\(\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx\) [325]

Optimal result

Integrand size = 21, antiderivative size = 96 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=-\frac {F^{a+\frac {b}{(c+d x)^2}} \left (24 (c+d x)^8-24 b (c+d x)^6 \log (F)+12 b^2 (c+d x)^4 \log ^2(F)-4 b^3 (c+d x)^2 \log ^3(F)+b^4 \log ^4(F)\right )}{2 b^5 d (c+d x)^8 \log ^5(F)} \]

[Out]

-1/2*F^(a+b/(d*x+c)^2)*(24*(d*x+c)^8-24*b*(d*x+c)^6*ln(F)+12*b^2*(d*x+c)^4*ln(F)^2-4*b^3*(d*x+c)^2*ln(F)^3+b^4
*ln(F)^4)/b^5/d/(d*x+c)^8/ln(F)^5

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2249} \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=-\frac {F^{a+\frac {b}{(c+d x)^2}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)^2+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^6+24 (c+d x)^8\right )}{2 b^5 d \log ^5(F) (c+d x)^8} \]

[In]

Int[F^(a + b/(c + d*x)^2)/(c + d*x)^11,x]

[Out]

-1/2*(F^(a + b/(c + d*x)^2)*(24*(c + d*x)^8 - 24*b*(c + d*x)^6*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 - 4*b^3*(c
+ d*x)^2*Log[F]^3 + b^4*Log[F]^4))/(b^5*d*(c + d*x)^8*Log[F]^5)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+\frac {b}{(c+d x)^2}} \left (24 (c+d x)^8-24 b (c+d x)^6 \log (F)+12 b^2 (c+d x)^4 \log ^2(F)-4 b^3 (c+d x)^2 \log ^3(F)+b^4 \log ^4(F)\right )}{2 b^5 d (c+d x)^8 \log ^5(F)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)} \]

[In]

Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^11,x]

[Out]

-1/2*(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^2)])/(b^5*d*Log[F]^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(94)=188\).

Time = 2.78 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.55

method	result	size
risch	\(-\frac {\left (-24 \ln \left (F \right ) b \,c^{6}+48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x -8 \ln \left (F \right )^{3} b^{3} c d x +24 c^{8}+b^{4} \ln \left (F \right )^{4}-4 \ln \left (F \right )^{3} b^{3} d^{2} x^{2}-4 \ln \left (F \right )^{3} b^{3} c^{2}+12 \ln \left (F \right )^{2} b^{2} c^{4}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-144 \ln \left (F \right ) b c \,d^{5} x^{5}-360 \ln \left (F \right ) b \,c^{2} d^{4} x^{4}-480 \ln \left (F \right ) b \,c^{3} d^{3} x^{3}-360 \ln \left (F \right ) b \,c^{4} d^{2} x^{2}-144 \ln \left (F \right ) b \,c^{5} d x +24 d^{8} x^{8}-24 \ln \left (F \right ) b \,d^{6} x^{6}+192 c \,d^{7} x^{7}+672 c^{2} d^{6} x^{6}+1344 c^{3} d^{5} x^{5}+1680 c^{4} d^{4} x^{4}+1344 c^{5} d^{3} x^{3}+672 c^{6} d^{2} x^{2}+192 c^{7} d x \right ) F^{\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}}}{2 b^{5} \ln \left (F \right )^{5} d \left (d x +c \right )^{8}}\)	\(341\)
norman	\(\frac {-\frac {12 d^{9} x^{10} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {c \left (b^{4} \ln \left (F \right )^{4}-8 \ln \left (F \right )^{3} b^{3} c^{2}+36 \ln \left (F \right )^{2} b^{2} c^{4}-96 \ln \left (F \right ) b \,c^{6}+120 c^{8}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{b^{5} \ln \left (F \right )^{5}}-\frac {d \left (b^{4} \ln \left (F \right )^{4}-24 \ln \left (F \right )^{3} b^{3} c^{2}+180 \ln \left (F \right )^{2} b^{2} c^{4}-672 \ln \left (F \right ) b \,c^{6}+1080 c^{8}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{5} b^{5}}+\frac {2 d^{3} \left (\ln \left (F \right )^{3} b^{3}-45 \ln \left (F \right )^{2} b^{2} c^{2}+420 \ln \left (F \right ) b \,c^{4}-1260 c^{6}\right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {6 d^{5} \left (\ln \left (F \right )^{2} b^{2}-56 \ln \left (F \right ) b \,c^{2}+420 c^{4}\right ) x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {12 d^{7} \left (b \ln \left (F \right )-45 c^{2}\right ) x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {120 d^{8} c \,x^{9} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {\left (b^{4} \ln \left (F \right )^{4}-4 \ln \left (F \right )^{3} b^{3} c^{2}+12 \ln \left (F \right )^{2} b^{2} c^{4}-24 \ln \left (F \right ) b \,c^{6}+24 c^{8}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 b^{5} \ln \left (F \right )^{5} d}+\frac {8 c \,d^{2} \left (\ln \left (F \right )^{3} b^{3}-15 \ln \left (F \right )^{2} b^{2} c^{2}+84 \ln \left (F \right ) b \,c^{4}-180 c^{6}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {12 c \,d^{4} \left (3 \ln \left (F \right )^{2} b^{2}-56 \ln \left (F \right ) b \,c^{2}+252 c^{4}\right ) x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {96 c \,d^{6} \left (b \ln \left (F \right )-15 c^{2}\right ) x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}}{\left (d x +c \right )^{10}}\)	\(609\)
parallelrisch	\(\frac {-24 d^{21} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{8}-192 d^{20} c \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{7}-672 x^{6} F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{2} d^{19}-1344 x^{5} F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{3} d^{18}-1680 x^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{4} d^{17}-1344 x^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{5} d^{16}-672 x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{6} d^{15}-192 x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{7} d^{14}-\ln \left (F \right )^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{4} d^{13}+144 \ln \left (F \right ) x^{5} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b c \,d^{18}+360 \ln \left (F \right ) x^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{2} d^{17}-48 \ln \left (F \right )^{2} x^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c \,d^{16}+480 \ln \left (F \right ) x^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{3} d^{16}-72 \ln \left (F \right )^{2} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c^{2} d^{15}+360 \ln \left (F \right ) x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{4} d^{15}+8 \ln \left (F \right )^{3} x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{3} c \,d^{14}-48 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c^{3} d^{14}+144 \ln \left (F \right ) x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{5} d^{14}-24 F^{a +\frac {b}{\left (d x +c \right )^{2}}} c^{8} d^{13}+24 \ln \left (F \right ) x^{6} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,d^{19}-12 \ln \left (F \right )^{2} x^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} d^{17}+4 \ln \left (F \right )^{3} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{3} d^{15}+4 \ln \left (F \right )^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{3} c^{2} d^{13}-12 \ln \left (F \right )^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c^{4} d^{13}+24 \ln \left (F \right ) F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{6} d^{13}}{2 \left (d x +c \right )^{8} \ln \left (F \right )^{5} b^{5} d^{14}}\)	\(659\)

[In]

int(F^(a+b/(d*x+c)^2)/(d*x+c)^11,x,method=_RETURNVERBOSE)

[Out]

-1/2*(-24*ln(F)*b*c^6+48*d^3*c*x^3*b^2*ln(F)^2+72*ln(F)^2*b^2*c^2*d^2*x^2+48*ln(F)^2*b^2*c^3*d*x-8*ln(F)^3*b^3
*c*d*x+24*c^8+b^4*ln(F)^4-4*ln(F)^3*b^3*d^2*x^2-4*ln(F)^3*b^3*c^2+12*ln(F)^2*b^2*c^4+12*d^4*x^4*b^2*ln(F)^2-14
4*ln(F)*b*c*d^5*x^5-360*ln(F)*b*c^2*d^4*x^4-480*ln(F)*b*c^3*d^3*x^3-360*ln(F)*b*c^4*d^2*x^2-144*ln(F)*b*c^5*d*
x+24*d^8*x^8-24*ln(F)*b*d^6*x^6+192*c*d^7*x^7+672*c^2*d^6*x^6+1344*c^3*d^5*x^5+1680*c^4*d^4*x^4+1344*c^5*d^3*x
^3+672*c^6*d^2*x^2+192*c^7*d*x)/b^5/ln(F)^5/d/(d*x+c)^8*F^((a*d^2*x^2+2*a*c*d*x+a*c^2+b)/(d*x+c)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (94) = 188\).

Time = 0.29 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.38 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=-\frac {{\left (24 \, d^{8} x^{8} + 192 \, c d^{7} x^{7} + 672 \, c^{2} d^{6} x^{6} + 1344 \, c^{3} d^{5} x^{5} + 1680 \, c^{4} d^{4} x^{4} + 1344 \, c^{5} d^{3} x^{3} + 672 \, c^{6} d^{2} x^{2} + 192 \, c^{7} d x + 24 \, c^{8} + b^{4} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{5} d^{9} x^{8} + 8 \, b^{5} c d^{8} x^{7} + 28 \, b^{5} c^{2} d^{7} x^{6} + 56 \, b^{5} c^{3} d^{6} x^{5} + 70 \, b^{5} c^{4} d^{5} x^{4} + 56 \, b^{5} c^{5} d^{4} x^{3} + 28 \, b^{5} c^{6} d^{3} x^{2} + 8 \, b^{5} c^{7} d^{2} x + b^{5} c^{8} d\right )} \log \left (F\right )^{5}} \]

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^11,x, algorithm="fricas")

[Out]

-1/2*(24*d^8*x^8 + 192*c*d^7*x^7 + 672*c^2*d^6*x^6 + 1344*c^3*d^5*x^5 + 1680*c^4*d^4*x^4 + 1344*c^5*d^3*x^3 +
672*c^6*d^2*x^2 + 192*c^7*d*x + 24*c^8 + b^4*log(F)^4 - 4*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*log(F)^3 + 12*
(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(F)^2 - 24*(b*d^6*x^6 + 6*b*c
*d^5*x^5 + 15*b*c^2*d^4*x^4 + 20*b*c^3*d^3*x^3 + 15*b*c^4*d^2*x^2 + 6*b*c^5*d*x + b*c^6)*log(F))*F^((a*d^2*x^2
+ 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))/((b^5*d^9*x^8 + 8*b^5*c*d^8*x^7 + 28*b^5*c^2*d^7*x^6 + 56
*b^5*c^3*d^6*x^5 + 70*b^5*c^4*d^5*x^4 + 56*b^5*c^5*d^4*x^3 + 28*b^5*c^6*d^3*x^2 + 8*b^5*c^7*d^2*x + b^5*c^8*d)
*log(F)^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (95) = 190\).

Time = 0.35 (sec) , antiderivative size = 518, normalized size of antiderivative = 5.40 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=\frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{4} \log {\left (F \right )}^{4} + 4 b^{3} c^{2} \log {\left (F \right )}^{3} + 8 b^{3} c d x \log {\left (F \right )}^{3} + 4 b^{3} d^{2} x^{2} \log {\left (F \right )}^{3} - 12 b^{2} c^{4} \log {\left (F \right )}^{2} - 48 b^{2} c^{3} d x \log {\left (F \right )}^{2} - 72 b^{2} c^{2} d^{2} x^{2} \log {\left (F \right )}^{2} - 48 b^{2} c d^{3} x^{3} \log {\left (F \right )}^{2} - 12 b^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 24 b c^{6} \log {\left (F \right )} + 144 b c^{5} d x \log {\left (F \right )} + 360 b c^{4} d^{2} x^{2} \log {\left (F \right )} + 480 b c^{3} d^{3} x^{3} \log {\left (F \right )} + 360 b c^{2} d^{4} x^{4} \log {\left (F \right )} + 144 b c d^{5} x^{5} \log {\left (F \right )} + 24 b d^{6} x^{6} \log {\left (F \right )} - 24 c^{8} - 192 c^{7} d x - 672 c^{6} d^{2} x^{2} - 1344 c^{5} d^{3} x^{3} - 1680 c^{4} d^{4} x^{4} - 1344 c^{3} d^{5} x^{5} - 672 c^{2} d^{6} x^{6} - 192 c d^{7} x^{7} - 24 d^{8} x^{8}\right )}{2 b^{5} c^{8} d \log {\left (F \right )}^{5} + 16 b^{5} c^{7} d^{2} x \log {\left (F \right )}^{5} + 56 b^{5} c^{6} d^{3} x^{2} \log {\left (F \right )}^{5} + 112 b^{5} c^{5} d^{4} x^{3} \log {\left (F \right )}^{5} + 140 b^{5} c^{4} d^{5} x^{4} \log {\left (F \right )}^{5} + 112 b^{5} c^{3} d^{6} x^{5} \log {\left (F \right )}^{5} + 56 b^{5} c^{2} d^{7} x^{6} \log {\left (F \right )}^{5} + 16 b^{5} c d^{8} x^{7} \log {\left (F \right )}^{5} + 2 b^{5} d^{9} x^{8} \log {\left (F \right )}^{5}} \]

[In]

integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**11,x)

[Out]

F**(a + b/(c + d*x)**2)*(-b**4*log(F)**4 + 4*b**3*c**2*log(F)**3 + 8*b**3*c*d*x*log(F)**3 + 4*b**3*d**2*x**2*l
og(F)**3 - 12*b**2*c**4*log(F)**2 - 48*b**2*c**3*d*x*log(F)**2 - 72*b**2*c**2*d**2*x**2*log(F)**2 - 48*b**2*c*
d**3*x**3*log(F)**2 - 12*b**2*d**4*x**4*log(F)**2 + 24*b*c**6*log(F) + 144*b*c**5*d*x*log(F) + 360*b*c**4*d**2
*x**2*log(F) + 480*b*c**3*d**3*x**3*log(F) + 360*b*c**2*d**4*x**4*log(F) + 144*b*c*d**5*x**5*log(F) + 24*b*d**
6*x**6*log(F) - 24*c**8 - 192*c**7*d*x - 672*c**6*d**2*x**2 - 1344*c**5*d**3*x**3 - 1680*c**4*d**4*x**4 - 1344
*c**3*d**5*x**5 - 672*c**2*d**6*x**6 - 192*c*d**7*x**7 - 24*d**8*x**8)/(2*b**5*c**8*d*log(F)**5 + 16*b**5*c**7
*d**2*x*log(F)**5 + 56*b**5*c**6*d**3*x**2*log(F)**5 + 112*b**5*c**5*d**4*x**3*log(F)**5 + 140*b**5*c**4*d**5*
x**4*log(F)**5 + 112*b**5*c**3*d**6*x**5*log(F)**5 + 56*b**5*c**2*d**7*x**6*log(F)**5 + 16*b**5*c*d**8*x**7*lo
g(F)**5 + 2*b**5*d**9*x**8*log(F)**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (94) = 188\).

Time = 0.27 (sec) , antiderivative size = 526, normalized size of antiderivative = 5.48 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=-\frac {{\left (24 \, F^{a} d^{8} x^{8} + 192 \, F^{a} c d^{7} x^{7} + 24 \, F^{a} c^{8} - 24 \, F^{a} b c^{6} \log \left (F\right ) + 12 \, F^{a} b^{2} c^{4} \log \left (F\right )^{2} - 4 \, F^{a} b^{3} c^{2} \log \left (F\right )^{3} + F^{a} b^{4} \log \left (F\right )^{4} + 24 \, {\left (28 \, F^{a} c^{2} d^{6} - F^{a} b d^{6} \log \left (F\right )\right )} x^{6} + 48 \, {\left (28 \, F^{a} c^{3} d^{5} - 3 \, F^{a} b c d^{5} \log \left (F\right )\right )} x^{5} + 12 \, {\left (140 \, F^{a} c^{4} d^{4} - 30 \, F^{a} b c^{2} d^{4} \log \left (F\right ) + F^{a} b^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 48 \, {\left (28 \, F^{a} c^{5} d^{3} - 10 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + F^{a} b^{2} c d^{3} \log \left (F\right )^{2}\right )} x^{3} + 4 \, {\left (168 \, F^{a} c^{6} d^{2} - 90 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + 18 \, F^{a} b^{2} c^{2} d^{2} \log \left (F\right )^{2} - F^{a} b^{3} d^{2} \log \left (F\right )^{3}\right )} x^{2} + 8 \, {\left (24 \, F^{a} c^{7} d - 18 \, F^{a} b c^{5} d \log \left (F\right ) + 6 \, F^{a} b^{2} c^{3} d \log \left (F\right )^{2} - F^{a} b^{3} c d \log \left (F\right )^{3}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{5} d^{9} x^{8} \log \left (F\right )^{5} + 8 \, b^{5} c d^{8} x^{7} \log \left (F\right )^{5} + 28 \, b^{5} c^{2} d^{7} x^{6} \log \left (F\right )^{5} + 56 \, b^{5} c^{3} d^{6} x^{5} \log \left (F\right )^{5} + 70 \, b^{5} c^{4} d^{5} x^{4} \log \left (F\right )^{5} + 56 \, b^{5} c^{5} d^{4} x^{3} \log \left (F\right )^{5} + 28 \, b^{5} c^{6} d^{3} x^{2} \log \left (F\right )^{5} + 8 \, b^{5} c^{7} d^{2} x \log \left (F\right )^{5} + b^{5} c^{8} d \log \left (F\right )^{5}\right )}} \]

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^11,x, algorithm="maxima")

[Out]

-1/2*(24*F^a*d^8*x^8 + 192*F^a*c*d^7*x^7 + 24*F^a*c^8 - 24*F^a*b*c^6*log(F) + 12*F^a*b^2*c^4*log(F)^2 - 4*F^a*
b^3*c^2*log(F)^3 + F^a*b^4*log(F)^4 + 24*(28*F^a*c^2*d^6 - F^a*b*d^6*log(F))*x^6 + 48*(28*F^a*c^3*d^5 - 3*F^a*
b*c*d^5*log(F))*x^5 + 12*(140*F^a*c^4*d^4 - 30*F^a*b*c^2*d^4*log(F) + F^a*b^2*d^4*log(F)^2)*x^4 + 48*(28*F^a*c
^5*d^3 - 10*F^a*b*c^3*d^3*log(F) + F^a*b^2*c*d^3*log(F)^2)*x^3 + 4*(168*F^a*c^6*d^2 - 90*F^a*b*c^4*d^2*log(F)
+ 18*F^a*b^2*c^2*d^2*log(F)^2 - F^a*b^3*d^2*log(F)^3)*x^2 + 8*(24*F^a*c^7*d - 18*F^a*b*c^5*d*log(F) + 6*F^a*b^
2*c^3*d*log(F)^2 - F^a*b^3*c*d*log(F)^3)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^5*d^9*x^8*log(F)^5 + 8*b^5*c*d^
8*x^7*log(F)^5 + 28*b^5*c^2*d^7*x^6*log(F)^5 + 56*b^5*c^3*d^6*x^5*log(F)^5 + 70*b^5*c^4*d^5*x^4*log(F)^5 + 56*
b^5*c^5*d^4*x^3*log(F)^5 + 28*b^5*c^6*d^3*x^2*log(F)^5 + 8*b^5*c^7*d^2*x*log(F)^5 + b^5*c^8*d*log(F)^5)

Giac [F]

\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{11}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^11,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^11, x)

Mupad [B] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.45 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx=-\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {b^4\,{\ln \left (F\right )}^4-4\,b^3\,c^2\,{\ln \left (F\right )}^3+12\,b^2\,c^4\,{\ln \left (F\right )}^2-24\,b\,c^6\,\ln \left (F\right )+24\,c^8}{2\,b^5\,d^9\,{\ln \left (F\right )}^5}+\frac {12\,x^8}{b^5\,d\,{\ln \left (F\right )}^5}+\frac {96\,c\,x^7}{b^5\,d^2\,{\ln \left (F\right )}^5}-\frac {2\,x^2\,\left (b^3\,{\ln \left (F\right )}^3-18\,b^2\,c^2\,{\ln \left (F\right )}^2+90\,b\,c^4\,\ln \left (F\right )-168\,c^6\right )}{b^5\,d^7\,{\ln \left (F\right )}^5}+\frac {6\,x^4\,\left (b^2\,{\ln \left (F\right )}^2-30\,b\,c^2\,\ln \left (F\right )+140\,c^4\right )}{b^5\,d^5\,{\ln \left (F\right )}^5}-\frac {12\,x^6\,\left (b\,\ln \left (F\right )-28\,c^2\right )}{b^5\,d^3\,{\ln \left (F\right )}^5}+\frac {24\,c\,x^3\,\left (b^2\,{\ln \left (F\right )}^2-10\,b\,c^2\,\ln \left (F\right )+28\,c^4\right )}{b^5\,d^6\,{\ln \left (F\right )}^5}-\frac {24\,c\,x^5\,\left (3\,b\,\ln \left (F\right )-28\,c^2\right )}{b^5\,d^4\,{\ln \left (F\right )}^5}-\frac {4\,c\,x\,\left (b^3\,{\ln \left (F\right )}^3-6\,b^2\,c^2\,{\ln \left (F\right )}^2+18\,b\,c^4\,\ln \left (F\right )-24\,c^6\right )}{b^5\,d^8\,{\ln \left (F\right )}^5}\right )}{x^8+\frac {c^8}{d^8}+\frac {8\,c\,x^7}{d}+\frac {8\,c^7\,x}{d^7}+\frac {28\,c^2\,x^6}{d^2}+\frac {56\,c^3\,x^5}{d^3}+\frac {70\,c^4\,x^4}{d^4}+\frac {56\,c^5\,x^3}{d^5}+\frac {28\,c^6\,x^2}{d^6}} \]

[In]

int(F^(a + b/(c + d*x)^2)/(c + d*x)^11,x)

[Out]

-(F^a*F^(b/(c^2 + d^2*x^2 + 2*c*d*x))*((b^4*log(F)^4 + 24*c^8 - 24*b*c^6*log(F) + 12*b^2*c^4*log(F)^2 - 4*b^3*
c^2*log(F)^3)/(2*b^5*d^9*log(F)^5) + (12*x^8)/(b^5*d*log(F)^5) + (96*c*x^7)/(b^5*d^2*log(F)^5) - (2*x^2*(b^3*l
og(F)^3 - 168*c^6 + 90*b*c^4*log(F) - 18*b^2*c^2*log(F)^2))/(b^5*d^7*log(F)^5) + (6*x^4*(b^2*log(F)^2 + 140*c^
4 - 30*b*c^2*log(F)))/(b^5*d^5*log(F)^5) - (12*x^6*(b*log(F) - 28*c^2))/(b^5*d^3*log(F)^5) + (24*c*x^3*(b^2*lo
g(F)^2 + 28*c^4 - 10*b*c^2*log(F)))/(b^5*d^6*log(F)^5) - (24*c*x^5*(3*b*log(F) - 28*c^2))/(b^5*d^4*log(F)^5) -
(4*c*x*(b^3*log(F)^3 - 24*c^6 + 18*b*c^4*log(F) - 6*b^2*c^2*log(F)^2))/(b^5*d^8*log(F)^5)))/(x^8 + c^8/d^8 +
(8*c*x^7)/d + (8*c^7*x)/d^7 + (28*c^2*x^6)/d^2 + (56*c^3*x^5)/d^3 + (70*c^4*x^4)/d^4 + (56*c^5*x^3)/d^5 + (28*
c^6*x^2)/d^6)