3.268 \(\int F^{a+b (c+d x)^2} (c+d x)^{10} \, dx\)

Optimal. Leaf size=49 \[ -\frac {F^a (c+d x)^{11} \Gamma \left (\frac {11}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{11/2}} \]

[Out]

-1/2*F^a*(d*x+c)^11*(1048576/61836869254970658257624840625*GAMMA(51/2,-b*(d*x+c)^2*ln(F))-1048576/618368692549
70658257624840625*(-b*(d*x+c)^2*ln(F))^(49/2)*exp(b*(d*x+c)^2*ln(F))-524288/1261976923570829760359690625*(-b*(
d*x+c)^2*ln(F))^(47/2)*exp(b*(d*x+c)^2*ln(F))-262144/26850572841932548092759375*(-b*(d*x+c)^2*ln(F))^(45/2)*ex
p(b*(d*x+c)^2*ln(F))-131072/596679396487389957616875*(-b*(d*x+c)^2*ln(F))^(43/2)*exp(b*(d*x+c)^2*ln(F))-65536/
13876265034590464130625*(-b*(d*x+c)^2*ln(F))^(41/2)*exp(b*(d*x+c)^2*ln(F))-32768/338445488648547905625*(-b*(d*
x+c)^2*ln(F))^(39/2)*exp(b*(d*x+c)^2*ln(F))-16384/8678089452526869375*(-b*(d*x+c)^2*ln(F))^(37/2)*exp(b*(d*x+c
)^2*ln(F))-8192/234542958176401875*(-b*(d*x+c)^2*ln(F))^(35/2)*exp(b*(d*x+c)^2*ln(F))-4096/6701227376468625*(-
b*(d*x+c)^2*ln(F))^(33/2)*exp(b*(d*x+c)^2*ln(F))-2048/203067496256625*(-b*(d*x+c)^2*ln(F))^(31/2)*exp(b*(d*x+c
)^2*ln(F))-1024/6550564395375*(-b*(d*x+c)^2*ln(F))^(29/2)*exp(b*(d*x+c)^2*ln(F))-512/225881530875*(-b*(d*x+c)^
2*ln(F))^(27/2)*exp(b*(d*x+c)^2*ln(F))-256/8365982625*(-b*(d*x+c)^2*ln(F))^(25/2)*exp(b*(d*x+c)^2*ln(F))-128/3
34639305*(-b*(d*x+c)^2*ln(F))^(23/2)*exp(b*(d*x+c)^2*ln(F))-64/14549535*(-b*(d*x+c)^2*ln(F))^(21/2)*exp(b*(d*x
+c)^2*ln(F))-32/692835*(-b*(d*x+c)^2*ln(F))^(19/2)*exp(b*(d*x+c)^2*ln(F))-16/36465*(-b*(d*x+c)^2*ln(F))^(17/2)
*exp(b*(d*x+c)^2*ln(F))-8/2145*(-b*(d*x+c)^2*ln(F))^(15/2)*exp(b*(d*x+c)^2*ln(F))-4/143*(-b*(d*x+c)^2*ln(F))^(
13/2)*exp(b*(d*x+c)^2*ln(F))-2/11*(-b*(d*x+c)^2*ln(F))^(11/2)*exp(b*(d*x+c)^2*ln(F)))/d/(-b*(d*x+c)^2*ln(F))^(
11/2)

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Rubi [A]  time = 0.07, 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 = {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*}

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Mathematica [A]  time = 0.03, size = 49, normalized size = 1.00 \[ -\frac {F^a (c+d x)^{11} \Gamma \left (\frac {11}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^10,x]

[Out]

-1/2*(F^a*(c + d*x)^11*Gamma[11/2, -(b*(c + d*x)^2*Log[F])])/(d*(-(b*(c + d*x)^2*Log[F]))^(11/2))

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fricas [A]  time = 0.53, size = 456, normalized size = 9.31 \[ \frac {945 \, \sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) + 2 \, {\left (16 \, {\left (b^{5} d^{10} x^{9} + 9 \, b^{5} c d^{9} x^{8} + 36 \, b^{5} c^{2} d^{8} x^{7} + 84 \, b^{5} c^{3} d^{7} x^{6} + 126 \, b^{5} c^{4} d^{6} x^{5} + 126 \, b^{5} c^{5} d^{5} x^{4} + 84 \, b^{5} c^{6} d^{4} x^{3} + 36 \, b^{5} c^{7} d^{3} x^{2} + 9 \, b^{5} c^{8} d^{2} x + b^{5} c^{9} d\right )} \log \relax (F)^{5} - 72 \, {\left (b^{4} d^{8} x^{7} + 7 \, b^{4} c d^{7} x^{6} + 21 \, b^{4} c^{2} d^{6} x^{5} + 35 \, b^{4} c^{3} d^{5} x^{4} + 35 \, b^{4} c^{4} d^{4} x^{3} + 21 \, b^{4} c^{5} d^{3} x^{2} + 7 \, b^{4} c^{6} d^{2} x + b^{4} c^{7} d\right )} \log \relax (F)^{4} + 252 \, {\left (b^{3} d^{6} x^{5} + 5 \, b^{3} c d^{5} x^{4} + 10 \, b^{3} c^{2} d^{4} x^{3} + 10 \, b^{3} c^{3} d^{3} x^{2} + 5 \, b^{3} c^{4} d^{2} x + b^{3} c^{5} d\right )} \log \relax (F)^{3} - 630 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \relax (F)^{2} + 945 \, {\left (b d^{2} x + b c d\right )} \log \relax (F)\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{64 \, b^{6} d^{2} \log \relax (F)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^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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giac [A]  time = 0.35, size = 174, normalized size = 3.55 \[ \frac {{\left (16 \, b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{9} \log \relax (F)^{4} - 72 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{7} \log \relax (F)^{3} + 252 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{5} \log \relax (F)^{2} - 630 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{3} \log \relax (F) + 945 \, x + \frac {945 \, c}{d}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{32 \, b^{5} \log \relax (F)^{5}} + \frac {945 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{64 \, \sqrt {-b \log \relax (F)} b^{5} d \log \relax (F)^{5}} \]

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)

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maple [B]  time = 0.25, size = 1359, normalized size = 27.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+(d*x+c)^2*b)*(d*x+c)^10,x)

[Out]

1/2*d^8/ln(F)/b*x^9*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a+945/32/d*c/ln(F)^5/b^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-3
15/16/d*c^3/ln(F)^4/b^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+1/2/d*c^9/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x
)*F^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a+9/2*c^8/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c
^2)*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^(b*c^2)*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^(b*c^2)*F^a+42*d^2*c^6/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+18*d*c^7/ln(F)/b*
x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c^2
)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a+9/2*d^7*c/l
n(F)/b*x^8*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*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^(b*c
^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*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^(b*c^2)*F^a+945/32/ln(F)^5/b^5*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*
F^(b*c^2)*F^a+945/64/d/ln(F)^5/b^5*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(1/(-b*ln(F))^(1/2)*b*c*ln(F)-(-b*ln(F))^(
1/2)*d*x)

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maxima [B]  time = 16.75, size = 4471, normalized size = 91.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^10,x, algorithm="maxima")

[Out]

-5*(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^9/sqrt(b*log(F)) + 45/2*(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*lo
g(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^8*d/sqrt(b*lo
g(F)) - 60*(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*lo
g(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*lo
g(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^7*d^2/sqrt(b*log(F)) + 105*(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^6*d^3/sqrt(b*log(F)) - 126*(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))*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^5*d^4/sqrt(b*log(F)) + 105*(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*lo
g(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) - 1
5*(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^4*d^5/sqrt(b*log(F)
) - 60*(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*l
og(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))*l
og(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^3*d^6/sqrt(b*log(F)) + 45/2*(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*gamma(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^2*d^7/sqrt(b*log(F)) - 5*(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*lo
g(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*gamma(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*d^8/sqrt(b*log(F)) + 1/2*(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))^(2
1/2)*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*gamma(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*d^9/sqrt(b*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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mupad [B]  time = 4.13, size = 730, normalized size = 14.90 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {b^4\,c^9\,{\ln \relax (F)}^4}{2}-\frac {9\,b^3\,c^7\,{\ln \relax (F)}^3}{4}+\frac {63\,b^2\,c^5\,{\ln \relax (F)}^2}{8}-\frac {315\,b\,c^3\,\ln \relax (F)}{16}+\frac {945\,c}{32}\right )}{b^5\,d\,{\ln \relax (F)}^5}-\frac {945\,F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )}{64\,b^5\,{\ln \relax (F)}^5\,\sqrt {b\,d^2\,\ln \relax (F)}}+\frac {63\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^4\,\left (8\,b^2\,c^5\,d^3\,{\ln \relax (F)}^2-10\,b\,c^3\,d^3\,\ln \relax (F)+5\,c\,d^3\right )}{8\,b^3\,{\ln \relax (F)}^3}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^8\,x^9}{2\,b\,\ln \relax (F)}-\frac {9\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^2\,\left (-32\,d\,b^3\,c^7\,{\ln \relax (F)}^3+84\,d\,b^2\,c^5\,{\ln \relax (F)}^2-140\,d\,b\,c^3\,\ln \relax (F)+105\,d\,c\right )}{16\,b^4\,{\ln \relax (F)}^4}+\frac {63\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^5\,\left (8\,b^2\,c^4\,d^4\,{\ln \relax (F)}^2-6\,b\,c^2\,d^4\,\ln \relax (F)+d^4\right )}{8\,b^3\,{\ln \relax (F)}^3}-\frac {21\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^6\,\left (3\,c\,d^5-8\,b\,c^3\,d^5\,\ln \relax (F)\right )}{4\,b^2\,{\ln \relax (F)}^2}-\frac {21\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^3\,\left (-32\,b^3\,c^6\,d^2\,{\ln \relax (F)}^3+60\,b^2\,c^4\,d^2\,{\ln \relax (F)}^2-60\,b\,c^2\,d^2\,\ln \relax (F)+15\,d^2\right )}{16\,b^4\,{\ln \relax (F)}^4}+\frac {9\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (16\,b^4\,c^8\,{\ln \relax (F)}^4-56\,b^3\,c^6\,{\ln \relax (F)}^3+140\,b^2\,c^4\,{\ln \relax (F)}^2-210\,b\,c^2\,\ln \relax (F)+105\right )}{32\,b^5\,{\ln \relax (F)}^5}+\frac {9\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,c\,d^7\,x^8}{2\,b\,\ln \relax (F)}+\frac {9\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^6\,x^7\,\left (8\,b\,c^2\,\ln \relax (F)-1\right )}{4\,b^2\,{\ln \relax (F)}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^2)*(c + d*x)^10,x)

[Out]

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*((945*c)/32 - (315*b*c^3*log(F))/16 + (63*b^2*c^5*log(F)^2)/8 - (9*
b^3*c^7*log(F)^3)/4 + (b^4*c^9*log(F)^4)/2))/(b^5*d*log(F)^5) - (945*F^a*pi^(1/2)*erfi((b*c*d*log(F) + b*d^2*x
*log(F))/(b*d^2*log(F))^(1/2)))/(64*b^5*log(F)^5*(b*d^2*log(F))^(1/2)) + (63*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*
b*c*d*x)*x^4*(5*c*d^3 + 8*b^2*c^5*d^3*log(F)^2 - 10*b*c^3*d^3*log(F)))/(8*b^3*log(F)^3) + (F^(b*d^2*x^2)*F^a*F
^(b*c^2)*F^(2*b*c*d*x)*d^8*x^9)/(2*b*log(F)) - (9*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x^2*(105*c*d + 84*
b^2*c^5*d*log(F)^2 - 32*b^3*c^7*d*log(F)^3 - 140*b*c^3*d*log(F)))/(16*b^4*log(F)^4) + (63*F^(b*d^2*x^2)*F^a*F^
(b*c^2)*F^(2*b*c*d*x)*x^5*(d^4 + 8*b^2*c^4*d^4*log(F)^2 - 6*b*c^2*d^4*log(F)))/(8*b^3*log(F)^3) - (21*F^(b*d^2
*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x^6*(3*c*d^5 - 8*b*c^3*d^5*log(F)))/(4*b^2*log(F)^2) - (21*F^(b*d^2*x^2)*F^a
*F^(b*c^2)*F^(2*b*c*d*x)*x^3*(15*d^2 + 60*b^2*c^4*d^2*log(F)^2 - 32*b^3*c^6*d^2*log(F)^3 - 60*b*c^2*d^2*log(F)
))/(16*b^4*log(F)^4) + (9*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x*(140*b^2*c^4*log(F)^2 - 210*b*c^2*log(F)
- 56*b^3*c^6*log(F)^3 + 16*b^4*c^8*log(F)^4 + 105))/(32*b^5*log(F)^5) + (9*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b
*c*d*x)*c*d^7*x^8)/(2*b*log(F)) + (9*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*d^6*x^7*(8*b*c^2*log(F) - 1))/(
4*b^2*log(F)^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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