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

Optimal. Leaf size=341 \[ \frac {3 b c-a d}{5 a^4 x^5}-\frac {c}{8 a^3 x^8}-\frac {a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{9 \sqrt {3} a^{17/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac {x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]

[Out]

-1/8*c/a^3/x^8+1/5*(-a*d+3*b*c)/a^4/x^5+1/2*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x^2-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c
)*x/a^4/(b*x^3+a)^2-1/18*(-5*a^3*f+11*a^2*b*e-17*a*b^2*d+23*b^3*c)*x/a^5/(b*x^3+a)-1/27*(-5*a^3*f+20*a^2*b*e-4
4*a*b^2*d+77*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(17/3)/b^(1/3)+1/54*(-5*a^3*f+20*a^2*b*e-44*a*b^2*d+77*b^3*c)*ln(a
^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)/b^(1/3)+1/27*(-5*a^3*f+20*a^2*b*e-44*a*b^2*d+77*b^3*c)*arctan(1
/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(17/3)/b^(1/3)*3^(1/2)

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Rubi [A]  time = 0.55, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1829, 1834, 200, 31, 634, 617, 204, 628} \[ -\frac {x \left (11 a^2 b e-5 a^3 f-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{9 \sqrt {3} a^{17/3} \sqrt [3]{b}}-\frac {a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {c}{8 a^3 x^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(8*a^3*x^8) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(2*a^5*x^2) - ((b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x)/(6*a^4*(a + b*x^3)^2) - ((23*b^3*c - 17*a*b^2*d + 11*a^2*b*e - 5*a^3*f)*x)/(18*a^5*(a + b*x^3
)) + ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqr
t[3]*a^(17/3)*b^(1/3)) - ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(17/3
)*b^(1/3)) + ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
54*a^(17/3)*b^(1/3))

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 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

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^9 \left (a+b x^3\right )^3} \, dx &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^3 c+6 b^3 \left (\frac {b c}{a}-d\right ) x^3-\frac {6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac {5 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\int \frac {18 b^6 c-18 b^6 \left (\frac {2 b c}{a}-d\right ) x^3+18 b^6 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^6-\frac {2 b^6 \left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )} \, dx}{18 a^2 b^6}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^6 c}{a x^9}+\frac {18 b^6 (-3 b c+a d)}{a^2 x^6}+\frac {18 b^6 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^3}+\frac {2 b^6 \left (-77 b^3 c+44 a b^2 d-20 a^2 b e+5 a^3 f\right )}{a^3 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^6}\\ &=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{9 a^5}\\ &=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{17/3}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\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}{27 a^{17/3}}\\ &=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{16/3}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\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}{54 a^{17/3} \sqrt [3]{b}}\\ &=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{17/3} \sqrt [3]{b}}\\ &=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 324, normalized size = 0.95 \[ \frac {-\frac {216 a^{5/3} (a d-3 b c)}{x^5}-\frac {135 a^{8/3} c}{x^8}-\frac {540 a^{2/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^2}+\frac {40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-20 a^2 b e+44 a b^2 d-77 b^3 c\right )}{\sqrt [3]{b}}+\frac {40 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}+\frac {180 a^{5/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac {60 a^{2/3} x \left (5 a^3 f-11 a^2 b e+17 a b^2 d-23 b^3 c\right )}{a+b x^3}+\frac {20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}}{1080 a^{17/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-135*a^(8/3)*c)/x^8 - (216*a^(5/3)*(-3*b*c + a*d))/x^5 - (540*a^(2/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x^2 + (18
0*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3)^2 + (60*a^(2/3)*(-23*b^3*c + 17*a*b^2*d - 11*a
^2*b*e + 5*a^3*f)*x)/(a + b*x^3) + (40*Sqrt[3]*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b
^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-77*b^3*c + 44*a*b^2*d - 20*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/
3)*x])/b^(1/3) + (20*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/b^(1/3))/(1080*a^(17/3))

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fricas [B]  time = 0.62, size = 1317, normalized size = 3.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/1080*(60*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^12 + 96*(77*a^3*b^4*c - 44*a^4*b^3*d
+ 20*a^5*b^2*e - 5*a^6*b*f)*x^9 + 135*a^6*b*c + 27*(77*a^4*b^3*c - 44*a^5*b^2*d + 20*a^6*b*e)*x^6 - 54*(7*a^5
*b^2*c - 4*a^6*b*d)*x^3 + 60*sqrt(1/3)*((77*a*b^6*c - 44*a^2*b^5*d + 20*a^3*b^4*e - 5*a^4*b^3*f)*x^14 + 2*(77*
a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^11 + (77*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*
a^6*b*f)*x^8)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^
2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 20*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3
*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*b^3*c - 44*a^
3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*((77*
b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*
f)*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/
(a^7*b^3*x^14 + 2*a^8*b^2*x^11 + a^9*b*x^8), -1/1080*(60*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b
^2*f)*x^12 + 96*(77*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^9 + 135*a^6*b*c + 27*(77*a^4*b^3*c
- 44*a^5*b^2*d + 20*a^6*b*e)*x^6 - 54*(7*a^5*b^2*c - 4*a^6*b*d)*x^3 + 120*sqrt(1/3)*((77*a*b^6*c - 44*a^2*b^5*
d + 20*a^3*b^4*e - 5*a^4*b^3*f)*x^14 + 2*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^11 + (77
*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^8)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(
2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 20*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f
)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d + 20*a^4
*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*((77*b^5*c - 44*a*b^4
*d + 20*a^2*b^3*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^
2*b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^7*b^3*x^14 +
2*a^8*b^2*x^11 + a^9*b*x^8)]

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giac [A]  time = 0.23, size = 394, normalized size = 1.16 \[ \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d - 5 \, a^{3} f + 20 \, a^{2} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{6}} - \frac {\sqrt {3} {\left (77 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} b} - \frac {{\left (77 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} b} - \frac {23 \, b^{4} c x^{4} - 17 \, a b^{3} d x^{4} - 5 \, a^{3} b f x^{4} + 11 \, a^{2} b^{2} x^{4} e + 26 \, a b^{3} c x - 20 \, a^{2} b^{2} d x - 8 \, a^{4} f x + 14 \, a^{3} b x e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{5}} - \frac {120 \, b^{2} c x^{6} - 60 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 24 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{5} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(77*b^3*c - 44*a*b^2*d - 5*a^3*f + 20*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 - 1/27*sqrt(3)
*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*b^2)^(1/3)*a*b^2*d - 5*(-a*b^2)^(1/3)*a^3*f + 20*(-a*b^2)^(1/3)*a^2*b*e)*ar
ctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b) - 1/54*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*b^2)^(1/3
)*a*b^2*d - 5*(-a*b^2)^(1/3)*a^3*f + 20*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^6*
b) - 1/18*(23*b^4*c*x^4 - 17*a*b^3*d*x^4 - 5*a^3*b*f*x^4 + 11*a^2*b^2*x^4*e + 26*a*b^3*c*x - 20*a^2*b^2*d*x -
8*a^4*f*x + 14*a^3*b*x*e)/((b*x^3 + a)^2*a^5) - 1/40*(120*b^2*c*x^6 - 60*a*b*d*x^6 + 20*a^2*x^6*e - 24*a*b*c*x
^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^5*x^8)

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maple [B]  time = 0.06, size = 603, normalized size = 1.77 \[ \frac {5 b f \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {11 b^{2} e \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {17 b^{3} d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{4}}-\frac {23 b^{4} c \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{5}}+\frac {4 f x}{9 \left (b \,x^{3}+a \right )^{2} a}-\frac {7 b e x}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {10 b^{2} d x}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {13 b^{3} c x}{9 \left (b \,x^{3}+a \right )^{2} a^{4}}+\frac {5 \sqrt {3}\, f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {5 f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {5 f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {20 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {20 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {10 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {44 \sqrt {3}\, b d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {44 b d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {22 b d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {77 \sqrt {3}\, b^{2} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}-\frac {77 b^{2} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}+\frac {77 b^{2} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}-\frac {e}{2 a^{3} x^{2}}+\frac {3 b d}{2 a^{4} x^{2}}-\frac {3 b^{2} c}{a^{5} x^{2}}-\frac {d}{5 a^{3} x^{5}}+\frac {3 b c}{5 a^{4} x^{5}}-\frac {c}{8 a^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/27/a^3*e/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-3/a^5/x^2*b^2*c+3/5/a^4/x^5*b*c+4/9/a/(b*x^3+a)^2*f*
x-20/27/a^3*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+3/2/a^4/x^2*b*d-1/2/a^3/x^2*e+5/27/a^2*f/b/(a/b)^(2/3)*3^(1/2)*arc
tan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+44/27/a^4*b*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))
-77/27/a^5*b^2*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/5/a^3/x^5*d+5/18/a^2/(b*x^3+a)^
2*x^4*b*f-11/18/a^3/(b*x^3+a)^2*x^4*b^2*e+17/18/a^4/(b*x^3+a)^2*x^4*b^3*d-23/18/a^5/(b*x^3+a)^2*x^4*b^4*c-20/2
7/a^3*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+44/27/a^4*b*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3)
)-22/27/a^4*b*d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-77/27/a^5*b^2*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+77
/54/a^5*b^2*c/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+5/27/a^2*f/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-5/54/a^
2*f/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+10/9/a^3/(b*x^3+a)^2*b^2*d*x-13/9/a^4/(b*x^3+a)^2*b^3*c*x-
7/9/a^2/(b*x^3+a)^2*b*e*x-1/8*c/a^3/x^8

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maxima [A]  time = 2.95, size = 343, normalized size = 1.01 \[ -\frac {20 \, {\left (77 \, b^{4} c - 44 \, a b^{3} d + 20 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 32 \, {\left (77 \, a b^{3} c - 44 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} + 9 \, {\left (77 \, a^{2} b^{2} c - 44 \, a^{3} b d + 20 \, a^{4} e\right )} x^{6} + 45 \, a^{4} c - 18 \, {\left (7 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}}{360 \, {\left (a^{5} b^{2} x^{14} + 2 \, a^{6} b x^{11} + a^{7} x^{8}\right )}} - \frac {\sqrt {3} {\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360*(20*(77*b^4*c - 44*a*b^3*d + 20*a^2*b^2*e - 5*a^3*b*f)*x^12 + 32*(77*a*b^3*c - 44*a^2*b^2*d + 20*a^3*b*
e - 5*a^4*f)*x^9 + 9*(77*a^2*b^2*c - 44*a^3*b*d + 20*a^4*e)*x^6 + 45*a^4*c - 18*(7*a^3*b*c - 4*a^4*d)*x^3)/(a^
5*b^2*x^14 + 2*a^6*b*x^11 + a^7*x^8) - 1/27*sqrt(3)*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*arctan(1/3*
sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^5*b*(a/b)^(2/3)) + 1/54*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^
3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*b*(a/b)^(2/3)) - 1/27*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5
*a^3*f)*log(x + (a/b)^(1/3))/(a^5*b*(a/b)^(2/3))

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mupad [B]  time = 5.22, size = 321, normalized size = 0.94 \[ -\frac {\frac {c}{8\,a}+\frac {4\,x^9\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{45\,a^4}+\frac {x^3\,\left (4\,a\,d-7\,b\,c\right )}{20\,a^2}+\frac {x^6\,\left (20\,e\,a^2-44\,d\,a\,b+77\,c\,b^2\right )}{40\,a^3}+\frac {b\,x^{12}\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{18\,a^5}}{a^2\,x^8+2\,a\,b\,x^{11}+b^2\,x^{14}}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*
a^2*b*e))/(27*a^(17/3)*b^(1/3)) - (log(b^(1/3)*x + a^(1/3))*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(2
7*a^(17/3)*b^(1/3)) - (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(77*b^3*c - 5*a^
3*f - 44*a*b^2*d + 20*a^2*b*e))/(27*a^(17/3)*b^(1/3)) - (c/(8*a) + (4*x^9*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 2
0*a^2*b*e))/(45*a^4) + (x^3*(4*a*d - 7*b*c))/(20*a^2) + (x^6*(77*b^2*c + 20*a^2*e - 44*a*b*d))/(40*a^3) + (b*x
^12*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(18*a^5))/(a^2*x^8 + b^2*x^14 + 2*a*b*x^11)

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

[Out]

Timed out

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