3.268 \(\int F^{a+b (c+d x)^2} (c+d x)^{10} \, dx\)
Optimal. Leaf size=49 \[ -\frac{F^a (c+d x)^{11} \text{Gamma}\left (\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{11/2}} \]
[Out]
-(F^a*(c + d*x)^11*Gamma[11/2, -(b*(c + d*x)^2*Log[F])])/(2*d*(-(b*(c + d*x)^2*Log[F]))^(11/2))
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Rubi [A] time = 0.0669821, 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} \[ -\frac{F^a (c+d x)^{11} \text{Gamma}\left (\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{11/2}} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b*(c + d*x)^2)*(c + d*x)^10,x]
[Out]
-(F^a*(c + d*x)^11*Gamma[11/2, -(b*(c + d*x)^2*Log[F])])/(2*d*(-(b*(c + d*x)^2*Log[F]))^(11/2))
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)^{10} \, dx &=-\frac{F^a (c+d x)^{11} \Gamma \left (\frac{11}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b (c+d x)^2 \log (F)\right )^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0249053, size = 49, normalized size = 1. \[ -\frac{F^a (c+d x)^{11} \text{Gamma}\left (\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{11/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^10,x]
[Out]
-(F^a*(c + d*x)^11*Gamma[11/2, -(b*(c + d*x)^2*Log[F])])/(2*d*(-(b*(c + d*x)^2*Log[F]))^(11/2))
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Maple [B] time = 0.222, size = 1359, normalized size = 27.7 \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)^10,x)
[Out]
42*d^2*c^6/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+18*d*c^7/ln(F)/b*x^2*F^(b*d^2*x^2)*F^(2*b*c*d
*x)*F^(c^2*b)*F^a-189/4*d*c^5/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-315/4*d^2*c^4/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+315/8*d^3*c/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-945/16*d*c/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-63/4*d^5*c/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+9/2*d^7*c/ln(F)/b*x^8*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+18*d^6*c
^2/ln(F)/b*x^7*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-189/4*d^4*c^2/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+315/4*d^2*c^2/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+315/4*d*c^3/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+42*d^5*c^3/ln(F)/b*x^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b
)*F^a-315/4*d^3*c^3/ln(F)^2/b^2*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+63*d^4*c^4/ln(F)/b*x^5*F^(b*d^2*
x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+63*d^3*c^5/ln(F)/b*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+9/2*c^8/ln(F
)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-63/4*c^6/ln(F)^2/b^2*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F
^a+315/8*c^4/ln(F)^3/b^3*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-945/16*c^2/ln(F)^4/b^4*x*F^(b*d^2*x^2)*F^
(2*b*c*d*x)*F^(c^2*b)*F^a+63/8*d^4/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-315/16*d^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+1/2*d^8/ln(F)/b*x^9*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*
F^a+1/2/d*c^9/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-9/4/d*c^7/ln(F)^2/b^2*F^(b*d^2*x^2)*F^(2*b*c*d
*x)*F^(c^2*b)*F^a+63/8/d*c^5/ln(F)^3/b^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-315/16/d*c^3/ln(F)^4/b^4*F^
(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+945/32/d*c/ln(F)^5/b^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-9/4*d
^6/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+945/32/ln(F)^5/b^5*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*
F^(c^2*b)*F^a+945/64/d/ln(F)^5/b^5*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 = 3.16986, size = 6310, normalized size = 128.78 \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)^10,x, algorithm="maxima")
[Out]
-5*(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^9*d/sqrt(b*d^2*log(F)) + 45/2*(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^8*d^2/sqrt(b*d^2*log(F)) - 60*(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^7*d^3/sqrt(b*d^2*log(F)
) + 105*(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^6*d^
4/sqrt(b*d^2*log(F)) - 126*(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^5*d^5/sqrt(b*d^2*log(F)) + 105*(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^4*d^6/sqrt(b*d^2*log(F)) - 60*(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))^(1
5/2)*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^3*d^7/sqrt(b*d^2*log(F)) + 45/2*(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^2*d^
8/sqrt(b*d^2*log(F)) - 5*(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)) + 8
4*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*l
og(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*d^9/sqrt(b*d^2*log(
F)) + 1/2*(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*ga
mma(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*d^10/sqrt(b*d^2*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^10*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 = 1.59994, size = 992, normalized size = 20.24 \begin{align*} \frac{945 \, \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 (16 \,{\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} - 72 \,{\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} + 252 \,{\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} - 630 \,{\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} + 945 \,{\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}}{64 \, b^{6} d^{2} \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)^10,x, algorithm="fricas")
[Out]
1/64*(945*sqrt(pi)*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) + 2*(16*(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 - 72*(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)*l
og(F)^4 + 252*(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 - 630*(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 + 945*(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^6*d^2*log(F)^6)
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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)**10,x)
[Out]
Timed out
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Giac [A] time = 1.22319, size = 235, normalized size = 4.8 \begin{align*} \frac{{\left (16 \, b^{4} d^{8}{\left (x + \frac{c}{d}\right )}^{9} \log \left (F\right )^{4} - 72 \, b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{7} \log \left (F\right )^{3} + 252 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{5} \log \left (F\right )^{2} - 630 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{3} \log \left (F\right ) + 945 \, x + \frac{945 \, 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 )}}{32 \, b^{5} \log \left (F\right )^{5}} + \frac{945 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right )}{64 \, \sqrt{-b \log \left (F\right )} b^{5} d \log \left (F\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^10,x, algorithm="giac")
[Out]
1/32*(16*b^4*d^8*(x + c/d)^9*log(F)^4 - 72*b^3*d^6*(x + c/d)^7*log(F)^3 + 252*b^2*d^4*(x + c/d)^5*log(F)^2 - 6
30*b*d^2*(x + c/d)^3*log(F) + 945*x + 945*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^5*log(F)^5) + 945/64*sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*d*(x + c/d))/(sqrt(-b*log(F))*b^5*d*log(F)^5)