3.255 \(\int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx\)
Optimal. Leaf size=105 \[ -\frac {F^{a+b (c+d x)^2} \left (-b^5 \log ^5(F) (c+d x)^{10}+5 b^4 \log ^4(F) (c+d x)^8-20 b^3 \log ^3(F) (c+d x)^6+60 b^2 \log ^2(F) (c+d x)^4-120 b \log (F) (c+d x)^2+120\right )}{2 b^6 d \log ^6(F)} \]
[Out]
-1/2*F^(a+b*(d*x+c)^2)*(120-120*b*(d*x+c)^2*ln(F)+60*b^2*(d*x+c)^4*ln(F)^2-20*b^3*(d*x+c)^6*ln(F)^3+5*b^4*(d*x
+c)^8*ln(F)^4-b^5*(d*x+c)^10*ln(F)^5)/b^6/d/ln(F)^6
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Rubi [C] time = 0.07, antiderivative size = 31, normalized size of antiderivative = 0.30,
number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used =
{2218} \[ -\frac {F^a \text {Gamma}\left (6,-b \log (F) (c+d x)^2\right )}{2 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b*(c + d*x)^2)*(c + d*x)^11,x]
[Out]
-(F^a*Gamma[6, -(b*(c + d*x)^2*Log[F])])/(2*b^6*d*Log[F]^6)
Rule 2218
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx &=-\frac {F^a \Gamma \left (6,-b (c+d x)^2 \log (F)\right )}{2 b^6 d \log ^6(F)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.30 \[ -\frac {F^a \Gamma \left (6,-b (c+d x)^2 \log (F)\right )}{2 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^11,x]
[Out]
-1/2*(F^a*Gamma[6, -(b*(c + d*x)^2*Log[F])])/(b^6*d*Log[F]^6)
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fricas [B] time = 0.47, size = 468, normalized size = 4.46 \[ \frac {{\left ({\left (b^{5} d^{10} x^{10} + 10 \, b^{5} c d^{9} x^{9} + 45 \, b^{5} c^{2} d^{8} x^{8} + 120 \, b^{5} c^{3} d^{7} x^{7} + 210 \, b^{5} c^{4} d^{6} x^{6} + 252 \, b^{5} c^{5} d^{5} x^{5} + 210 \, b^{5} c^{6} d^{4} x^{4} + 120 \, b^{5} c^{7} d^{3} x^{3} + 45 \, b^{5} c^{8} d^{2} x^{2} + 10 \, b^{5} c^{9} d x + b^{5} c^{10}\right )} \log \relax (F)^{5} - 5 \, {\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \relax (F)^{4} + 20 \, {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \relax (F)^{3} - 60 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \relax (F)^{2} + 120 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F) - 120\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{6} d \log \relax (F)^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x, algorithm="fricas")
[Out]
1/2*((b^5*d^10*x^10 + 10*b^5*c*d^9*x^9 + 45*b^5*c^2*d^8*x^8 + 120*b^5*c^3*d^7*x^7 + 210*b^5*c^4*d^6*x^6 + 252*
b^5*c^5*d^5*x^5 + 210*b^5*c^6*d^4*x^4 + 120*b^5*c^7*d^3*x^3 + 45*b^5*c^8*d^2*x^2 + 10*b^5*c^9*d*x + b^5*c^10)*
log(F)^5 - 5*(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 + 5
6*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 + 20*(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 - 60*(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 + 120*(b*d^2*x^2 + 2*b*c*d*x
+ b*c^2)*log(F) - 120)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)/(b^6*d*log(F)^6)
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giac [A] time = 0.34, size = 145, normalized size = 1.38 \[ \frac {{\left (b^{5} d^{10} {\left (x + \frac {c}{d}\right )}^{10} \log \relax (F)^{5} - 5 \, b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{8} \log \relax (F)^{4} + 20 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \relax (F)^{3} - 60 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \relax (F)^{2} + 120 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \relax (F) - 120\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{2 \, b^{6} d \log \relax (F)^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x, algorithm="giac")
[Out]
1/2*(b^5*d^10*(x + c/d)^10*log(F)^5 - 5*b^4*d^8*(x + c/d)^8*log(F)^4 + 20*b^3*d^6*(x + c/d)^6*log(F)^3 - 60*b^
2*d^4*(x + c/d)^4*log(F)^2 + 120*b*d^2*(x + c/d)^2*log(F) - 120)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^
2*log(F) + a*log(F))/(b^6*d*log(F)^6)
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maple [B] time = 0.02, size = 579, normalized size = 5.51 \[ \frac {\left (b^{5} d^{10} x^{10} \ln \relax (F )^{5}+10 b^{5} c \,d^{9} x^{9} \ln \relax (F )^{5}+45 b^{5} c^{2} d^{8} x^{8} \ln \relax (F )^{5}+120 b^{5} c^{3} d^{7} x^{7} \ln \relax (F )^{5}+210 b^{5} c^{4} d^{6} x^{6} \ln \relax (F )^{5}+252 b^{5} c^{5} d^{5} x^{5} \ln \relax (F )^{5}+210 b^{5} c^{6} d^{4} x^{4} \ln \relax (F )^{5}-5 b^{4} d^{8} x^{8} \ln \relax (F )^{4}+120 b^{5} c^{7} d^{3} x^{3} \ln \relax (F )^{5}-40 b^{4} c \,d^{7} x^{7} \ln \relax (F )^{4}+45 b^{5} c^{8} d^{2} x^{2} \ln \relax (F )^{5}-140 b^{4} c^{2} d^{6} x^{6} \ln \relax (F )^{4}+10 b^{5} c^{9} d x \ln \relax (F )^{5}-280 b^{4} c^{3} d^{5} x^{5} \ln \relax (F )^{4}+b^{5} c^{10} \ln \relax (F )^{5}-350 b^{4} c^{4} d^{4} x^{4} \ln \relax (F )^{4}-280 b^{4} c^{5} d^{3} x^{3} \ln \relax (F )^{4}-140 b^{4} c^{6} d^{2} x^{2} \ln \relax (F )^{4}+20 b^{3} d^{6} x^{6} \ln \relax (F )^{3}-40 b^{4} c^{7} d x \ln \relax (F )^{4}+120 b^{3} c \,d^{5} x^{5} \ln \relax (F )^{3}-5 b^{4} c^{8} \ln \relax (F )^{4}+300 b^{3} c^{2} d^{4} x^{4} \ln \relax (F )^{3}+400 b^{3} c^{3} d^{3} x^{3} \ln \relax (F )^{3}+300 b^{3} c^{4} d^{2} x^{2} \ln \relax (F )^{3}+120 b^{3} c^{5} d x \ln \relax (F )^{3}+20 b^{3} c^{6} \ln \relax (F )^{3}-60 b^{2} d^{4} x^{4} \ln \relax (F )^{2}-240 b^{2} c \,d^{3} x^{3} \ln \relax (F )^{2}-360 b^{2} c^{2} d^{2} x^{2} \ln \relax (F )^{2}-240 b^{2} c^{3} d x \ln \relax (F )^{2}-60 b^{2} c^{4} \ln \relax (F )^{2}+120 b \,d^{2} x^{2} \ln \relax (F )+240 b c d x \ln \relax (F )+120 b \,c^{2} \ln \relax (F )-120\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{6} d \ln \relax (F )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a+(d*x+c)^2*b)*(d*x+c)^11,x)
[Out]
1/2*(-120+120*ln(F)*b*c^2+120*ln(F)*b*d^2*x^2+240*ln(F)*b*c*d*x+ln(F)^5*b^5*c^10+20*ln(F)^3*b^3*c^6-60*ln(F)^2
*b^2*c^4-5*ln(F)^4*b^4*c^8+d^10*x^10*b^5*ln(F)^5-5*d^8*x^8*b^4*ln(F)^4+20*d^6*x^6*b^3*ln(F)^3-60*d^4*x^4*b^2*l
n(F)^2+400*ln(F)^3*b^3*c^3*d^3*x^3+300*ln(F)^3*b^3*c^4*d^2*x^2+120*ln(F)^3*b^3*c^5*d*x-240*d^3*c*x^3*b^2*ln(F)
^2-360*ln(F)^2*b^2*c^2*d^2*x^2-240*ln(F)^2*b^2*c^3*d*x+252*ln(F)^5*b^5*c^5*d^5*x^5+210*ln(F)^5*b^5*c^6*d^4*x^4
+120*ln(F)^5*b^5*c^7*d^3*x^3-40*c*d^7*x^7*b^4*ln(F)^4+45*ln(F)^5*b^5*c^8*d^2*x^2-140*ln(F)^4*b^4*c^2*d^6*x^6+1
0*ln(F)^5*b^5*c^9*d*x-280*ln(F)^4*b^4*c^3*d^5*x^5-350*ln(F)^4*b^4*c^4*d^4*x^4-280*ln(F)^4*b^4*c^5*d^3*x^3-140*
ln(F)^4*b^4*c^6*d^2*x^2-40*ln(F)^4*b^4*c^7*d*x+120*c*d^5*x^5*b^3*ln(F)^3+300*ln(F)^3*b^3*c^2*d^4*x^4+210*ln(F)
^5*b^5*c^4*d^6*x^6+10*d^9*c*x^9*b^5*ln(F)^5+45*ln(F)^5*b^5*c^2*d^8*x^8+120*ln(F)^5*b^5*c^3*d^7*x^7)*F^(b*d^2*x
^2+2*b*c*d*x+b*c^2+a)/b^6/ln(F)^6/d
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maxima [C] time = 15.45, size = 5261, normalized size = 50.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x, algorithm="maxima")
[Out]
-11/2*(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^10/sqrt(b*log(F)) + 55/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*lo
g(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*log(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^9*d/sqrt(
b*log(F)) - 165/2*(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*log(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*log(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^8*d^2/sqrt(b*log(F)) + 165*(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^7*d^3/sqrt(b*log(F)) - 231*(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))*lo
g(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^6*d^4/sqrt(b*log(F)) + 231*(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^5*d^5/sqrt(b
*log(F)) - 165*(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^4*d^6/sqrt(b*log(F)) + 165/2*(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*ga
mma(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^3*d^7/sqrt(b*log(F))
- 55/2*(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*log(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*g
amma(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*log(F)/(b*d^2))^(9/2)))*F^a*c^2*d^8/sqrt(b*log(F)) + 11/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^10*c^10
*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^11/((b*log(F))^(21/2)*d^11*sqrt(-(b*d^2*x + b*c*d
)^2*log(F)/(b*d^2))) - 10*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^10*c^9*log(F)^10/((b*log(F))^(21/2)*d^10) - 45*(b*
d^2*x + b*c*d)^3*b^8*c^8*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*log(F))^(21/2)*d^13*(-(
b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 120*b^9*c^7*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9
/((b*log(F))^(21/2)*d^10) - 252*b^8*c^5*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*log(F))^(21
/2)*d^10) - 210*(b*d^2*x + b*c*d)^5*b^6*c^6*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*log(
F))^(21/2)*d^15*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + 120*b^7*c^3*gamma(4, -(b*d^2*x + b*c*d)^2*log(F
)/(b*d^2))*log(F)^7/((b*log(F))^(21/2)*d^10) - 210*(b*d^2*x + b*c*d)^7*b^4*c^4*gamma(7/2, -(b*d^2*x + b*c*d)^2
*log(F)/(b*d^2))*log(F)^11/((b*log(F))^(21/2)*d^17*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)) - 10*b^6*c*gam
ma(5, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(21/2)*d^10) - 45*(b*d^2*x + b*c*d)^9*b^2*c^2*
gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*log(F))^(21/2)*d^19*(-(b*d^2*x + b*c*d)^2*log(F)
/(b*d^2))^(9/2)) - (b*d^2*x + b*c*d)^11*gamma(11/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*log(F))
^(21/2)*d^21*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(11/2)))*F^a*c*d^9/sqrt(b*log(F)) - 1/2*(sqrt(pi)*(b*d^2*x
+ b*c*d)*b^11*c^11*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^12/((b*log(F))^(23/2)*d^12*sqrt
(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 11*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^11*c^10*log(F)^11/((b*log(F))^(2
3/2)*d^11) - 55*(b*d^2*x + b*c*d)^3*b^9*c^9*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*log(
F))^(23/2)*d^14*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 165*b^10*c^8*gamma(2, -(b*d^2*x + b*c*d)^2*log(
F)/(b*d^2))*log(F)^10/((b*log(F))^(23/2)*d^11) - 462*b^9*c^6*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log
(F)^9/((b*log(F))^(23/2)*d^11) - 330*(b*d^2*x + b*c*d)^5*b^7*c^7*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2
))*log(F)^12/((b*log(F))^(23/2)*d^16*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + 330*b^8*c^4*gamma(4, -(b*d
^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*log(F))^(23/2)*d^11) - 462*(b*d^2*x + b*c*d)^7*b^5*c^5*gamma(7/2,
-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*log(F))^(23/2)*d^18*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^
(7/2)) - 55*b^7*c^2*gamma(5, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(23/2)*d^11) - 165*(b*d
^2*x + b*c*d)^9*b^3*c^3*gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*log(F))^(23/2)*d^20*(-(b
*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)) + b^6*gamma(6, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(
F))^(23/2)*d^11) - 11*(b*d^2*x + b*c*d)^11*b*c*gamma(11/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*
log(F))^(23/2)*d^22*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(11/2)))*F^a*d^10/sqrt(b*log(F)) + 1/2*sqrt(pi)*F^(b
*c^2 + a)*c^11*erf(sqrt(-b*log(F))*d*x - b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)
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mupad [B] time = 4.15, size = 553, normalized size = 5.27 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (c^{10}+10\,c^9\,d\,x+45\,c^8\,d^2\,x^2+120\,c^7\,d^3\,x^3+210\,c^6\,d^4\,x^4+252\,c^5\,d^5\,x^5+210\,c^4\,d^6\,x^6+120\,c^3\,d^7\,x^7+45\,c^2\,d^8\,x^8+10\,c\,d^9\,x^9+d^{10}\,x^{10}\right )}{2\,b\,d\,\ln \relax (F)}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (5\,c^8+40\,c^7\,d\,x+140\,c^6\,d^2\,x^2+280\,c^5\,d^3\,x^3+350\,c^4\,d^4\,x^4+280\,c^3\,d^5\,x^5+140\,c^2\,d^6\,x^6+40\,c\,d^7\,x^7+5\,d^8\,x^8\right )}{2\,b^2\,d\,{\ln \relax (F)}^2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (60\,c^4+240\,c^3\,d\,x+360\,c^2\,d^2\,x^2+240\,c\,d^3\,x^3+60\,d^4\,x^4\right )}{2\,b^4\,d\,{\ln \relax (F)}^4}-\frac {60\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}}{b^6\,d\,{\ln \relax (F)}^6}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (20\,c^6+120\,c^5\,d\,x+300\,c^4\,d^2\,x^2+400\,c^3\,d^3\,x^3+300\,c^2\,d^4\,x^4+120\,c\,d^5\,x^5+20\,d^6\,x^6\right )}{2\,b^3\,d\,{\ln \relax (F)}^3}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (120\,c^2+240\,c\,d\,x+120\,d^2\,x^2\right )}{2\,b^5\,d\,{\ln \relax (F)}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a + b*(c + d*x)^2)*(c + d*x)^11,x)
[Out]
(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(c^10 + d^10*x^10 + 10*c*d^9*x^9 + 45*c^8*d^2*x^2 + 120*c^7*d^3*x^3
+ 210*c^6*d^4*x^4 + 252*c^5*d^5*x^5 + 210*c^4*d^6*x^6 + 120*c^3*d^7*x^7 + 45*c^2*d^8*x^8 + 10*c^9*d*x))/(2*b*
d*log(F)) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(5*c^8 + 5*d^8*x^8 + 40*c*d^7*x^7 + 140*c^6*d^2*x^2 + 2
80*c^5*d^3*x^3 + 350*c^4*d^4*x^4 + 280*c^3*d^5*x^5 + 140*c^2*d^6*x^6 + 40*c^7*d*x))/(2*b^2*d*log(F)^2) - (F^(b
*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(60*c^4 + 60*d^4*x^4 + 240*c*d^3*x^3 + 360*c^2*d^2*x^2 + 240*c^3*d*x))/(
2*b^4*d*log(F)^4) - (60*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x))/(b^6*d*log(F)^6) + (F^(b*d^2*x^2)*F^a*F^(b*
c^2)*F^(2*b*c*d*x)*(20*c^6 + 20*d^6*x^6 + 120*c*d^5*x^5 + 300*c^4*d^2*x^2 + 400*c^3*d^3*x^3 + 300*c^2*d^4*x^4
+ 120*c^5*d*x))/(2*b^3*d*log(F)^3) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(120*c^2 + 120*d^2*x^2 + 240*c
*d*x))/(2*b^5*d*log(F)^5)
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sympy [A] time = 0.55, size = 796, normalized size = 7.58 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b^{5} c^{10} \log {\relax (F )}^{5} + 10 b^{5} c^{9} d x \log {\relax (F )}^{5} + 45 b^{5} c^{8} d^{2} x^{2} \log {\relax (F )}^{5} + 120 b^{5} c^{7} d^{3} x^{3} \log {\relax (F )}^{5} + 210 b^{5} c^{6} d^{4} x^{4} \log {\relax (F )}^{5} + 252 b^{5} c^{5} d^{5} x^{5} \log {\relax (F )}^{5} + 210 b^{5} c^{4} d^{6} x^{6} \log {\relax (F )}^{5} + 120 b^{5} c^{3} d^{7} x^{7} \log {\relax (F )}^{5} + 45 b^{5} c^{2} d^{8} x^{8} \log {\relax (F )}^{5} + 10 b^{5} c d^{9} x^{9} \log {\relax (F )}^{5} + b^{5} d^{10} x^{10} \log {\relax (F )}^{5} - 5 b^{4} c^{8} \log {\relax (F )}^{4} - 40 b^{4} c^{7} d x \log {\relax (F )}^{4} - 140 b^{4} c^{6} d^{2} x^{2} \log {\relax (F )}^{4} - 280 b^{4} c^{5} d^{3} x^{3} \log {\relax (F )}^{4} - 350 b^{4} c^{4} d^{4} x^{4} \log {\relax (F )}^{4} - 280 b^{4} c^{3} d^{5} x^{5} \log {\relax (F )}^{4} - 140 b^{4} c^{2} d^{6} x^{6} \log {\relax (F )}^{4} - 40 b^{4} c d^{7} x^{7} \log {\relax (F )}^{4} - 5 b^{4} d^{8} x^{8} \log {\relax (F )}^{4} + 20 b^{3} c^{6} \log {\relax (F )}^{3} + 120 b^{3} c^{5} d x \log {\relax (F )}^{3} + 300 b^{3} c^{4} d^{2} x^{2} \log {\relax (F )}^{3} + 400 b^{3} c^{3} d^{3} x^{3} \log {\relax (F )}^{3} + 300 b^{3} c^{2} d^{4} x^{4} \log {\relax (F )}^{3} + 120 b^{3} c d^{5} x^{5} \log {\relax (F )}^{3} + 20 b^{3} d^{6} x^{6} \log {\relax (F )}^{3} - 60 b^{2} c^{4} \log {\relax (F )}^{2} - 240 b^{2} c^{3} d x \log {\relax (F )}^{2} - 360 b^{2} c^{2} d^{2} x^{2} \log {\relax (F )}^{2} - 240 b^{2} c d^{3} x^{3} \log {\relax (F )}^{2} - 60 b^{2} d^{4} x^{4} \log {\relax (F )}^{2} + 120 b c^{2} \log {\relax (F )} + 240 b c d x \log {\relax (F )} + 120 b d^{2} x^{2} \log {\relax (F )} - 120\right )}{2 b^{6} d \log {\relax (F )}^{6}} & \text {for}\: 2 b^{6} d \log {\relax (F )}^{6} \neq 0 \\c^{11} x + \frac {11 c^{10} d x^{2}}{2} + \frac {55 c^{9} d^{2} x^{3}}{3} + \frac {165 c^{8} d^{3} x^{4}}{4} + 66 c^{7} d^{4} x^{5} + 77 c^{6} d^{5} x^{6} + 66 c^{5} d^{6} x^{7} + \frac {165 c^{4} d^{7} x^{8}}{4} + \frac {55 c^{3} d^{8} x^{9}}{3} + \frac {11 c^{2} d^{9} x^{10}}{2} + c d^{10} x^{11} + \frac {d^{11} x^{12}}{12} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**11,x)
[Out]
Piecewise((F**(a + b*(c + d*x)**2)*(b**5*c**10*log(F)**5 + 10*b**5*c**9*d*x*log(F)**5 + 45*b**5*c**8*d**2*x**2
*log(F)**5 + 120*b**5*c**7*d**3*x**3*log(F)**5 + 210*b**5*c**6*d**4*x**4*log(F)**5 + 252*b**5*c**5*d**5*x**5*l
og(F)**5 + 210*b**5*c**4*d**6*x**6*log(F)**5 + 120*b**5*c**3*d**7*x**7*log(F)**5 + 45*b**5*c**2*d**8*x**8*log(
F)**5 + 10*b**5*c*d**9*x**9*log(F)**5 + b**5*d**10*x**10*log(F)**5 - 5*b**4*c**8*log(F)**4 - 40*b**4*c**7*d*x*
log(F)**4 - 140*b**4*c**6*d**2*x**2*log(F)**4 - 280*b**4*c**5*d**3*x**3*log(F)**4 - 350*b**4*c**4*d**4*x**4*lo
g(F)**4 - 280*b**4*c**3*d**5*x**5*log(F)**4 - 140*b**4*c**2*d**6*x**6*log(F)**4 - 40*b**4*c*d**7*x**7*log(F)**
4 - 5*b**4*d**8*x**8*log(F)**4 + 20*b**3*c**6*log(F)**3 + 120*b**3*c**5*d*x*log(F)**3 + 300*b**3*c**4*d**2*x**
2*log(F)**3 + 400*b**3*c**3*d**3*x**3*log(F)**3 + 300*b**3*c**2*d**4*x**4*log(F)**3 + 120*b**3*c*d**5*x**5*log
(F)**3 + 20*b**3*d**6*x**6*log(F)**3 - 60*b**2*c**4*log(F)**2 - 240*b**2*c**3*d*x*log(F)**2 - 360*b**2*c**2*d*
*2*x**2*log(F)**2 - 240*b**2*c*d**3*x**3*log(F)**2 - 60*b**2*d**4*x**4*log(F)**2 + 120*b*c**2*log(F) + 240*b*c
*d*x*log(F) + 120*b*d**2*x**2*log(F) - 120)/(2*b**6*d*log(F)**6), Ne(2*b**6*d*log(F)**6, 0)), (c**11*x + 11*c*
*10*d*x**2/2 + 55*c**9*d**2*x**3/3 + 165*c**8*d**3*x**4/4 + 66*c**7*d**4*x**5 + 77*c**6*d**5*x**6 + 66*c**5*d*
*6*x**7 + 165*c**4*d**7*x**8/4 + 55*c**3*d**8*x**9/3 + 11*c**2*d**9*x**10/2 + c*d**10*x**11 + d**11*x**12/12,
True))
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