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]

-(F^(a + b*(c + d*x)^2)*(120 - 120*b*(c + d*x)^2*Log[F] + 60*b^2*(c + d*x)^4*Log[F]^2 - 20*b^3*(c + d*x)^6*Log
[F]^3 + 5*b^4*(c + d*x)^8*Log[F]^4 - b^5*(c + d*x)^10*Log[F]^5))/(2*b^6*d*Log[F]^6)

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Rubi [C]  time = 0.0726111, antiderivative size = 31, normalized size of antiderivative = 0.3, 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*}

Mathematica [C]  time = 0.0083566, size = 31, normalized size = 0.3 \[ -\frac{F^a \text{Gamma}\left (6,-b \log (F) (c+d x)^2\right )}{2 b^6 d \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[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)

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Maple [B]  time = 0.021, size = 579, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-120+120*ln(F)*b*d^2*x^2+20*ln(F)^3*b^3*c^6-60*ln(F)^2*b^2*c^4+120*ln(F)*b*c^2+240*ln(F)*b*c*d*x-360*ln(F
)^2*b^2*c^2*d^2*x^2-240*ln(F)^2*b^2*c^3*d*x-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+10*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+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+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+210*ln(F)^5*b^5*c^4*d^6*x^6+
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^3-5*ln(F)^4*b^4*c^8+ln(F)^5*
b^5*c^10+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*ln(F)^2)*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 = 3.57992, size = 7426, normalized size = 70.72 \begin{align*} \text{result too large to display} \end{align*}

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*d*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^2/((b*d^2*
log(F))^(3/2)*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*d^2*log(F)/(b*d^2
*log(F))^(3/2))*F^a*c^10*d/sqrt(b*d^2*log(F)) + 55/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*d^2*(erf(sqrt(-(b*d^2
*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*d^2*log(F))^(5/2)*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*d^3*log(F)^2/(b*d^2*log(F))^(5/2) - (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*d^2*log(F))^(5/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)
))*F^a*c^9*d^2/sqrt(b*d^2*log(F)) - 165/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*d^3*(erf(sqrt(-(b*d^2*x + b*c*d)
^2*log(F)/(b*d^2))) - 1)*log(F)^4/((b*d^2*log(F))^(7/2)*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*d^4*log(F)^3/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x + b*c*d)^3*b*c*d*gamma(3/2, -(
b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*d^2*log(F))^(7/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2))
+ b^2*d^4*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^2/(b*d^2*log(F))^(7/2))*F^a*c^8*d^3/sqrt(b*d^2
*log(F)) + 165*(sqrt(pi)*(b*d^2*x + b*c*d)*b^4*c^4*d^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*lo
g(F)^5/((b*d^2*log(F))^(9/2)*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*d^5*log(F)^4/(b*d^2*log(F))^(9/2) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*d^2*gamma(3/2, -(b*d^2*x + b*c*d)^2*lo
g(F)/(b*d^2))*log(F)^5/((b*d^2*log(F))^(9/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 4*b^3*c*d^5*gamma(
2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/(b*d^2*log(F))^(9/2) - (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*d^2*log(F))^(9/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a
*c^7*d^4/sqrt(b*d^2*log(F)) - 231*(sqrt(pi)*(b*d^2*x + b*c*d)*b^5*c^5*d^5*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F
)/(b*d^2))) - 1)*log(F)^6/((b*d^2*log(F))^(11/2)*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*d^6*log(F)^5/(b*d^2*log(F))^(11/2) - 10*(b*d^2*x + b*c*d)^3*b^3*c^3*d^3*gamma(3/2, -
(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*d^2*log(F))^(11/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2
)) + 10*b^4*c^2*d^6*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(11/2) - b^3*d^6*gam
ma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/(b*d^2*log(F))^(11/2) - 5*(b*d^2*x + b*c*d)^5*b*c*d*gamma(
5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*d^2*log(F))^(11/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)
)^(5/2)))*F^a*c^6*d^5/sqrt(b*d^2*log(F)) + 231*(sqrt(pi)*(b*d^2*x + b*c*d)*b^6*c^6*d^6*(erf(sqrt(-(b*d^2*x + b
*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^7/((b*d^2*log(F))^(13/2)*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*d^7*log(F)^6/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x + b*c*d)^3*b^4*c^4*d^4
*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*d^2*log(F))^(13/2)*(-(b*d^2*x + b*c*d)^2*log(F)/
(b*d^2))^(3/2)) + 20*b^5*c^3*d^7*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/(b*d^2*log(F))^(13/2)
- 6*b^4*c*d^7*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x + b*c
*d)^5*b^2*c^2*d^2*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*d^2*log(F))^(13/2)*(-(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*d^2*log(F))^(13/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a*c^5*d^6/sqrt(b*d^2*log(F)) - 165*(s
qrt(pi)*(b*d^2*x + b*c*d)*b^7*c^7*d^7*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^8/((b*d^2*lo
g(F))^(15/2)*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*d^8*log(F)
^7/(b*d^2*log(F))^(15/2) - 21*(b*d^2*x + b*c*d)^3*b^5*c^5*d^5*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*
log(F)^8/((b*d^2*log(F))^(15/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 35*b^6*c^4*d^8*gamma(2, -(b*d^2
*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/(b*d^2*log(F))^(15/2) - 21*b^5*c^2*d^8*gamma(3, -(b*d^2*x + b*c*d)^2*lo
g(F)/(b*d^2))*log(F)^5/(b*d^2*log(F))^(15/2) - 35*(b*d^2*x + b*c*d)^5*b^3*c^3*d^3*gamma(5/2, -(b*d^2*x + b*c*d
)^2*log(F)/(b*d^2))*log(F)^8/((b*d^2*log(F))^(15/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + b^4*d^8*gam
ma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(15/2) - 7*(b*d^2*x + b*c*d)^7*b*c*d*gamma(
7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*d^2*log(F))^(15/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)
)^(7/2)))*F^a*c^4*d^7/sqrt(b*d^2*log(F)) + 165/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^8*c^8*d^8*(erf(sqrt(-(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^9/((b*d^2*log(F))^(17/2)*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*d^9*log(F)^8/(b*d^2*log(F))^(17/2) - 28*(b*d^2*x + b*c*d)^3*b^6*c^6*d
^6*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*d^2*log(F))^(17/2)*(-(b*d^2*x + b*c*d)^2*log(F
)/(b*d^2))^(3/2)) + 56*b^7*c^5*d^9*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/(b*d^2*log(F))^(17/2
) - 56*b^6*c^3*d^9*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/(b*d^2*log(F))^(17/2) - 70*(b*d^2*x
+ b*c*d)^5*b^4*c^4*d^4*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*d^2*log(F))^(17/2)*(-(b*d^
2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + 8*b^5*c*d^9*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/(b*
d^2*log(F))^(17/2) - 28*(b*d^2*x + b*c*d)^7*b^2*c^2*d^2*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)
^9/((b*d^2*log(F))^(17/2)*(-(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*d^2*log(F))^(17/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)))*F
^a*c^3*d^8/sqrt(b*d^2*log(F)) - 55/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^9*c^9*d^9*(erf(sqrt(-(b*d^2*x + b*c*d)^2*lo
g(F)/(b*d^2))) - 1)*log(F)^10/((b*d^2*log(F))^(19/2)*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*d^10*log(F)^9/(b*d^2*log(F))^(19/2) - 36*(b*d^2*x + b*c*d)^3*b^7*c^7*d^7*gamma(3
/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^10/((b*d^2*log(F))^(19/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)
)^(3/2)) + 84*b^8*c^6*d^10*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/(b*d^2*log(F))^(19/2) - 126*
b^7*c^4*d^10*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/(b*d^2*log(F))^(19/2) - 126*(b*d^2*x + b*c
*d)^5*b^5*c^5*d^5*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^10/((b*d^2*log(F))^(19/2)*(-(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))^(5/2)) + 36*b^6*c^2*d^10*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/(b*
d^2*log(F))^(19/2) - 84*(b*d^2*x + b*c*d)^7*b^3*c^3*d^3*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)
^10/((b*d^2*log(F))^(19/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)) - b^5*d^10*gamma(5, -(b*d^2*x + b*c*d)
^2*log(F)/(b*d^2))*log(F)^5/(b*d^2*log(F))^(19/2) - 9*(b*d^2*x + b*c*d)^9*b*c*d*gamma(9/2, -(b*d^2*x + b*c*d)^
2*log(F)/(b*d^2))*log(F)^10/((b*d^2*log(F))^(19/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)))*F^a*c^2*d^9/s
qrt(b*d^2*log(F)) + 11/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^10*c^10*d^10*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d
^2))) - 1)*log(F)^11/((b*d^2*log(F))^(21/2)*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*d^11*log(F)^10/(b*d^2*log(F))^(21/2) - 45*(b*d^2*x + b*c*d)^3*b^8*c^8*d^8*gamma(3/2, -(
b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*d^2*log(F))^(21/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2
)) + 120*b^9*c^7*d^11*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/(b*d^2*log(F))^(21/2) - 252*b^8*c
^5*d^11*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/(b*d^2*log(F))^(21/2) - 210*(b*d^2*x + b*c*d)^5
*b^6*c^6*d^6*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*d^2*log(F))^(21/2)*(-(b*d^2*x + b*c
*d)^2*log(F)/(b*d^2))^(5/2)) + 120*b^7*c^3*d^11*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/(b*d^2*
log(F))^(21/2) - 210*(b*d^2*x + b*c*d)^7*b^4*c^4*d^4*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11
/((b*d^2*log(F))^(21/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)) - 10*b^6*c*d^11*gamma(5, -(b*d^2*x + b*c*
d)^2*log(F)/(b*d^2))*log(F)^6/(b*d^2*log(F))^(21/2) - 45*(b*d^2*x + b*c*d)^9*b^2*c^2*d^2*gamma(9/2, -(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))*log(F)^11/((b*d^2*log(F))^(21/2)*(-(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*d^2*log(F))^(21/2)*(-(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))^(11/2)))*F^a*c*d^10/sqrt(b*d^2*log(F)) - 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^11*c^11
*d^11*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^12/((b*d^2*log(F))^(23/2)*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*d^12*log(F)^11/(b*d^2*log(F))^(23/2) -
55*(b*d^2*x + b*c*d)^3*b^9*c^9*d^9*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*d^2*log(F))^
(23/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 165*b^10*c^8*d^12*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(
b*d^2))*log(F)^10/(b*d^2*log(F))^(23/2) - 462*b^9*c^6*d^12*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F
)^9/(b*d^2*log(F))^(23/2) - 330*(b*d^2*x + b*c*d)^5*b^7*c^7*d^7*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)
)*log(F)^12/((b*d^2*log(F))^(23/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + 330*b^8*c^4*d^12*gamma(4, -(
b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/(b*d^2*log(F))^(23/2) - 462*(b*d^2*x + b*c*d)^7*b^5*c^5*d^5*gamma(
7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*d^2*log(F))^(23/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2
))^(7/2)) - 55*b^7*c^2*d^12*gamma(5, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/(b*d^2*log(F))^(23/2) - 165
*(b*d^2*x + b*c*d)^9*b^3*c^3*d^3*gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^12/((b*d^2*log(F))^(23
/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)) + b^6*d^12*gamma(6, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(
F)^6/(b*d^2*log(F))^(23/2) - 11*(b*d^2*x + b*c*d)^11*b*c*d*gamma(11/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*lo
g(F)^12/((b*d^2*log(F))^(23/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(11/2)))*F^a*d^11/sqrt(b*d^2*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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Fricas [B]  time = 1.54839, size = 999, normalized size = 9.51 \begin{align*} \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 \left (F\right )^{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 \left (F\right )^{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 \left (F\right )^{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 \left (F\right )^{2} + 120 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 120\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{6} d \log \left (F\right )^{6}} \end{align*}

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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Sympy [A]  time = 0.398578, size = 796, normalized size = 7.58 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [A]  time = 1.33087, size = 196, normalized size = 1.87 \begin{align*} \frac{{\left (b^{5} d^{10}{\left (x + \frac{c}{d}\right )}^{10} \log \left (F\right )^{5} - 5 \, b^{4} d^{8}{\left (x + \frac{c}{d}\right )}^{8} \log \left (F\right )^{4} + 20 \, b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{6} \log \left (F\right )^{3} - 60 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{4} \log \left (F\right )^{2} + 120 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{2} \log \left (F\right ) - 120\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^{6} d \log \left (F\right )^{6}} \end{align*}

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)