e}{2 a^{3} x^{2}}+\frac {b^{3} d}{2 a^{4} x^{2}}-\frac {b^{4} c}{2 a^{5} x^{2}}-\frac {f}{5 a \,x^{5}}+\frac {b e}{5 a^{2} x^{5}}-\frac {b^{2} d}{5 a^{3} x^{5}}+\frac {b^{3} c}{5 a^{4} x^{5}}-\frac {e}{8 a \,x^{8}}+\frac {b d}{8 a^{2} x^{8}}-\frac {b^{2} c}{8 a^{3} x^{8}}-\frac {d}{11 a \,x^{11}}+\frac {b c}{11 a^{2} x^{11}}-\frac {c}{14 a \,x^{14}} \]

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

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maxima [A]  time = 3.11, size = 307, normalized size = 0.97 \[ -\frac {\sqrt {3} {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1540 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 616 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 385 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 220 \, a^{4} c - 280 \, {\left (a^{3} b c - a^{4} d\right )} x^{3}}{3080 \, a^{5} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^15/(b*x^3+a),x, algorithm="maxima")

[Out]

-1/3*sqrt(3)*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^5*
(a/b)^(2/3)) + 1/6*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(
2/3)) - 1/3*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*log(x + (a/b)^(1/3))/(a^5*(a/b)^(2/3)) - 1/3080*(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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mupad [B]  time = 5.17, size = 287, normalized size = 0.91 \[ -\frac {\frac {c}{14\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{5\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{11\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{8\,a^3}+\frac {b\,x^{12}\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{2\,a^5}}{x^{14}}-\frac {b^{5/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{17/3}}-\frac {b^{5/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{17/3}}+\frac {b^{5/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{17/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^15*(a + b*x^3)),x)

[Out]

(b^(5/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b^3*c - a^3*f - a*b^2*d + a^2
*b*e))/(3*a^(17/3)) - (b^(5/3)*log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(17/3)) - (b
^(5/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(b^3*c - a^3*f - a*b^2*d + a^2*b
*e))/(3*a^(17/3)) - (c/(14*a) - (x^9*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(5*a^4) + (x^3*(a*d - b*c))/(11*a^2)
+ (x^6*(b^2*c + a^2*e - a*b*d))/(8*a^3) + (b*x^12*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(2*a^5))/x^14

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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