3.3.67 \(\int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx\) [267]
Optimal. Leaf size=49 \[ -\frac {F^a (c+d x)^{13} \Gamma \left (\frac {13}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b (c+d x)^2 \log (F)\right )^{13/2}} \]
[Out]
-1/2*F^a*(d*x+c)^13*(524288/5621533568633696205238621875*GAMMA(51/2,-b*(d*x+c)^2*ln(F))-524288/562153356863369
6205238621875*(-b*(d*x+c)^2*ln(F))^(49/2)*exp(b*(d*x+c)^2*ln(F))-262144/114725174870075432759971875*(-b*(d*x+c
)^2*ln(F))^(47/2)*exp(b*(d*x+c)^2*ln(F))-131072/2440961167448413462978125*(-b*(d*x+c)^2*ln(F))^(45/2)*exp(b*(d
*x+c)^2*ln(F))-65536/54243581498853632510625*(-b*(d*x+c)^2*ln(F))^(43/2)*exp(b*(d*x+c)^2*ln(F))-32768/12614786
39508224011875*(-b*(d*x+c)^2*ln(F))^(41/2)*exp(b*(d*x+c)^2*ln(F))-16384/30767771695322536875*(-b*(d*x+c)^2*ln(
F))^(39/2)*exp(b*(d*x+c)^2*ln(F))-8192/788917222956988125*(-b*(d*x+c)^2*ln(F))^(37/2)*exp(b*(d*x+c)^2*ln(F))-4
096/21322087106945625*(-b*(d*x+c)^2*ln(F))^(35/2)*exp(b*(d*x+c)^2*ln(F))-2048/609202488769875*(-b*(d*x+c)^2*ln
(F))^(33/2)*exp(b*(d*x+c)^2*ln(F))-1024/18460681477875*(-b*(d*x+c)^2*ln(F))^(31/2)*exp(b*(d*x+c)^2*ln(F))-512/
595505854125*(-b*(d*x+c)^2*ln(F))^(29/2)*exp(b*(d*x+c)^2*ln(F))-256/20534684625*(-b*(d*x+c)^2*ln(F))^(27/2)*ex
p(b*(d*x+c)^2*ln(F))-128/760543875*(-b*(d*x+c)^2*ln(F))^(25/2)*exp(b*(d*x+c)^2*ln(F))-64/30421755*(-b*(d*x+c)^
2*ln(F))^(23/2)*exp(b*(d*x+c)^2*ln(F))-32/1322685*(-b*(d*x+c)^2*ln(F))^(21/2)*exp(b*(d*x+c)^2*ln(F))-16/62985*
(-b*(d*x+c)^2*ln(F))^(19/2)*exp(b*(d*x+c)^2*ln(F))-8/3315*(-b*(d*x+c)^2*ln(F))^(17/2)*exp(b*(d*x+c)^2*ln(F))-4
/195*(-b*(d*x+c)^2*ln(F))^(15/2)*exp(b*(d*x+c)^2*ln(F))-2/13*(-b*(d*x+c)^2*ln(F))^(13/2)*exp(b*(d*x+c)^2*ln(F)
))/d/(-b*(d*x+c)^2*ln(F))^(13/2)
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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {2250}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[F^(a + b*(c + d*x)^2)*(c + d*x)^12,x]
[Out]
-1/2*(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(d*(-(b*(c + d*x)^2*Log[F]))^(13/2))
Rule 2250
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{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*}
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Mathematica [A]
time = 0.67, size = 49, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^12,x]
[Out]
-1/2*(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(d*(-(b*(c + d*x)^2*Log[F]))^(13/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1895\) vs.
\(2(578)=1156\).
time = 0.34, size = 1896, normalized size = 38.69
| | |
| method |
result |
size |
| | |
| risch |
\(\text {Expression too large to display}\) |
\(1896\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a+b*(d*x+c)^2)*(d*x+c)^12,x,method=_RETURNVERBOSE)
[Out]
55/2*d*c^9/ln(F)/b*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+11/2*c^10/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*
x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a-346
5/16*c^4/ln(F)^4/b^4*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a-10395/64/d*c/ln(F)^6/b^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*
F^a+1/2/d*c^11/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a-10395/128/d/ln(F)^6/b^6*P
i^(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))-10395/64/ln(F)^6/b^6*x*F^(b
*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a+165/2*d^7*c^3/ln(F)/b*x
^8*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2
)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a-231*d^5*c^3/l
n(F)^2/b^2*x^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)
*F^a-693/2*d^3*c^5/ln(F)^2/b^2*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a+1/2*d^10/ln(F)/b*x^11*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(
b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 6135 vs.
\(2 (559) = 1118\).
time = 1.61, size = 6135, normalized size = 125.20 \begin {gather*} \text {Too large to display} \end {gather*}
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*(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^11/sqrt(b*log(F)) + 33*(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*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^10*d/sqrt(b*lo
g(F)) - 110*(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*l
og(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*l
og(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^9*d^2/sqrt(b*log(F)) + 495/2*(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^8*d^3/sqrt(b*log(F)) - 396*(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) - 1
0*(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^7*d^4/sqrt(b*log(F)) + 462*(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^6*d^5/sqrt(b*log
(F)) - 396*(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*l
og(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^5*d^6/sqrt(b*log(F)) + 495/2*(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...
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Fricas [A]
time = 0.09, size = 617, normalized size = 12.59 \begin {gather*} -\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 {gather*}
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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]
Exception raised: SystemError >> excessive stack use: stack is 3060 deep
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Giac [A]
time = 1.45, size = 195, normalized size = 3.98 \begin {gather*} \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 {gather*}
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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Mupad [B]
time = 4.02, size = 209, normalized size = 4.27 \begin {gather*} \frac {\frac {F^a\,\left (\frac {10395\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (F\right )\,\left (c+d\,x\right )}{\sqrt {b\,\ln \left (F\right )}}\right )}{128}-\frac {10395\,F^{b\,{\left (c+d\,x\right )}^2}\,\sqrt {b\,\ln \left (F\right )}\,\left (c+d\,x\right )}{64}\right )}{\sqrt {b\,\ln \left (F\right )}}-\frac {693\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^5}{16}+\frac {99\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^7}{8}-\frac {11\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^9}{4}+\frac {F^{a+b\,{\left (c+d\,x\right )}^2}\,b^5\,{\ln \left (F\right )}^5\,{\left (c+d\,x\right )}^{11}}{2}+\frac {3465\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}{32}}{b^6\,d\,{\ln \left (F\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a + b*(c + d*x)^2)*(c + d*x)^12,x)
[Out]
((F^a*((10395*pi^(1/2)*erfi((b*log(F)*(c + d*x))/(b*log(F))^(1/2)))/128 - (10395*F^(b*(c + d*x)^2)*(b*log(F))^
(1/2)*(c + d*x))/64))/(b*log(F))^(1/2) - (693*F^(a + b*(c + d*x)^2)*b^2*log(F)^2*(c + d*x)^5)/16 + (99*F^(a +
b*(c + d*x)^2)*b^3*log(F)^3*(c + d*x)^7)/8 - (11*F^(a + b*(c + d*x)^2)*b^4*log(F)^4*(c + d*x)^9)/4 + (F^(a + b
*(c + d*x)^2)*b^5*log(F)^5*(c + d*x)^11)/2 + (3465*F^(a + b*(c + d*x)^2)*b*log(F)*(c + d*x)^3)/32)/(b^6*d*log(
F)^6)
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