\(\int F^{a+b (c+d x)^2} (c+d x)^9 \, dx\) [256]

Optimal result

Integrand size = 21, antiderivative size = 88 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {F^{a+b (c+d x)^2} \left (24-24 b (c+d x)^2 \log (F)+12 b^2 (c+d x)^4 \log ^2(F)-4 b^3 (c+d x)^6 \log ^3(F)+b^4 (c+d x)^8 \log ^4(F)\right )}{2 b^5 d \log ^5(F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 88, 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} \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {F^{a+b (c+d x)^2} \left (b^4 \log ^4(F) (c+d x)^8-4 b^3 \log ^3(F) (c+d x)^6+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^2+24\right )}{2 b^5 d \log ^5(F)} \]

[In]

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]

[Out]

(F^(a + b*(c + d*x)^2)*(24 - 24*b*(c + d*x)^2*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 - 4*b^3*(c + d*x)^6*Log[F]^
3 + b^4*(c + d*x)^8*Log[F]^4))/(2*b^5*d*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+b (c+d x)^2} \left (24-24 b (c+d x)^2 \log (F)+12 b^2 (c+d x)^4 \log ^2(F)-4 b^3 (c+d x)^6 \log ^3(F)+b^4 (c+d x)^8 \log ^4(F)\right )}{2 b^5 d \log ^5(F)} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {F^a \Gamma \left (5,-b (c+d x)^2 \log (F)\right )}{2 b^5 d \log ^5(F)} \]

[In]

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(86)=172\).

Time = 0.68 (sec) , antiderivative size = 396, normalized size of antiderivative = 4.50

method	result	size
gosper	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}+8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}+28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x -24 \ln \left (F \right ) b \,d^{2} x^{2}-48 \ln \left (F \right ) b c d x +\ln \left (F \right )^{4} b^{4} c^{8}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{5} \ln \left (F \right )^{5} d}\)	\(396\)
risch	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}+8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}+28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x -24 \ln \left (F \right ) b \,d^{2} x^{2}-48 \ln \left (F \right ) b c d x +\ln \left (F \right )^{4} b^{4} c^{8}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{5} \ln \left (F \right )^{5} d}\)	\(396\)
norman	\(\frac {d^{3} \left (35 \ln \left (F \right )^{2} b^{2} c^{4}-30 \ln \left (F \right ) b \,c^{2}+6\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {\left (\ln \left (F \right )^{4} b^{4} c^{8}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}-24 \ln \left (F \right ) b \,c^{2}+24\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 b^{5} \ln \left (F \right )^{5} d}+\frac {d^{7} x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}+\frac {4 c \left (\ln \left (F \right )^{3} b^{3} c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,c^{2}-6\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {2 d^{5} \left (7 \ln \left (F \right ) b \,c^{2}-1\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {2 d \left (7 \ln \left (F \right )^{3} b^{3} c^{6}-15 \ln \left (F \right )^{2} b^{2} c^{4}+18 \ln \left (F \right ) b \,c^{2}-6\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {4 d^{6} c \,x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {4 c \,d^{4} \left (7 \ln \left (F \right ) b \,c^{2}-3\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {4 d^{2} c \left (7 \ln \left (F \right )^{2} b^{2} c^{4}-10 \ln \left (F \right ) b \,c^{2}+6\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}\)	\(441\)
parallelrisch	\(\frac {\ln \left (F \right )^{4} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{8}-4 \ln \left (F \right )^{3} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{6}+12 \ln \left (F \right )^{2} F^{a +b \left (d x +c \right )^{2}} b^{2} c^{4}-24 \ln \left (F \right ) F^{a +b \left (d x +c \right )^{2}} b \,c^{2}+24 F^{a +b \left (d x +c \right )^{2}}+8 c \,d^{7} F^{a +b \left (d x +c \right )^{2}} x^{7} \ln \left (F \right )^{4} b^{4}+28 \ln \left (F \right )^{4} x^{6} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{2} d^{6}+56 \ln \left (F \right )^{4} x^{5} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{3} d^{5}+70 \ln \left (F \right )^{4} x^{4} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{4} d^{4}+56 \ln \left (F \right )^{4} x^{3} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{5} d^{3}+28 \ln \left (F \right )^{4} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{6} d^{2}+8 \ln \left (F \right )^{4} x \,F^{a +b \left (d x +c \right )^{2}} b^{4} c^{7} d -24 c \,d^{5} F^{a +b \left (d x +c \right )^{2}} x^{5} \ln \left (F \right )^{3} b^{3}-60 \ln \left (F \right )^{3} x^{4} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{2} d^{4}-80 \ln \left (F \right )^{3} x^{3} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{3} d^{3}-60 \ln \left (F \right )^{3} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{4} d^{2}-24 \ln \left (F \right )^{3} x \,F^{a +b \left (d x +c \right )^{2}} b^{3} c^{5} d +48 d^{3} c \,F^{a +b \left (d x +c \right )^{2}} x^{3} \ln \left (F \right )^{2} b^{2}+72 \ln \left (F \right )^{2} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{2} c^{2} d^{2}+48 \ln \left (F \right )^{2} x \,F^{a +b \left (d x +c \right )^{2}} b^{2} c^{3} d -48 c \,F^{a +b \left (d x +c \right )^{2}} x \ln \left (F \right ) b d +d^{8} F^{a +b \left (d x +c \right )^{2}} x^{8} \ln \left (F \right )^{4} b^{4}-4 d^{6} F^{a +b \left (d x +c \right )^{2}} x^{6} \ln \left (F \right )^{3} b^{3}+12 d^{4} F^{a +b \left (d x +c \right )^{2}} x^{4} \ln \left (F \right )^{2} b^{2}-24 d^{2} F^{a +b \left (d x +c \right )^{2}} x^{2} \ln \left (F \right ) b}{2 b^{5} \ln \left (F \right )^{5} d}\)	\(699\)

[In]

int(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x,method=_RETURNVERBOSE)

[Out]

1/2*(24-24*ln(F)*b*c^2+8*c*d^7*x^7*b^4*ln(F)^4+28*ln(F)^4*b^4*c^2*d^6*x^6+56*ln(F)^4*b^4*c^3*d^5*x^5+70*ln(F)^
4*b^4*c^4*d^4*x^4+56*ln(F)^4*b^4*c^5*d^3*x^3+28*ln(F)^4*b^4*c^6*d^2*x^2+8*ln(F)^4*b^4*c^7*d*x-24*c*d^5*x^5*b^3
*ln(F)^3-60*ln(F)^3*b^3*c^2*d^4*x^4-80*ln(F)^3*b^3*c^3*d^3*x^3-60*ln(F)^3*b^3*c^4*d^2*x^2-24*ln(F)^3*b^3*c^5*d
*x+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-24*ln(F)*b*d^2*x^2-48*ln(F)*b*c*
d*x+ln(F)^4*b^4*c^8-4*ln(F)^3*b^3*c^6+12*ln(F)^2*b^2*c^4+d^8*x^8*b^4*ln(F)^4-4*d^6*x^6*b^3*ln(F)^3+12*d^4*x^4*
b^2*ln(F)^2)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)/b^5/ln(F)^5/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (86) = 172\).

Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.68 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {{\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{5} d \log \left (F\right )^{5}} \]

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (87) = 174\).

Time = 0.20 (sec) , antiderivative size = 556, normalized size of antiderivative = 6.32 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b^{4} c^{8} \log {\left (F \right )}^{4} + 8 b^{4} c^{7} d x \log {\left (F \right )}^{4} + 28 b^{4} c^{6} d^{2} x^{2} \log {\left (F \right )}^{4} + 56 b^{4} c^{5} d^{3} x^{3} \log {\left (F \right )}^{4} + 70 b^{4} c^{4} d^{4} x^{4} \log {\left (F \right )}^{4} + 56 b^{4} c^{3} d^{5} x^{5} \log {\left (F \right )}^{4} + 28 b^{4} c^{2} d^{6} x^{6} \log {\left (F \right )}^{4} + 8 b^{4} c d^{7} x^{7} \log {\left (F \right )}^{4} + b^{4} d^{8} x^{8} \log {\left (F \right )}^{4} - 4 b^{3} c^{6} \log {\left (F \right )}^{3} - 24 b^{3} c^{5} d x \log {\left (F \right )}^{3} - 60 b^{3} c^{4} d^{2} x^{2} \log {\left (F \right )}^{3} - 80 b^{3} c^{3} d^{3} x^{3} \log {\left (F \right )}^{3} - 60 b^{3} c^{2} d^{4} x^{4} \log {\left (F \right )}^{3} - 24 b^{3} c d^{5} x^{5} \log {\left (F \right )}^{3} - 4 b^{3} d^{6} x^{6} \log {\left (F \right )}^{3} + 12 b^{2} c^{4} \log {\left (F \right )}^{2} + 48 b^{2} c^{3} d x \log {\left (F \right )}^{2} + 72 b^{2} c^{2} d^{2} x^{2} \log {\left (F \right )}^{2} + 48 b^{2} c d^{3} x^{3} \log {\left (F \right )}^{2} + 12 b^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 24 b c^{2} \log {\left (F \right )} - 48 b c d x \log {\left (F \right )} - 24 b d^{2} x^{2} \log {\left (F \right )} + 24\right )}{2 b^{5} d \log {\left (F \right )}^{5}} & \text {for}\: b^{5} d \log {\left (F \right )}^{5} \neq 0 \\c^{9} x + \frac {9 c^{8} d x^{2}}{2} + 12 c^{7} d^{2} x^{3} + 21 c^{6} d^{3} x^{4} + \frac {126 c^{5} d^{4} x^{5}}{5} + 21 c^{4} d^{5} x^{6} + 12 c^{3} d^{6} x^{7} + \frac {9 c^{2} d^{7} x^{8}}{2} + c d^{8} x^{9} + \frac {d^{9} x^{10}}{10} & \text {otherwise} \end {cases} \]

[In]

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**9,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**2)*(b**4*c**8*log(F)**4 + 8*b**4*c**7*d*x*log(F)**4 + 28*b**4*c**6*d**2*x**2*l
og(F)**4 + 56*b**4*c**5*d**3*x**3*log(F)**4 + 70*b**4*c**4*d**4*x**4*log(F)**4 + 56*b**4*c**3*d**5*x**5*log(F)
**4 + 28*b**4*c**2*d**6*x**6*log(F)**4 + 8*b**4*c*d**7*x**7*log(F)**4 + b**4*d**8*x**8*log(F)**4 - 4*b**3*c**6
*log(F)**3 - 24*b**3*c**5*d*x*log(F)**3 - 60*b**3*c**4*d**2*x**2*log(F)**3 - 80*b**3*c**3*d**3*x**3*log(F)**3
- 60*b**3*c**2*d**4*x**4*log(F)**3 - 24*b**3*c*d**5*x**5*log(F)**3 - 4*b**3*d**6*x**6*log(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 + 1
2*b**2*d**4*x**4*log(F)**2 - 24*b*c**2*log(F) - 48*b*c*d*x*log(F) - 24*b*d**2*x**2*log(F) + 24)/(2*b**5*d*log(
F)**5), Ne(b**5*d*log(F)**5, 0)), (c**9*x + 9*c**8*d*x**2/2 + 12*c**7*d**2*x**3 + 21*c**6*d**3*x**4 + 126*c**5
*d**4*x**5/5 + 21*c**4*d**5*x**6 + 12*c**3*d**6*x**7 + 9*c**2*d**7*x**8/2 + c*d**8*x**9 + d**9*x**10/10, True)
)

Maxima [C] (verification not implemented)

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

Time = 1.45 (sec) , antiderivative size = 3727, normalized size of antiderivative = 42.35 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\text {Too large to display} \]

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x, algorithm="maxima")

[Out]

-9/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^2/((b*log(F))
^(3/2)*d^2*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*log(F)/((b*log(F))^(
3/2)*d))*F^a*c^8/sqrt(b*log(F)) + 18*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)
/(b*d^2))) - 1)*log(F)^3/((b*log(F))^(5/2)*d^3*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^((b*d^2*x + b*
c*d)^2/(b*d^2))*b^2*c*log(F)^2/((b*log(F))^(5/2)*d^2) - (b*d^2*x + b*c*d)^3*gamma(3/2, -(b*d^2*x + b*c*d)^2*lo
g(F)/(b*d^2))*log(F)^3/((b*log(F))^(5/2)*d^5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*c^7*d/sqrt(b*lo
g(F)) - 42*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^4/(
(b*log(F))^(7/2)*d^4*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^3*c^2*lo
g(F)^3/((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(
F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x + b*c*d)^2*lo
g(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^6*d^2/sqrt(b*log(F)) + 63*(sqrt(pi)*(b*d^2*x + b*c*d)*b^4
*c^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^5/((b*log(F))^(9/2)*d^5*sqrt(-(b*d^2*x + b*c*
d)^2*log(F)/(b*d^2))) - 4*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^4*c^3*log(F)^4/((b*log(F))^(9/2)*d^4) - 6*(b*d^2*x
+ b*c*d)^3*b^2*c^2*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(9/2)*d^7*(-(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 4*b^3*c*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F))
^(9/2)*d^4) - (b*d^2*x + b*c*d)^5*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(9/2)*d
^9*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*c^5*d^3/sqrt(b*log(F)) - 63*(sqrt(pi)*(b*d^2*x + b*c*d)*b
^5*c^5*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^6/((b*log(F))^(11/2)*d^6*sqrt(-(b*d^2*x + b
*c*d)^2*log(F)/(b*d^2))) - 5*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^5*c^4*log(F)^5/((b*log(F))^(11/2)*d^5) - 10*(b*
d^2*x + b*c*d)^3*b^3*c^3*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(11/2)*d^8*(-(b*
d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 10*b^4*c^2*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((
b*log(F))^(11/2)*d^5) - b^3*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F))^(11/2)*d^5) - 5
*(b*d^2*x + b*c*d)^5*b*c*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(11/2)*d^10*(-(b
*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*c^4*d^4/sqrt(b*log(F)) + 42*(sqrt(pi)*(b*d^2*x + b*c*d)*b^6*c^6*
(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^7/((b*log(F))^(13/2)*d^7*sqrt(-(b*d^2*x + b*c*d)^2
*log(F)/(b*d^2))) - 6*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^6*c^5*log(F)^6/((b*log(F))^(13/2)*d^6) - 15*(b*d^2*x +
b*c*d)^3*b^4*c^4*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(13/2)*d^9*(-(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 20*b^5*c^3*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F
))^(13/2)*d^6) - 6*b^4*c*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(13/2)*d^6) - 15*(
b*d^2*x + b*c*d)^5*b^2*c^2*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(13/2)*d^11*(-
(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) - (b*d^2*x + b*c*d)^7*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2
))*log(F)^7/((b*log(F))^(13/2)*d^13*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a*c^3*d^5/sqrt(b*log(F)) -
18*(sqrt(pi)*(b*d^2*x + b*c*d)*b^7*c^7*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^8/((b*log(
F))^(15/2)*d^8*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 7*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^7*c^6*log(F)^7
/((b*log(F))^(15/2)*d^7) - 21*(b*d^2*x + b*c*d)^3*b^5*c^5*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(
F)^8/((b*log(F))^(15/2)*d^10*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 35*b^6*c^4*gamma(2, -(b*d^2*x + b*
c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(15/2)*d^7) - 21*b^5*c^2*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d
^2))*log(F)^5/((b*log(F))^(15/2)*d^7) - 35*(b*d^2*x + b*c*d)^5*b^3*c^3*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/
(b*d^2))*log(F)^8/((b*log(F))^(15/2)*d^12*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + b^4*gamma(4, -(b*d^2*
x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(15/2)*d^7) - 7*(b*d^2*x + b*c*d)^7*b*c*gamma(7/2, -(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*log(F))^(15/2)*d^14*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a
*c^2*d^6/sqrt(b*log(F)) + 9/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^8*c^8*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2
))) - 1)*log(F)^9/((b*log(F))^(17/2)*d^9*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 8*F^((b*d^2*x + b*c*d)^2
/(b*d^2))*b^8*c^7*log(F)^8/((b*log(F))^(17/2)*d^8) - 28*(b*d^2*x + b*c*d)^3*b^6*c^6*gamma(3/2, -(b*d^2*x + b*c
*d)^2*log(F)/(b*d^2))*log(F)^9/((b*log(F))^(17/2)*d^11*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 56*b^7*c
^5*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(17/2)*d^8) - 56*b^6*c^3*gamma(3, -(b*d^
2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(17/2)*d^8) - 70*(b*d^2*x + b*c*d)^5*b^4*c^4*gamma(5/2, -(
b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*log(F))^(17/2)*d^13*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2
)) + 8*b^5*c*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(17/2)*d^8) - 28*(b*d^2*x + b*
c*d)^7*b^2*c^2*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*log(F))^(17/2)*d^15*(-(b*d^2*x + b
*c*d)^2*log(F)/(b*d^2))^(7/2)) - (b*d^2*x + b*c*d)^9*gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/
((b*log(F))^(17/2)*d^17*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)))*F^a*c*d^7/sqrt(b*log(F)) - 1/2*(sqrt(pi)
*(b*d^2*x + b*c*d)*b^9*c^9*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^10/((b*log(F))^(19/2)*d
^10*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 9*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^9*c^8*log(F)^9/((b*log(F)
)^(19/2)*d^9) - 36*(b*d^2*x + b*c*d)^3*b^7*c^7*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^10/((b*l
og(F))^(19/2)*d^12*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 84*b^8*c^6*gamma(2, -(b*d^2*x + b*c*d)^2*log
(F)/(b*d^2))*log(F)^8/((b*log(F))^(19/2)*d^9) - 126*b^7*c^4*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(
F)^7/((b*log(F))^(19/2)*d^9) - 126*(b*d^2*x + b*c*d)^5*b^5*c^5*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))
*log(F)^10/((b*log(F))^(19/2)*d^14*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + 36*b^6*c^2*gamma(4, -(b*d^2*
x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(19/2)*d^9) - 84*(b*d^2*x + b*c*d)^7*b^3*c^3*gamma(7/2, -(b*
d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^10/((b*log(F))^(19/2)*d^16*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)
) - b^5*gamma(5, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(19/2)*d^9) - 9*(b*d^2*x + b*c*d)^9
*b*c*gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^10/((b*log(F))^(19/2)*d^18*(-(b*d^2*x + b*c*d)^2*l
og(F)/(b*d^2))^(9/2)))*F^a*d^8/sqrt(b*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^9*erf(sqrt(-b*log(F))*d*x - b*c*l
og(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.41 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {{\left (b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{8} \log \left (F\right )^{4} - 4 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \left (F\right )^{3} + 12 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} - 24 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) + 24\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{5} d \log \left (F\right )^{5}} \]

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 391, normalized size of antiderivative = 4.44 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {12\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^3\,{\ln \left (F\right )}^3\,\left (4\,c^6+24\,c^5\,d\,x+60\,c^4\,d^2\,x^2+80\,c^3\,d^3\,x^3+60\,c^2\,d^4\,x^4+24\,c\,d^5\,x^5+4\,d^6\,x^6\right )}{2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^4\,{\ln \left (F\right )}^4\,\left (c^8+8\,c^7\,d\,x+28\,c^6\,d^2\,x^2+56\,c^5\,d^3\,x^3+70\,c^4\,d^4\,x^4+56\,c^3\,d^5\,x^5+28\,c^2\,d^6\,x^6+8\,c\,d^7\,x^7+d^8\,x^8\right )}{2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b\,\ln \left (F\right )\,\left (24\,c^2+48\,c\,d\,x+24\,d^2\,x^2\right )}{2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^2\,{\ln \left (F\right )}^2\,\left (12\,c^4+48\,c^3\,d\,x+72\,c^2\,d^2\,x^2+48\,c\,d^3\,x^3+12\,d^4\,x^4\right )}{2}}{b^5\,d\,{\ln \left (F\right )}^5} \]

[In]

int(F^(a + b*(c + d*x)^2)*(c + d*x)^9,x)

[Out]

(12*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*b^3*log(F)^3*(4*c^6
+ 4*d^6*x^6 + 24*c*d^5*x^5 + 60*c^4*d^2*x^2 + 80*c^3*d^3*x^3 + 60*c^2*d^4*x^4 + 24*c^5*d*x))/2 + (F^(b*d^2*x^
2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*b^4*log(F)^4*(c^8 + d^8*x^8 + 8*c*d^7*x^7 + 28*c^6*d^2*x^2 + 56*c^5*d^3*x^3 + 7
0*c^4*d^4*x^4 + 56*c^3*d^5*x^5 + 28*c^2*d^6*x^6 + 8*c^7*d*x))/2 - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*b
*log(F)*(24*c^2 + 24*d^2*x^2 + 48*c*d*x))/2 + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*b^2*log(F)^2*(12*c^4
+ 12*d^4*x^4 + 48*c*d^3*x^3 + 72*c^2*d^2*x^2 + 48*c^3*d*x))/2)/(b^5*d*log(F)^5)