3.256 \(\int F^{a+b (c+d x)^2} (c+d x)^9 \, dx\)

Optimal. Leaf size=88 \[ \frac{F^{a+b (c+d x)^2} \left (b^4 \log ^4(F) (c+d x)^8-4 b^3 \log ^3(F) (c+d x)^6+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^2+24\right )}{2 b^5 d \log ^5(F)} \]

[Out]

(F^(a + b*(c + d*x)^2)*(24 - 24*b*(c + d*x)^2*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 - 4*b^3*(c + d*x)^6*Log[F]^
3 + b^4*(c + d*x)^8*Log[F]^4))/(2*b^5*d*Log[F]^5)

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Rubi [C]  time = 0.0693239, antiderivative size = 31, normalized size of antiderivative = 0.35, 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} \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^2\right )}{2 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]

[Out]

(F^a*Gamma[5, -(b*(c + d*x)^2*Log[F])])/(2*b^5*d*Log[F]^5)

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)^9 \, dx &=\frac{F^a \Gamma \left (5,-b (c+d x)^2 \log (F)\right )}{2 b^5 d \log ^5(F)}\\ \end{align*}

Mathematica [C]  time = 0.0077248, size = 31, normalized size = 0.35 \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^2\right )}{2 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]

[Out]

(F^a*Gamma[5, -(b*(c + d*x)^2*Log[F])])/(2*b^5*d*Log[F]^5)

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Maple [B]  time = 0.015, size = 396, normalized size = 4.5 \begin{align*}{\frac{ \left ( 24-24\,\ln \left ( F \right ) b{d}^{2}{x}^{2}-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{6}+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{4}-24\,\ln \left ( F \right ) b{c}^{2}-48\,\ln \left ( F \right ) bcdx+72\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{d}^{2}{x}^{2}+48\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{3}dx+8\,c{d}^{7}{x}^{7}{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}+28\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{2}{d}^{6}{x}^{6}+56\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{3}{d}^{5}{x}^{5}+70\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}{d}^{4}{x}^{4}+56\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{5}{d}^{3}{x}^{3}+28\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{6}{d}^{2}{x}^{2}+8\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{7}dx-24\,c{d}^{5}{x}^{5}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}-60\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{2}{d}^{4}{x}^{4}-80\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{3}{x}^{3}-60\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{4}{d}^{2}{x}^{2}-24\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{5}dx+48\,{d}^{3}c{x}^{3}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{8}+{d}^{8}{x}^{8}{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}-4\,{d}^{6}{x}^{6}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+12\,{d}^{4}{x}^{4}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a}}{2\,{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x)

[Out]

1/2*(24-24*ln(F)*b*d^2*x^2-4*ln(F)^3*b^3*c^6+12*ln(F)^2*b^2*c^4-24*ln(F)*b*c^2-48*ln(F)*b*c*d*x+72*ln(F)^2*b^2
*c^2*d^2*x^2+48*ln(F)^2*b^2*c^3*d*x+8*c*d^7*x^7*b^4*ln(F)^4+28*ln(F)^4*b^4*c^2*d^6*x^6+56*ln(F)^4*b^4*c^3*d^5*
x^5+70*ln(F)^4*b^4*c^4*d^4*x^4+56*ln(F)^4*b^4*c^5*d^3*x^3+28*ln(F)^4*b^4*c^6*d^2*x^2+8*ln(F)^4*b^4*c^7*d*x-24*
c*d^5*x^5*b^3*ln(F)^3-60*ln(F)^3*b^3*c^2*d^4*x^4-80*ln(F)^3*b^3*c^3*d^3*x^3-60*ln(F)^3*b^3*c^4*d^2*x^2-24*ln(F
)^3*b^3*c^5*d*x+48*d^3*c*x^3*b^2*ln(F)^2+ln(F)^4*b^4*c^8+d^8*x^8*b^4*ln(F)^4-4*d^6*x^6*b^3*ln(F)^3+12*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^5/ln(F)^5/d

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Maxima [C]  time = 3.09527, size = 5261, normalized size = 59.78 \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)^9,x, algorithm="maxima")

[Out]

-9/2*(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*l
og(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^8*d/sqrt(b*d^2*log(F)) + 18*(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^7*d^2/sqrt(b*d^2*log(F)) - 42*(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^6*d^3/sqrt(b*d^2*log(F)
) + 63*(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^5*d^4
/sqrt(b*d^2*log(F)) - 63*(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^4*d^5/sqrt(b*d^2*log(F)) + 42*(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*lo
g(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^3*d^6/sqrt(b*d^2*log(F)) - 18*(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*l
og(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^2*d^7/sqrt(b*d^2*log(F)) + 9/2*(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*lo
g(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*d^8/sqrt
(b*d^2*log(F)) - 1/2*(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*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*gam
ma(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*d^9/sqrt(b*d^2*log(F)) +
1/2*sqrt(pi)*F^(b*c^2 + a)*c^9*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 = 1.5642, size = 687, normalized size = 7.81 \begin{align*} \frac{{\left ({\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} - 4 \,{\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} + 12 \,{\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} - 24 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, 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)^9,x, algorithm="fricas")

[Out]

1/2*((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)*log(F)^4 - 4*(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 + 12*(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 - 24*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*
log(F) + 24)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)/(b^5*d*log(F)^5)

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Sympy [A]  time = 0.318528, size = 558, normalized size = 6.34 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )^{2}} \left (b^{4} c^{8} \log{\left (F \right )}^{4} + 8 b^{4} c^{7} d x \log{\left (F \right )}^{4} + 28 b^{4} c^{6} d^{2} x^{2} \log{\left (F \right )}^{4} + 56 b^{4} c^{5} d^{3} x^{3} \log{\left (F \right )}^{4} + 70 b^{4} c^{4} d^{4} x^{4} \log{\left (F \right )}^{4} + 56 b^{4} c^{3} d^{5} x^{5} \log{\left (F \right )}^{4} + 28 b^{4} c^{2} d^{6} x^{6} \log{\left (F \right )}^{4} + 8 b^{4} c d^{7} x^{7} \log{\left (F \right )}^{4} + b^{4} d^{8} x^{8} \log{\left (F \right )}^{4} - 4 b^{3} c^{6} \log{\left (F \right )}^{3} - 24 b^{3} c^{5} d x \log{\left (F \right )}^{3} - 60 b^{3} c^{4} d^{2} x^{2} \log{\left (F \right )}^{3} - 80 b^{3} c^{3} d^{3} x^{3} \log{\left (F \right )}^{3} - 60 b^{3} c^{2} d^{4} x^{4} \log{\left (F \right )}^{3} - 24 b^{3} c d^{5} x^{5} \log{\left (F \right )}^{3} - 4 b^{3} d^{6} x^{6} \log{\left (F \right )}^{3} + 12 b^{2} c^{4} \log{\left (F \right )}^{2} + 48 b^{2} c^{3} d x \log{\left (F \right )}^{2} + 72 b^{2} c^{2} d^{2} x^{2} \log{\left (F \right )}^{2} + 48 b^{2} c d^{3} x^{3} \log{\left (F \right )}^{2} + 12 b^{2} d^{4} x^{4} \log{\left (F \right )}^{2} - 24 b c^{2} \log{\left (F \right )} - 48 b c d x \log{\left (F \right )} - 24 b d^{2} x^{2} \log{\left (F \right )} + 24\right )}{2 b^{5} d \log{\left (F \right )}^{5}} & \text{for}\: 2 b^{5} d \log{\left (F \right )}^{5} \neq 0 \\c^{9} x + \frac{9 c^{8} d x^{2}}{2} + 12 c^{7} d^{2} x^{3} + 21 c^{6} d^{3} x^{4} + \frac{126 c^{5} d^{4} x^{5}}{5} + 21 c^{4} d^{5} x^{6} + 12 c^{3} d^{6} x^{7} + \frac{9 c^{2} d^{7} x^{8}}{2} + c d^{8} x^{9} + \frac{d^{9} x^{10}}{10} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**9,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**2)*(b**4*c**8*log(F)**4 + 8*b**4*c**7*d*x*log(F)**4 + 28*b**4*c**6*d**2*x**2*l
og(F)**4 + 56*b**4*c**5*d**3*x**3*log(F)**4 + 70*b**4*c**4*d**4*x**4*log(F)**4 + 56*b**4*c**3*d**5*x**5*log(F)
**4 + 28*b**4*c**2*d**6*x**6*log(F)**4 + 8*b**4*c*d**7*x**7*log(F)**4 + b**4*d**8*x**8*log(F)**4 - 4*b**3*c**6
*log(F)**3 - 24*b**3*c**5*d*x*log(F)**3 - 60*b**3*c**4*d**2*x**2*log(F)**3 - 80*b**3*c**3*d**3*x**3*log(F)**3
- 60*b**3*c**2*d**4*x**4*log(F)**3 - 24*b**3*c*d**5*x**5*log(F)**3 - 4*b**3*d**6*x**6*log(F)**3 + 12*b**2*c**4
*log(F)**2 + 48*b**2*c**3*d*x*log(F)**2 + 72*b**2*c**2*d**2*x**2*log(F)**2 + 48*b**2*c*d**3*x**3*log(F)**2 + 1
2*b**2*d**4*x**4*log(F)**2 - 24*b*c**2*log(F) - 48*b*c*d*x*log(F) - 24*b*d**2*x**2*log(F) + 24)/(2*b**5*d*log(
F)**5), Ne(2*b**5*d*log(F)**5, 0)), (c**9*x + 9*c**8*d*x**2/2 + 12*c**7*d**2*x**3 + 21*c**6*d**3*x**4 + 126*c*
*5*d**4*x**5/5 + 21*c**4*d**5*x**6 + 12*c**3*d**6*x**7 + 9*c**2*d**7*x**8/2 + c*d**8*x**9 + d**9*x**10/10, Tru
e))

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Giac [A]  time = 1.31809, size = 167, normalized size = 1.9 \begin{align*} \frac{{\left (b^{4} d^{8}{\left (x + \frac{c}{d}\right )}^{8} \log \left (F\right )^{4} - 4 \, b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{6} \log \left (F\right )^{3} + 12 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{4} \log \left (F\right )^{2} - 24 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{2} \log \left (F\right ) + 24\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^{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)^9,x, algorithm="giac")

[Out]

1/2*(b^4*d^8*(x + c/d)^8*log(F)^4 - 4*b^3*d^6*(x + c/d)^6*log(F)^3 + 12*b^2*d^4*(x + c/d)^4*log(F)^2 - 24*b*d^
2*(x + c/d)^2*log(F) + 24)*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*d*log(F)^5)