\(\int \frac {x^9 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\) [288]

Optimal result

Integrand size = 30, antiderivative size = 375 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^{10}}{10 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{19/3}}-\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}} \]

[Out]

(-10*a^3*f+6*a^2*b*e-3*a*b^2*d+b^3*c)*x/b^6+1/4*(6*a^2*f-3*a*b*e+b^2*d)*x^4/b^5+1/7*(-3*a*f+b*e)*x^7/b^4+1/10*
f*x^10/b^3-1/6*a^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^6/(b*x^3+a)^2+1/18*a*(-31*a^3*f+25*a^2*b*e-19*a*b^2*d+13
*b^3*c)*x/b^6/(b*x^3+a)-1/27*a^(1/3)*(-104*a^3*f+65*a^2*b*e-35*a*b^2*d+14*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/b^(19/3
)+1/54*a^(1/3)*(-104*a^3*f+65*a^2*b*e-35*a*b^2*d+14*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(19/3)+
1/27*a^(1/3)*(-104*a^3*f+65*a^2*b*e-35*a*b^2*d+14*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/b^(
19/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1842, 1872, 1901, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{9 \sqrt {3} b^{19/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac {a x \left (-31 a^3 f+25 a^2 b e-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac {x \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{b^6}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac {x^7 (b e-3 a f)}{7 b^4}+\frac {f x^{10}}{10 b^3} \]

[In]

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

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^4)/(4*b^5) + ((b*e - 3*a*f
)*x^7)/(7*b^4) + (f*x^10)/(10*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^6*(a + b*x^3)^2) + (a*(1
3*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(18*b^6*(a + b*x^3)) + (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65*a
^2*b*e - 104*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(19/3)) - (a^(1/3)*(14*b^3
*c - 35*a*b^2*d + 65*a^2*b*e - 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(19/3)) + (a^(1/3)*(14*b^3*c - 35*a*
b^2*d + 65*a^2*b*e - 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(19/3))

Rule 31

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

Rule 206

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 210

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

Rule 631

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 642

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 648

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 1842

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(
q - 1)/n] + 1))), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m,
0]

Rule 1872

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1901

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}-\frac {\int \frac {-a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+6 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-6 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^9-6 a b^4 (b e-a f) x^{12}-6 a b^5 f x^{15}}{\left (a+b x^3\right )^2} \, dx}{6 a b^6} \\ & = -\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac {\int \frac {-2 a^3 b^5 \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )+18 a^2 b^6 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+18 a^2 b^7 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+18 a^2 b^8 (b e-2 a f) x^9+18 a^2 b^9 f x^{12}}{a+b x^3} \, dx}{18 a^2 b^{11}} \\ & = -\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac {\int \left (18 a^2 b^5 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right )+18 a^2 b^6 \left (b^2 d-3 a b e+6 a^2 f\right ) x^3+18 a^2 b^7 (b e-3 a f) x^6+18 a^2 b^8 f x^9+\frac {2 \left (-14 a^3 b^8 c+35 a^4 b^7 d-65 a^5 b^6 e+104 a^6 b^5 f\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^{11}} \\ & = \frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^{10}}{10 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac {\left (a \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{9 b^6} \\ & = \frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^{10}}{10 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac {\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^6}-\frac {\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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}{27 b^6} \\ & = \frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^{10}}{10 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac {\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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}{54 b^{19/3}}-\frac {\left (a^{2/3} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^6} \\ & = \frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^{10}}{10 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}}-\frac {\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 b^{19/3}} \\ & = \frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^{10}}{10 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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} b^{19/3}}-\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac {\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.97 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {3780 \sqrt [3]{b} \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x+945 b^{4/3} \left (b^2 d-3 a b e+6 a^2 f\right ) x^4+540 b^{7/3} (b e-3 a f) x^7+378 b^{10/3} f x^{10}+\frac {630 a^2 \sqrt [3]{b} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{\left (a+b x^3\right )^2}+\frac {210 a \sqrt [3]{b} \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{a+b x^3}-140 \sqrt {3} \sqrt [3]{a} \left (-14 b^3 c+35 a b^2 d-65 a^2 b e+104 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+140 \sqrt [3]{a} \left (-14 b^3 c+35 a b^2 d-65 a^2 b e+104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-70 \sqrt [3]{a} \left (-14 b^3 c+35 a b^2 d-65 a^2 b e+104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{3780 b^{19/3}} \]

[In]

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

[Out]

(3780*b^(1/3)*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x + 945*b^(4/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^4 + 540
*b^(7/3)*(b*e - 3*a*f)*x^7 + 378*b^(10/3)*f*x^10 + (630*a^2*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/
(a + b*x^3)^2 + (210*a*b^(1/3)*(13*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(a + b*x^3) - 140*Sqrt[3]*a^
(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1
/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-14*b^3*c + 35*a*
b^2*d - 65*a^2*b*e + 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3780*b^(19/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.54 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {f \,x^{10}}{10 b^{3}}-\frac {3 x^{7} f a}{7 b^{4}}+\frac {x^{7} e}{7 b^{3}}+\frac {3 x^{4} f \,a^{2}}{2 b^{5}}-\frac {3 x^{4} a e}{4 b^{4}}+\frac {d \,x^{4}}{4 b^{3}}-\frac {10 x f \,a^{3}}{b^{6}}+\frac {6 x \,a^{2} e}{b^{5}}-\frac {3 x a d}{b^{4}}+\frac {x c}{b^{3}}+\frac {\left (-\frac {31}{18} a^{4} b f +\frac {25}{18} a^{3} b^{2} e -\frac {19}{18} a^{2} b^{3} d +\frac {13}{18} a \,b^{4} c \right ) x^{4}-\frac {a^{2} \left (14 f \,a^{3}-11 a^{2} b e +8 a \,b^{2} d -5 b^{3} c \right ) x}{9}}{b^{6} \left (b \,x^{3}+a \right )^{2}}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (104 f \,a^{3}-65 a^{2} b e +35 a \,b^{2} d -14 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 b^{7}}\)	\(233\)
default	\(-\frac {-\frac {1}{10} b^{3} f \,x^{10}+\frac {3}{7} x^{7} a \,b^{2} f -\frac {1}{7} x^{7} b^{3} e -\frac {3}{2} a^{2} b f \,x^{4}+\frac {3}{4} a \,b^{2} e \,x^{4}-\frac {1}{4} d \,x^{4} b^{3}+10 f \,a^{3} x -6 a^{2} b e x +3 a \,b^{2} d x -b^{3} c x}{b^{6}}+\frac {a \left (\frac {\left (-\frac {31}{18} a^{3} b f +\frac {25}{18} a^{2} e \,b^{2}-\frac {19}{18} a \,b^{3} d +\frac {13}{18} b^{4} c \right ) x^{4}-\frac {a \left (14 f \,a^{3}-11 a^{2} b e +8 a \,b^{2} d -5 b^{3} c \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (104 f \,a^{3}-65 a^{2} b e +35 a \,b^{2} d -14 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}\right )}{b^{6}}\)	\(296\)

[In]

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

[Out]

1/10*f*x^10/b^3-3/7/b^4*x^7*f*a+1/7/b^3*x^7*e+3/2/b^5*x^4*f*a^2-3/4/b^4*x^4*a*e+1/4/b^3*d*x^4-10/b^6*x*f*a^3+6
/b^5*x*a^2*e-3/b^4*x*a*d+1/b^3*x*c+((-31/18*a^4*b*f+25/18*a^3*b^2*e-19/18*a^2*b^3*d+13/18*a*b^4*c)*x^4-1/9*a^2
*(14*a^3*f-11*a^2*b*e+8*a*b^2*d-5*b^3*c)*x)/b^6/(b*x^3+a)^2+1/27/b^7*a*sum((104*a^3*f-65*a^2*b*e+35*a*b^2*d-14
*b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.61 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {378 \, b^{5} f x^{16} + 108 \, {\left (5 \, b^{5} e - 8 \, a b^{4} f\right )} x^{13} + 27 \, {\left (35 \, b^{5} d - 65 \, a b^{4} e + 104 \, a^{2} b^{3} f\right )} x^{10} + 270 \, {\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{7} + 735 \, {\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{4} - 140 \, \sqrt {3} {\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \, {\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 70 \, {\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \, {\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 140 \, {\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \, {\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 420 \, {\left (14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f\right )} x}{3780 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \]

[In]

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

[Out]

1/3780*(378*b^5*f*x^16 + 108*(5*b^5*e - 8*a*b^4*f)*x^13 + 27*(35*b^5*d - 65*a*b^4*e + 104*a^2*b^3*f)*x^10 + 27
0*(14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a^3*b^2*f)*x^7 + 735*(14*a*b^4*c - 35*a^2*b^3*d + 65*a^3*b^2*e -
104*a^4*b*f)*x^4 - 140*sqrt(3)*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a^3*b^2*f)*x^6 + 14*a^2*b^3*c - 3
5*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^2*b^3*d + 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(
1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3)*a)/a) + 70*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a
^3*b^2*f)*x^6 + 14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^2*b^3*d + 65*a^3*b
^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140*((14*b^5*c - 35*a*b^4*d + 65
*a^2*b^3*e - 104*a^3*b^2*f)*x^6 + 14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^
2*b^3*d + 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 420*(14*a^2*b^3*c - 35*a^3*b^2*d
+ 65*a^4*b*e - 104*a^5*f)*x)/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (13 \, a b^{4} c - 19 \, a^{2} b^{3} d + 25 \, a^{3} b^{2} e - 31 \, a^{4} b f\right )} x^{4} + 2 \, {\left (5 \, a^{2} b^{3} c - 8 \, a^{3} b^{2} d + 11 \, a^{4} b e - 14 \, a^{5} f\right )} x}{18 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} + \frac {14 \, b^{3} f x^{10} + 20 \, {\left (b^{3} e - 3 \, a b^{2} f\right )} x^{7} + 35 \, {\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{4} + 140 \, {\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x}{140 \, b^{6}} - \frac {\sqrt {3} {\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d + 65 \, a^{3} b e - 104 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d + 65 \, a^{3} b e - 104 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d + 65 \, a^{3} b e - 104 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

1/18*((13*a*b^4*c - 19*a^2*b^3*d + 25*a^3*b^2*e - 31*a^4*b*f)*x^4 + 2*(5*a^2*b^3*c - 8*a^3*b^2*d + 11*a^4*b*e
- 14*a^5*f)*x)/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6) + 1/140*(14*b^3*f*x^10 + 20*(b^3*e - 3*a*b^2*f)*x^7 + 35*(b^3
*d - 3*a*b^2*e + 6*a^2*b*f)*x^4 + 140*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 - 1/27*sqrt(3)*(14*a*b
^3*c - 35*a^2*b^2*d + 65*a^3*b*e - 104*a^4*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^7*(a/b)^(
2/3)) + 1/54*(14*a*b^3*c - 35*a^2*b^2*d + 65*a^3*b*e - 104*a^4*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^7*
(a/b)^(2/3)) - 1/27*(14*a*b^3*c - 35*a^2*b^2*d + 65*a^3*b*e - 104*a^4*f)*log(x + (a/b)^(1/3))/(b^7*(a/b)^(2/3)
)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.16 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 65 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 104 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{7}} + \frac {{\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d + 65 \, a^{3} b e - 104 \, a^{4} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{6}} - \frac {{\left (14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 65 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 104 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{7}} + \frac {13 \, a b^{4} c x^{4} - 19 \, a^{2} b^{3} d x^{4} + 25 \, a^{3} b^{2} e x^{4} - 31 \, a^{4} b f x^{4} + 10 \, a^{2} b^{3} c x - 16 \, a^{3} b^{2} d x + 22 \, a^{4} b e x - 28 \, a^{5} f x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{6}} + \frac {14 \, b^{27} f x^{10} + 20 \, b^{27} e x^{7} - 60 \, a b^{26} f x^{7} + 35 \, b^{27} d x^{4} - 105 \, a b^{26} e x^{4} + 210 \, a^{2} b^{25} f x^{4} + 140 \, b^{27} c x - 420 \, a b^{26} d x + 840 \, a^{2} b^{25} e x - 1400 \, a^{3} b^{24} f x}{140 \, b^{30}} \]

[In]

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

[Out]

-1/27*sqrt(3)*(14*(-a*b^2)^(1/3)*b^3*c - 35*(-a*b^2)^(1/3)*a*b^2*d + 65*(-a*b^2)^(1/3)*a^2*b*e - 104*(-a*b^2)^
(1/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^7 + 1/27*(14*a*b^3*c - 35*a^2*b^2*d + 65*
a^3*b*e - 104*a^4*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^6) - 1/54*(14*(-a*b^2)^(1/3)*b^3*c - 35*(-a*
b^2)^(1/3)*a*b^2*d + 65*(-a*b^2)^(1/3)*a^2*b*e - 104*(-a*b^2)^(1/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(
2/3))/b^7 + 1/18*(13*a*b^4*c*x^4 - 19*a^2*b^3*d*x^4 + 25*a^3*b^2*e*x^4 - 31*a^4*b*f*x^4 + 10*a^2*b^3*c*x - 16*
a^3*b^2*d*x + 22*a^4*b*e*x - 28*a^5*f*x)/((b*x^3 + a)^2*b^6) + 1/140*(14*b^27*f*x^10 + 20*b^27*e*x^7 - 60*a*b^
26*f*x^7 + 35*b^27*d*x^4 - 105*a*b^26*e*x^4 + 210*a^2*b^25*f*x^4 + 140*b^27*c*x - 420*a*b^26*d*x + 840*a^2*b^2
5*e*x - 1400*a^3*b^24*f*x)/b^30

Mupad [B] (verification not implemented)

Time = 9.63 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.12 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^7\,\left (\frac {e}{7\,b^3}-\frac {3\,a\,f}{7\,b^4}\right )+x\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )-x^4\,\left (\frac {3\,a^2\,f}{4\,b^5}-\frac {d}{4\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{4\,b}\right )-\frac {\left (\frac {31\,f\,a^4\,b}{18}-\frac {25\,e\,a^3\,b^2}{18}+\frac {19\,d\,a^2\,b^3}{18}-\frac {13\,c\,a\,b^4}{18}\right )\,x^4+\left (\frac {14\,f\,a^5}{9}-\frac {11\,e\,a^4\,b}{9}+\frac {8\,d\,a^3\,b^2}{9}-\frac {5\,c\,a^2\,b^3}{9}\right )\,x}{a^2\,b^6+2\,a\,b^7\,x^3+b^8\,x^6}+\frac {f\,x^{10}}{10\,b^3}-\frac {a^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-104\,f\,a^3+65\,e\,a^2\,b-35\,d\,a\,b^2+14\,c\,b^3\right )}{27\,b^{19/3}}-\frac {a^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-104\,f\,a^3+65\,e\,a^2\,b-35\,d\,a\,b^2+14\,c\,b^3\right )}{27\,b^{19/3}}+\frac {a^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-104\,f\,a^3+65\,e\,a^2\,b-35\,d\,a\,b^2+14\,c\,b^3\right )}{27\,b^{19/3}} \]

[In]

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

[Out]

x^7*(e/(7*b^3) - (3*a*f)/(7*b^4)) + x*(c/b^3 - (a^3*f)/b^6 - (3*a^2*(e/b^3 - (3*a*f)/b^4))/b^2 + (3*a*((3*a^2*
f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b) - x^4*((3*a^2*f)/(4*b^5) - d/(4*b^3) + (3*a*(e/b^3 - (3*a*
f)/b^4))/(4*b)) - (x*((14*a^5*f)/9 - (5*a^2*b^3*c)/9 + (8*a^3*b^2*d)/9 - (11*a^4*b*e)/9) + x^4*((19*a^2*b^3*d)
/18 - (25*a^3*b^2*e)/18 - (13*a*b^4*c)/18 + (31*a^4*b*f)/18))/(a^2*b^6 + b^8*x^6 + 2*a*b^7*x^3) + (f*x^10)/(10
*b^3) - (a^(1/3)*log(b^(1/3)*x + a^(1/3))*(14*b^3*c - 104*a^3*f - 35*a*b^2*d + 65*a^2*b*e))/(27*b^(19/3)) - (a
^(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)*(14*b^3*c - 104*a^3*f - 35*a*b^2
*d + 65*a^2*b*e))/(27*b^(19/3)) + (a^(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)*(14*b^3*c - 104*a^3*f - 35*a*b^2*d + 65*a^2*b*e))/(27*b^(19/3))