\(\int \frac {x^7 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\) [261]

Optimal result

Integrand size = 30, antiderivative size = 335 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{17/3}}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}} \]

[Out]

1/2*(-4*a^3*f+3*a^2*b*e-2*a*b^2*d+b^3*c)*x^2/b^5+1/5*(3*a^2*f-2*a*b*e+b^2*d)*x^5/b^4+1/8*(-2*a*f+b*e)*x^8/b^3+
1/11*f*x^11/b^2+1/3*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/b^5/(b*x^3+a)+1/9*a^(2/3)*(-14*a^3*f+11*a^2*b*e-8*a*b
^2*d+5*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/b^(17/3)-1/18*a^(2/3)*(-14*a^3*f+11*a^2*b*e-8*a*b^2*d+5*b^3*c)*ln(a^(2/3)-
a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(17/3)+1/9*a^(2/3)*(-14*a^3*f+11*a^2*b*e-8*a*b^2*d+5*b^3*c)*arctan(1/3*(a^(1/
3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/b^(17/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1842, 1865, 1850, 1502, 298, 31, 648, 631, 210, 642} \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac {x^2 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{3 \sqrt {3} b^{17/3}}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac {x^8 (b e-2 a f)}{8 b^3}+\frac {f x^{11}}{11 b^2} \]

[In]

Int[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^2)/(2*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^5)/(5*b^4) + ((b*e -
2*a*f)*x^8)/(8*b^3) + (f*x^11)/(11*b^2) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*b^5*(a + b*x^3)) + (a
^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqr
t[3]*b^(17/3)) + (a^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(17/3))
- (a^(2/3)*(5*b^3*c - 8*a*b^2*d + 11*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*
b^(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 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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 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 1502

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]

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 1850

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)
/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rule 1865

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x] /;
FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m
]]

Rubi steps \begin{align*} \text {integral}& = \frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^7-3 a b^4 (b e-a f) x^{10}-3 a b^5 f x^{13}}{a+b x^3} \, dx}{3 a b^6} \\ & = \frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {x \left (2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^6-3 a b^4 (b e-a f) x^9-3 a b^5 f x^{12}\right )}{a+b x^3} \, dx}{3 a b^6} \\ & = \frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {x \left (22 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-33 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-33 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^6-33 a b^5 (b e-2 a f) x^9\right )}{a+b x^3} \, dx}{33 a b^7} \\ & = \frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {x \left (176 a^2 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-264 a b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-264 a b^5 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6\right )}{a+b x^3} \, dx}{264 a b^8} \\ & = \frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \left (-264 a b^3 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x-264 a b^4 \left (b^2 d-2 a b e+3 a^2 f\right ) x^4-\frac {88 \left (-5 a^2 b^6 c+8 a^3 b^5 d-11 a^4 b^4 e+14 a^5 b^3 f\right ) x}{a+b x^3}\right ) \, dx}{264 a b^8} \\ & = \frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{3 b^5} \\ & = \frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^{16/3}}-\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^{16/3}} \\ & = \frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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}{18 b^{17/3}}-\frac {\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{16/3}} \\ & = \frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}-\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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 )}{3 b^{17/3}} \\ & = \frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{17/3}}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.95 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {1980 b^{2/3} \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2+792 b^{5/3} \left (b^2 d-2 a b e+3 a^2 f\right ) x^5+495 b^{8/3} (b e-2 a f) x^8+360 b^{11/3} f x^{11}+\frac {1320 a b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{a+b x^3}-440 \sqrt {3} a^{2/3} \left (-5 b^3 c+8 a b^2 d-11 a^2 b e+14 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-440 a^{2/3} \left (-5 b^3 c+8 a b^2 d-11 a^2 b e+14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+220 a^{2/3} \left (-5 b^3 c+8 a b^2 d-11 a^2 b e+14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{3960 b^{17/3}} \]

[In]

Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(1980*b^(2/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^2 + 792*b^(5/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^5 + 49
5*b^(8/3)*(b*e - 2*a*f)*x^8 + 360*b^(11/3)*f*x^11 + (1320*a*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(
a + b*x^3) - 440*Sqrt[3]*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1
/3))/Sqrt[3]] - 440*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*a^(2
/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3960*b^(17
/3))

Maple [C] (verified)

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

Time = 1.54 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.61

method	result	size
risch	\(\frac {f \,x^{11}}{11 b^{2}}-\frac {x^{8} f a}{4 b^{3}}+\frac {x^{8} e}{8 b^{2}}+\frac {3 x^{5} f \,a^{2}}{5 b^{4}}-\frac {2 x^{5} a e}{5 b^{3}}+\frac {d \,x^{5}}{5 b^{2}}-\frac {2 x^{2} f \,a^{3}}{b^{5}}+\frac {3 x^{2} a^{2} e}{2 b^{4}}-\frac {x^{2} a d}{b^{3}}+\frac {x^{2} c}{2 b^{2}}+\frac {\left (-\frac {1}{3} a^{4} f +\frac {1}{3} a^{3} b e -\frac {1}{3} a^{2} b^{2} d +\frac {1}{3} a \,b^{3} c \right ) x^{2}}{b^{5} \left (b \,x^{3}+a \right )}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (14 f \,a^{3}-11 a^{2} b e +8 a \,b^{2} d -5 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{9 b^{6}}\)	\(204\)
default	\(-\frac {-\frac {b^{3} f \,x^{11}}{11}+\frac {\left (2 f a \,b^{2}-b^{3} e \right ) x^{8}}{8}+\frac {\left (-3 f \,a^{2} b +2 a \,b^{2} e -b^{3} d \right ) x^{5}}{5}+\frac {x^{2} \left (4 f \,a^{3}-3 a^{2} b e +2 a \,b^{2} d -b^{3} c \right )}{2}}{b^{5}}+\frac {a \left (\frac {\left (-\frac {1}{3} f \,a^{3}+\frac {1}{3} a^{2} b e -\frac {1}{3} a \,b^{2} d +\frac {1}{3} b^{3} c \right ) x^{2}}{b \,x^{3}+a}+\left (\frac {14}{3} f \,a^{3}-\frac {11}{3} a^{2} b e +\frac {8}{3} a \,b^{2} d -\frac {5}{3} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{b^{5}}\)	\(258\)

[In]

int(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/11*f*x^11/b^2-1/4/b^3*x^8*f*a+1/8/b^2*x^8*e+3/5/b^4*x^5*f*a^2-2/5/b^3*x^5*a*e+1/5/b^2*d*x^5-2/b^5*x^2*f*a^3+
3/2/b^4*x^2*a^2*e-1/b^3*x^2*a*d+1/2/b^2*x^2*c+(-1/3*a^4*f+1/3*a^3*b*e-1/3*a^2*b^2*d+1/3*a*b^3*c)*x^2/b^5/(b*x^
3+a)+1/9/b^6*a*sum((14*a^3*f-11*a^2*b*e+8*a*b^2*d-5*b^3*c)/_R*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.36 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {360 \, b^{4} f x^{14} + 45 \, {\left (11 \, b^{4} e - 14 \, a b^{3} f\right )} x^{11} + 99 \, {\left (8 \, b^{4} d - 11 \, a b^{3} e + 14 \, a^{2} b^{2} f\right )} x^{8} + 396 \, {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{5} + 660 \, {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} x^{2} - 440 \, \sqrt {3} {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f + {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) + 220 \, {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f + {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 440 \, {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f + {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{3960 \, {\left (b^{6} x^{3} + a b^{5}\right )}} \]

[In]

integrate(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/3960*(360*b^4*f*x^14 + 45*(11*b^4*e - 14*a*b^3*f)*x^11 + 99*(8*b^4*d - 11*a*b^3*e + 14*a^2*b^2*f)*x^8 + 396*
(5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*a^3*b*f)*x^5 + 660*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f)*
x^2 - 440*sqrt(3)*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f + (5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*
a^3*b*f)*x^3)*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a^2/b^2)^(1/3) + sqrt(3)*a)/a) + 220*(5*a*b^3*c - 8
*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f + (5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*a^3*b*f)*x^3)*(-a^2/b^2)^(1/3)*l
og(a*x^2 - b*x*(-a^2/b^2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 440*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f +
(5*b^4*c - 8*a*b^3*d + 11*a^2*b^2*e - 14*a^3*b*f)*x^3)*(-a^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/3)))/(b^6*x
^3 + a*b^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(x**7*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.97 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{2}}{3 \, {\left (b^{6} x^{3} + a b^{5}\right )}} - \frac {\sqrt {3} {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {40 \, b^{3} f x^{11} + 55 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{8} + 88 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{5} + 220 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x^{2}}{440 \, b^{5}} - \frac {{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^2/(b^6*x^3 + a*b^5) - 1/9*sqrt(3)*(5*a*b^3*c - 8*a^2*b^2*d + 11*
a^3*b*e - 14*a^4*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(1/3)) + 1/440*(40*b^3*f*x^
11 + 55*(b^3*e - 2*a*b^2*f)*x^8 + 88*(b^3*d - 2*a*b^2*e + 3*a^2*b*f)*x^5 + 220*(b^3*c - 2*a*b^2*d + 3*a^2*b*e
- 4*a^3*f)*x^2)/b^5 - 1/18*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(
2/3))/(b^6*(a/b)^(1/3)) + 1/9*(5*a*b^3*c - 8*a^2*b^2*d + 11*a^3*b*e - 14*a^4*f)*log(x + (a/b)^(1/3))/(b^6*(a/b
)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.30 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (5 \, a b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 8 \, a^{2} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 11 \, a^{3} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{4} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 11 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{7}} + \frac {a b^{3} c x^{2} - a^{2} b^{2} d x^{2} + a^{3} b e x^{2} - a^{4} f x^{2}}{3 \, {\left (b x^{3} + a\right )} b^{5}} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 11 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{7}} + \frac {40 \, b^{20} f x^{11} + 55 \, b^{20} e x^{8} - 110 \, a b^{19} f x^{8} + 88 \, b^{20} d x^{5} - 176 \, a b^{19} e x^{5} + 264 \, a^{2} b^{18} f x^{5} + 220 \, b^{20} c x^{2} - 440 \, a b^{19} d x^{2} + 660 \, a^{2} b^{18} e x^{2} - 880 \, a^{3} b^{17} f x^{2}}{440 \, b^{22}} \]

[In]

integrate(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*(5*a*b^3*c*(-a/b)^(1/3) - 8*a^2*b^2*d*(-a/b)^(1/3) + 11*a^3*b*e*(-a/b)^(1/3) - 14*a^4*f*(-a/b)^(1/3))*(-a/
b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) + 1/9*sqrt(3)*(5*(-a*b^2)^(2/3)*b^3*c - 8*(-a*b^2)^(2/3)*a*b^2*d +
11*(-a*b^2)^(2/3)*a^2*b*e - 14*(-a*b^2)^(2/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^
7 + 1/3*(a*b^3*c*x^2 - a^2*b^2*d*x^2 + a^3*b*e*x^2 - a^4*f*x^2)/((b*x^3 + a)*b^5) - 1/18*(5*(-a*b^2)^(2/3)*b^3
*c - 8*(-a*b^2)^(2/3)*a*b^2*d + 11*(-a*b^2)^(2/3)*a^2*b*e - 14*(-a*b^2)^(2/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/b^7 + 1/440*(40*b^20*f*x^11 + 55*b^20*e*x^8 - 110*a*b^19*f*x^8 + 88*b^20*d*x^5 - 176*a*b^19*e*
x^5 + 264*a^2*b^18*f*x^5 + 220*b^20*c*x^2 - 440*a*b^19*d*x^2 + 660*a^2*b^18*e*x^2 - 880*a^3*b^17*f*x^2)/b^22

Mupad [B] (verification not implemented)

Time = 10.37 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.08 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^8\,\left (\frac {e}{8\,b^2}-\frac {a\,f}{4\,b^3}\right )-x^5\,\left (\frac {a^2\,f}{5\,b^4}-\frac {d}{5\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{5\,b}\right )+x^2\,\left (\frac {c}{2\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{2\,b^2}+\frac {a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )+\frac {f\,x^{11}}{11\,b^2}-\frac {x^2\,\left (\frac {f\,a^4}{3}-\frac {e\,a^3\,b}{3}+\frac {d\,a^2\,b^2}{3}-\frac {c\,a\,b^3}{3}\right )}{b^6\,x^3+a\,b^5}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-14\,f\,a^3+11\,e\,a^2\,b-8\,d\,a\,b^2+5\,c\,b^3\right )}{9\,b^{17/3}}-\frac {a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+11\,e\,a^2\,b-8\,d\,a\,b^2+5\,c\,b^3\right )}{9\,b^{17/3}}+\frac {a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+11\,e\,a^2\,b-8\,d\,a\,b^2+5\,c\,b^3\right )}{9\,b^{17/3}} \]

[In]

int((x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x)

[Out]

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