3.257 \(\int F^{a+b (c+d x)^2} (c+d x)^7 \, dx\)

Optimal. Leaf size=126 \[ -\frac{3 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}+\frac{3 (c+d x)^2 F^{a+b (c+d x)^2}}{b^3 d \log ^3(F)}-\frac{3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac{(c+d x)^6 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.25554, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ -\frac{3 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}+\frac{3 (c+d x)^2 F^{a+b (c+d x)^2}}{b^3 d \log ^3(F)}-\frac{3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac{(c+d x)^6 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^7,x]

[Out]

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

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx &=\frac{F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}-\frac{3 \int F^{a+b (c+d x)^2} (c+d x)^5 \, dx}{b \log (F)}\\ &=-\frac{3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}+\frac{6 \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{b^2 \log ^2(F)}\\ &=\frac{3 F^{a+b (c+d x)^2} (c+d x)^2}{b^3 d \log ^3(F)}-\frac{3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}-\frac{6 \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b^3 \log ^3(F)}\\ &=-\frac{3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac{3 F^{a+b (c+d x)^2} (c+d x)^2}{b^3 d \log ^3(F)}-\frac{3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0411262, size = 72, normalized size = 0.57 \[ \frac{F^{a+b (c+d x)^2} \left (b^3 \log ^3(F) (c+d x)^6-3 b^2 \log ^2(F) (c+d x)^4+6 b \log (F) (c+d x)^2-6\right )}{2 b^4 d \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^7,x]

[Out]

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

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Maple [B]  time = 0.009, size = 249, normalized size = 2. \begin{align*}{\frac{ \left ({d}^{6}{x}^{6}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+6\,c{d}^{5}{x}^{5}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+15\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{2}{d}^{4}{x}^{4}+20\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{3}{x}^{3}+15\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{4}{d}^{2}{x}^{2}+6\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{5}dx+ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{6}-3\,{d}^{4}{x}^{4}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-12\,{d}^{3}c{x}^{3}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-18\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{d}^{2}{x}^{2}-12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{3}dx-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{4}+6\,\ln \left ( F \right ) b{d}^{2}{x}^{2}+12\,\ln \left ( F \right ) bcdx+6\,\ln \left ( F \right ) b{c}^{2}-6 \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a}}{2\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}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)^7,x)

[Out]

1/2*(d^6*x^6*b^3*ln(F)^3+6*c*d^5*x^5*b^3*ln(F)^3+15*ln(F)^3*b^3*c^2*d^4*x^4+20*ln(F)^3*b^3*c^3*d^3*x^3+15*ln(F
)^3*b^3*c^4*d^2*x^2+6*ln(F)^3*b^3*c^5*d*x+ln(F)^3*b^3*c^6-3*d^4*x^4*b^2*ln(F)^2-12*d^3*c*x^3*b^2*ln(F)^2-18*ln
(F)^2*b^2*c^2*d^2*x^2-12*ln(F)^2*b^2*c^3*d*x-3*ln(F)^2*b^2*c^4+6*ln(F)*b*d^2*x^2+12*ln(F)*b*c*d*x+6*ln(F)*b*c^
2-6)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)/ln(F)^4/b^4/d

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Maxima [C]  time = 2.7074, size = 3461, normalized size = 27.47 \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)^7,x, algorithm="maxima")

[Out]

-7/2*(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^6*d/sqrt(b*d^2*log(F)) + 21/2*(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^5*d^2/sqrt(b*d^2*log(F)) - 35/2*(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^4*d^3/sqrt(b*d^2*lo
g(F)) + 35/2*(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
^3*d^4/sqrt(b*d^2*log(F)) - 21/2*(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*gamm
a(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^2*d^5/sqrt(b*d^2*log(F)) + 7/2*(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*d^6/sqrt(b*d^2*log(F)) - 1/2*(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*d^7/sqrt(b*d^2*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^7*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 = 1.57978, size = 446, normalized size = 3.54 \begin{align*} \frac{{\left ({\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} - 3 \,{\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} + 6 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 6\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{4} d \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/2*((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 - 3*(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 + 6*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) - 6)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)/(b^4*d*log(F)^4)

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Sympy [A]  time = 0.262146, size = 366, normalized size = 2.9 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )^{2}} \left (b^{3} c^{6} \log{\left (F \right )}^{3} + 6 b^{3} c^{5} d x \log{\left (F \right )}^{3} + 15 b^{3} c^{4} d^{2} x^{2} \log{\left (F \right )}^{3} + 20 b^{3} c^{3} d^{3} x^{3} \log{\left (F \right )}^{3} + 15 b^{3} c^{2} d^{4} x^{4} \log{\left (F \right )}^{3} + 6 b^{3} c d^{5} x^{5} \log{\left (F \right )}^{3} + b^{3} d^{6} x^{6} \log{\left (F \right )}^{3} - 3 b^{2} c^{4} \log{\left (F \right )}^{2} - 12 b^{2} c^{3} d x \log{\left (F \right )}^{2} - 18 b^{2} c^{2} d^{2} x^{2} \log{\left (F \right )}^{2} - 12 b^{2} c d^{3} x^{3} \log{\left (F \right )}^{2} - 3 b^{2} d^{4} x^{4} \log{\left (F \right )}^{2} + 6 b c^{2} \log{\left (F \right )} + 12 b c d x \log{\left (F \right )} + 6 b d^{2} x^{2} \log{\left (F \right )} - 6\right )}{2 b^{4} d \log{\left (F \right )}^{4}} & \text{for}\: 2 b^{4} d \log{\left (F \right )}^{4} \neq 0 \\c^{7} x + \frac{7 c^{6} d x^{2}}{2} + 7 c^{5} d^{2} x^{3} + \frac{35 c^{4} d^{3} x^{4}}{4} + 7 c^{3} d^{4} x^{5} + \frac{7 c^{2} d^{5} x^{6}}{2} + c d^{6} x^{7} + \frac{d^{7} x^{8}}{8} & \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)**7,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**2)*(b**3*c**6*log(F)**3 + 6*b**3*c**5*d*x*log(F)**3 + 15*b**3*c**4*d**2*x**2*l
og(F)**3 + 20*b**3*c**3*d**3*x**3*log(F)**3 + 15*b**3*c**2*d**4*x**4*log(F)**3 + 6*b**3*c*d**5*x**5*log(F)**3
+ b**3*d**6*x**6*log(F)**3 - 3*b**2*c**4*log(F)**2 - 12*b**2*c**3*d*x*log(F)**2 - 18*b**2*c**2*d**2*x**2*log(F
)**2 - 12*b**2*c*d**3*x**3*log(F)**2 - 3*b**2*d**4*x**4*log(F)**2 + 6*b*c**2*log(F) + 12*b*c*d*x*log(F) + 6*b*
d**2*x**2*log(F) - 6)/(2*b**4*d*log(F)**4), Ne(2*b**4*d*log(F)**4, 0)), (c**7*x + 7*c**6*d*x**2/2 + 7*c**5*d**
2*x**3 + 35*c**4*d**3*x**4/4 + 7*c**3*d**4*x**5 + 7*c**2*d**5*x**6/2 + c*d**6*x**7 + d**7*x**8/8, True))

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Giac [A]  time = 1.67317, size = 139, normalized size = 1.1 \begin{align*} \frac{{\left (b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{6} \log \left (F\right )^{3} - 3 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{4} \log \left (F\right )^{2} + 6 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{2} \log \left (F\right ) - 6\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^{4} d \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x, algorithm="giac")

[Out]

1/2*(b^3*d^6*(x + c/d)^6*log(F)^3 - 3*b^2*d^4*(x + c/d)^4*log(F)^2 + 6*b*d^2*(x + c/d)^2*log(F) - 6)*e^(b*d^2*
x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b^4*d*log(F)^4)