∫ Fa+b(c+dx)2(c + dx)12dx

3.267 \(\int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx\)

Optimal. Leaf size=49 \[ -\frac{F^a (c+d x)^{13} \text{Gamma}\left (\frac{13}{2},-b \log (F) (c+d x)^2\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{13/2}} \]

[Out]

-(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(2*d*(-(b*(c + d*x)^2*Log[F]))^(13/2))

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Rubi [A] time = 0.0656325, antiderivative size = 49, normalized size of antiderivative = 1., 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} \[ -\frac{F^a (c+d x)^{13} \text{Gamma}\left (\frac{13}{2},-b \log (F) (c+d x)^2\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{13/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^12,x]

[Out]

-(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(2*d*(-(b*(c + d*x)^2*Log[F]))^(13/2))

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)^{12} \, dx &=-\frac{F^a (c+d x)^{13} \Gamma \left (\frac{13}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b (c+d x)^2 \log (F)\right )^{13/2}}\\ \end{align*}

Mathematica [A] time = 0.0276344, size = 49, normalized size = 1. \[ -\frac{F^a (c+d x)^{13} \text{Gamma}\left (\frac{13}{2},-b \log (F) (c+d x)^2\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{13/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^12,x]

[Out]

-(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(2*d*(-(b*(c + d*x)^2*Log[F]))^(13/2))

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Maple [B] time = 0.504, size = 1896, normalized size = 38.7 \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)^12,x)

[Out]

165/2*d^7*c^3/ln(F)/b*x^8*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+3465/8*d^2*c^4/ln(F)^3/b^3*x^3*F^(b*d^2*x^

2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-693/2*d^4*c^4/ln(F)^2/b^2*x^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-231*d^5

*c^3/ln(F)^2/b^2*x^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+3465/8*d^3*c^3/ln(F)^3/b^3*x^4*F^(b*d^2*x^2)*F^

(2*b*c*d*x)*F^(c^2*b)*F^a-3465/8*d*c^3/ln(F)^4/b^4*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-99*d^6*c^2/ln

(F)^2/b^2*x^7*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-3465/8*d^2*c^2/ln(F)^4/b^4*x^3*F^(b*d^2*x^2)*F^(2*b*c*

d*x)*F^(c^2*b)*F^a+2079/8*d^4*c^2/ln(F)^3/b^3*x^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-99/4*d^7*c/ln(F)^2

/b^2*x^8*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+693/8*d^5*c/ln(F)^3/b^3*x^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(

c^2*b)*F^a-3465/16*d^3*c/ln(F)^4/b^4*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+10395/32*d*c/ln(F)^5/b^5*x^

2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+55/2*d^8*c^2/ln(F)/b*x^9*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a

+231*d^4*c^6/ln(F)/b*x^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+165*d^3*c^7/ln(F)/b*x^4*F^(b*d^2*x^2)*F^(2*

b*c*d*x)*F^(c^2*b)*F^a+165/2*d^2*c^8/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+55/2*d*c^9/ln(F)/b*

x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-99*d*c^7/ln(F)^2/b^2*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F

^a+11/2*d^9*c/ln(F)/b*x^10*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-231*d^2*c^6/ln(F)^2/b^2*x^3*F^(b*d^2*x^2)

*F^(2*b*c*d*x)*F^(c^2*b)*F^a+231*d^5*c^5/ln(F)/b*x^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-693/2*d^3*c^5/l

n(F)^2/b^2*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+2079/8*d*c^5/ln(F)^3/b^3*x^2*F^(b*d^2*x^2)*F^(2*b*c*d

*x)*F^(c^2*b)*F^a+165*d^6*c^4/ln(F)/b*x^7*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-10395/64/d*c/ln(F)^6/b^6*F

^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-11/4/d*c^9/ln(F)^2/b^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+99/8

/d*c^7/ln(F)^3/b^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-693/16/d*c^5/ln(F)^4/b^4*F^(b*d^2*x^2)*F^(2*b*c*d

*x)*F^(c^2*b)*F^a+3465/32/d*c^3/ln(F)^5/b^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+1/2/d*c^11/ln(F)/b*F^(b*

d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-3465/16*c^4/ln(F)^4/b^4*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+10395

/32*c^2/ln(F)^5/b^5*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+11/2*c^10/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x

)*F^(c^2*b)*F^a-99/4*c^8/ln(F)^2/b^2*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+693/8*c^6/ln(F)^3/b^3*x*F^(b*

d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+1/2*d^10/ln(F)/b*x^11*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-11/4*d^8/

ln(F)^2/b^2*x^9*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+99/8*d^6/ln(F)^3/b^3*x^7*F^(b*d^2*x^2)*F^(2*b*c*d*x)

*F^(c^2*b)*F^a+3465/32*d^2/ln(F)^5/b^5*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-693/16*d^4/ln(F)^4/b^4*x^

5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-10395/64/ln(F)^6/b^6*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-1

0395/128/d/ln(F)^6/b^6*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))

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Maxima [B] time = 3.66944, size = 8659, normalized size = 176.71 \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)^12,x, algorithm="maxima")

[Out]

-6*(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*lo

g(F))^(3/2))*F^a*c^11*d/sqrt(b*d^2*log(F)) + 33*(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^10*d^2/sqrt(b*d^2*log(F)) - 110*(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^9*d^3/sqrt(b*d^2*log(F

)) + 495/2*(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)*log(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*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))^(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^8

*d^4/sqrt(b*d^2*log(F)) - 396*(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*gamma(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^7*d^5/sqrt(b*d^2*log(F)) + 462*(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*gam

ma(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^6*d^6/sqrt(b*d^2*log(F)) - 396*(sqrt(

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*log(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*log(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*gamma(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^5*d^7/sqrt(b*d^2*log(F)) + 495/2*(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*g

amma(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

^4*d^8/sqrt(b*d^2*log(F)) - 110*(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*log(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*l

og(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*lo

g(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^3*d^9/sqrt(b

*d^2*log(F)) + 33*(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)) + 12

0*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*l

og(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*lo

g(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^2*d^10/sqrt(b*d^2*log(F)) - 6*(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))*log(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*c*d^11/sqrt(b*d^2*log(F)) + 1/2*(sq

rt(pi)*(b*d^2*x + b*c*d)*b^12*c^12*d^12*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^13/((b*d^2

*log(F))^(25/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 12*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^12*c^11*d^13

*log(F)^12/(b*d^2*log(F))^(25/2) - 66*(b*d^2*x + b*c*d)^3*b^10*c^10*d^10*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F

)/(b*d^2))*log(F)^13/((b*d^2*log(F))^(25/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 220*b^11*c^9*d^13*g

amma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^11/(b*d^2*log(F))^(25/2) - 792*b^10*c^7*d^13*gamma(3, -(b*

d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^10/(b*d^2*log(F))^(25/2) - 495*(b*d^2*x + b*c*d)^5*b^8*c^8*d^8*gamma(5

/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^13/((b*d^2*log(F))^(25/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)

)^(5/2)) + 792*b^9*c^5*d^13*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/(b*d^2*log(F))^(25/2) - 924

*(b*d^2*x + b*c*d)^7*b^6*c^6*d^6*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^13/((b*d^2*log(F))^(25

/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)) - 220*b^8*c^3*d^13*gamma(5, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^

2))*log(F)^8/(b*d^2*log(F))^(25/2) - 495*(b*d^2*x + b*c*d)^9*b^4*c^4*d^4*gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F

)/(b*d^2))*log(F)^13/((b*d^2*log(F))^(25/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)) + 12*b^7*c*d^13*gamma

(6, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/(b*d^2*log(F))^(25/2) - 66*(b*d^2*x + b*c*d)^11*b^2*c^2*d^2*

gamma(11/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^13/((b*d^2*log(F))^(25/2)*(-(b*d^2*x + b*c*d)^2*log(F)

/(b*d^2))^(11/2)) - (b*d^2*x + b*c*d)^13*gamma(13/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^13/((b*d^2*lo

g(F))^(25/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(13/2)))*F^a*d^12/sqrt(b*d^2*log(F)) + 1/2*sqrt(pi)*F^(b*c^

2 + a)*c^12*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 [A] time = 1.63886, size = 1355, normalized size = 27.65 \begin{align*} -\frac{10395 \, \sqrt{\pi } \sqrt{-b d^{2} \log \left (F\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) - 2 \,{\left (32 \,{\left (b^{6} d^{12} x^{11} + 11 \, b^{6} c d^{11} x^{10} + 55 \, b^{6} c^{2} d^{10} x^{9} + 165 \, b^{6} c^{3} d^{9} x^{8} + 330 \, b^{6} c^{4} d^{8} x^{7} + 462 \, b^{6} c^{5} d^{7} x^{6} + 462 \, b^{6} c^{6} d^{6} x^{5} + 330 \, b^{6} c^{7} d^{5} x^{4} + 165 \, b^{6} c^{8} d^{4} x^{3} + 55 \, b^{6} c^{9} d^{3} x^{2} + 11 \, b^{6} c^{10} d^{2} x + b^{6} c^{11} d\right )} \log \left (F\right )^{6} - 176 \,{\left (b^{5} d^{10} x^{9} + 9 \, b^{5} c d^{9} x^{8} + 36 \, b^{5} c^{2} d^{8} x^{7} + 84 \, b^{5} c^{3} d^{7} x^{6} + 126 \, b^{5} c^{4} d^{6} x^{5} + 126 \, b^{5} c^{5} d^{5} x^{4} + 84 \, b^{5} c^{6} d^{4} x^{3} + 36 \, b^{5} c^{7} d^{3} x^{2} + 9 \, b^{5} c^{8} d^{2} x + b^{5} c^{9} d\right )} \log \left (F\right )^{5} + 792 \,{\left (b^{4} d^{8} x^{7} + 7 \, b^{4} c d^{7} x^{6} + 21 \, b^{4} c^{2} d^{6} x^{5} + 35 \, b^{4} c^{3} d^{5} x^{4} + 35 \, b^{4} c^{4} d^{4} x^{3} + 21 \, b^{4} c^{5} d^{3} x^{2} + 7 \, b^{4} c^{6} d^{2} x + b^{4} c^{7} d\right )} \log \left (F\right )^{4} - 2772 \,{\left (b^{3} d^{6} x^{5} + 5 \, b^{3} c d^{5} x^{4} + 10 \, b^{3} c^{2} d^{4} x^{3} + 10 \, b^{3} c^{3} d^{3} x^{2} + 5 \, b^{3} c^{4} d^{2} x + b^{3} c^{5} d\right )} \log \left (F\right )^{3} + 6930 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2} - 10395 \,{\left (b d^{2} x + b c d\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{128 \, b^{7} d^{2} \log \left (F\right )^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^12,x, algorithm="fricas")

[Out]

-1/128*(10395*sqrt(pi)*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) - 2*(32*(b^6*d^12*x^11 + 1

1*b^6*c*d^11*x^10 + 55*b^6*c^2*d^10*x^9 + 165*b^6*c^3*d^9*x^8 + 330*b^6*c^4*d^8*x^7 + 462*b^6*c^5*d^7*x^6 + 46

2*b^6*c^6*d^6*x^5 + 330*b^6*c^7*d^5*x^4 + 165*b^6*c^8*d^4*x^3 + 55*b^6*c^9*d^3*x^2 + 11*b^6*c^10*d^2*x + b^6*c

^11*d)*log(F)^6 - 176*(b^5*d^10*x^9 + 9*b^5*c*d^9*x^8 + 36*b^5*c^2*d^8*x^7 + 84*b^5*c^3*d^7*x^6 + 126*b^5*c^4*

d^6*x^5 + 126*b^5*c^5*d^5*x^4 + 84*b^5*c^6*d^4*x^3 + 36*b^5*c^7*d^3*x^2 + 9*b^5*c^8*d^2*x + b^5*c^9*d)*log(F)^

5 + 792*(b^4*d^8*x^7 + 7*b^4*c*d^7*x^6 + 21*b^4*c^2*d^6*x^5 + 35*b^4*c^3*d^5*x^4 + 35*b^4*c^4*d^4*x^3 + 21*b^4

*c^5*d^3*x^2 + 7*b^4*c^6*d^2*x + b^4*c^7*d)*log(F)^4 - 2772*(b^3*d^6*x^5 + 5*b^3*c*d^5*x^4 + 10*b^3*c^2*d^4*x^

3 + 10*b^3*c^3*d^3*x^2 + 5*b^3*c^4*d^2*x + b^3*c^5*d)*log(F)^3 + 6930*(b^2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c

^2*d^2*x + b^2*c^3*d)*log(F)^2 - 10395*(b*d^2*x + b*c*d)*log(F))*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(b^7*d

^2*log(F)^7)

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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**12,x)

[Out]

Timed out

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Giac [A] time = 1.29933, size = 263, normalized size = 5.37 \begin{align*} \frac{{\left (32 \, b^{5} d^{10}{\left (x + \frac{c}{d}\right )}^{11} \log \left (F\right )^{5} - 176 \, b^{4} d^{8}{\left (x + \frac{c}{d}\right )}^{9} \log \left (F\right )^{4} + 792 \, b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{7} \log \left (F\right )^{3} - 2772 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{5} \log \left (F\right )^{2} + 6930 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{3} \log \left (F\right ) - 10395 \, x - \frac{10395 \, c}{d}\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 )}}{64 \, b^{6} \log \left (F\right )^{6}} - \frac{10395 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right )}{128 \, \sqrt{-b \log \left (F\right )} 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)^12,x, algorithm="giac")

[Out]

1/64*(32*b^5*d^10*(x + c/d)^11*log(F)^5 - 176*b^4*d^8*(x + c/d)^9*log(F)^4 + 792*b^3*d^6*(x + c/d)^7*log(F)^3

- 2772*b^2*d^4*(x + c/d)^5*log(F)^2 + 6930*b*d^2*(x + c/d)^3*log(F) - 10395*x - 10395*c/d)*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*log(F)^6) - 10395/128*sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*d*

(x + c/d))/(sqrt(-b*log(F))*b^6*d*log(F)^6)

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