\(\int F^{a+b (c+d x)^3} (c+d x)^8 \, dx\) [284]

Optimal result

Integrand size = 21, antiderivative size = 96 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2240} \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 (c+d x)^3 F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)}+\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]

[In]

Int[F^(a + b*(c + d*x)^3)*(c + d*x)^8,x]

[Out]

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

Rule 2240

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]

Rule 2243

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])

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.58 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\frac {F^{a+b (c+d x)^3} \left (2-2 b (c+d x)^3 \log (F)+b^2 (c+d x)^6 \log ^2(F)\right )}{3 b^3 d \log ^3(F)} \]

[In]

Integrate[F^(a + b*(c + d*x)^3)*(c + d*x)^8,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs. \(2(90)=180\).

Time = 0.57 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.08

method	result	size
gosper	\(\frac {\left (d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+6 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+15 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+20 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{5} d x +\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,d^{3} x^{3}-6 \ln \left (F \right ) b c \,d^{2} x^{2}-6 \ln \left (F \right ) b \,c^{2} d x -2 \ln \left (F \right ) b \,c^{3}+2\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{3} b^{3} d}\)	\(200\)
risch	\(\frac {\left (d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+6 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+15 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+20 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{5} d x +\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,d^{3} x^{3}-6 \ln \left (F \right ) b c \,d^{2} x^{2}-6 \ln \left (F \right ) b \,c^{2} d x -2 \ln \left (F \right ) b \,c^{3}+2\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{3} b^{3} d}\)	\(200\)
norman	\(\frac {c d \left (5 \ln \left (F \right ) b \,c^{3}-2\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {\left (\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,c^{3}+2\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{3} b^{3} d}+\frac {d^{5} x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right ) b}+\frac {2 c^{2} \left (\ln \left (F \right ) b \,c^{3}-1\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {2 d^{2} \left (10 \ln \left (F \right ) b \,c^{3}-1\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{2} b^{2}}+\frac {5 d^{3} c^{2} x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 d^{4} c \,x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}\)	\(259\)
parallelrisch	\(\frac {d^{6} F^{a +b \left (d x +c \right )^{3}} x^{6} b^{2} \ln \left (F \right )^{2}+6 c \,d^{5} F^{a +b \left (d x +c \right )^{3}} x^{5} b^{2} \ln \left (F \right )^{2}+15 c^{2} d^{4} F^{a +b \left (d x +c \right )^{3}} x^{4} b^{2} \ln \left (F \right )^{2}+20 \ln \left (F \right )^{2} x^{3} F^{a +b \left (d x +c \right )^{3}} b^{2} c^{3} d^{3}+15 \ln \left (F \right )^{2} x^{2} F^{a +b \left (d x +c \right )^{3}} b^{2} c^{4} d^{2}+6 \ln \left (F \right )^{2} x \,F^{a +b \left (d x +c \right )^{3}} b^{2} c^{5} d +\ln \left (F \right )^{2} F^{a +b \left (d x +c \right )^{3}} b^{2} c^{6}-2 d^{3} F^{a +b \left (d x +c \right )^{3}} x^{3} b \ln \left (F \right )-6 c \,d^{2} F^{a +b \left (d x +c \right )^{3}} x^{2} b \ln \left (F \right )-6 c^{2} F^{a +b \left (d x +c \right )^{3}} x b \ln \left (F \right ) d -2 \ln \left (F \right ) F^{a +b \left (d x +c \right )^{3}} b \,c^{3}+2 F^{a +b \left (d x +c \right )^{3}}}{3 \ln \left (F \right )^{3} b^{3} d}\)	\(322\)

[In]

int(F^(a+b*(d*x+c)^3)*(d*x+c)^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.79 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\frac {{\left ({\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 2\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{3} d \log \left (F\right )^{3}} \]

[In]

integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^8,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (83) = 166\).

Time = 0.16 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.17 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{2} c^{6} \log {\left (F \right )}^{2} + 6 b^{2} c^{5} d x \log {\left (F \right )}^{2} + 15 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} + 20 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} + 15 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 6 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} + b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} - 2 b c^{3} \log {\left (F \right )} - 6 b c^{2} d x \log {\left (F \right )} - 6 b c d^{2} x^{2} \log {\left (F \right )} - 2 b d^{3} x^{3} \log {\left (F \right )} + 2\right )}{3 b^{3} d \log {\left (F \right )}^{3}} & \text {for}\: b^{3} d \log {\left (F \right )}^{3} \neq 0 \\c^{8} x + 4 c^{7} d x^{2} + \frac {28 c^{6} d^{2} x^{3}}{3} + 14 c^{5} d^{3} x^{4} + 14 c^{4} d^{4} x^{5} + \frac {28 c^{3} d^{5} x^{6}}{3} + 4 c^{2} d^{6} x^{7} + c d^{7} x^{8} + \frac {d^{8} x^{9}}{9} & \text {otherwise} \end {cases} \]

[In]

integrate(F**(a+b*(d*x+c)**3)*(d*x+c)**8,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**3)*(b**2*c**6*log(F)**2 + 6*b**2*c**5*d*x*log(F)**2 + 15*b**2*c**4*d**2*x**2*l
og(F)**2 + 20*b**2*c**3*d**3*x**3*log(F)**2 + 15*b**2*c**2*d**4*x**4*log(F)**2 + 6*b**2*c*d**5*x**5*log(F)**2
+ b**2*d**6*x**6*log(F)**2 - 2*b*c**3*log(F) - 6*b*c**2*d*x*log(F) - 6*b*c*d**2*x**2*log(F) - 2*b*d**3*x**3*lo
g(F) + 2)/(3*b**3*d*log(F)**3), Ne(b**3*d*log(F)**3, 0)), (c**8*x + 4*c**7*d*x**2 + 28*c**6*d**2*x**3/3 + 14*c
**5*d**3*x**4 + 14*c**4*d**4*x**5 + 28*c**3*d**5*x**6/3 + 4*c**2*d**6*x**7 + c*d**7*x**8 + d**8*x**9/9, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (90) = 180\).

Time = 0.36 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.21 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\frac {{\left (F^{b c^{3} + a} b^{2} d^{6} x^{6} \log \left (F\right )^{2} + 6 \, F^{b c^{3} + a} b^{2} c d^{5} x^{5} \log \left (F\right )^{2} + 15 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} x^{4} \log \left (F\right )^{2} + F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} - 2 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 2 \, {\left (10 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} - F^{b c^{3} + a} b d^{3} \log \left (F\right )\right )} x^{3} + 3 \, {\left (5 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} - 2 \, F^{b c^{3} + a} b c d^{2} \log \left (F\right )\right )} x^{2} + 6 \, {\left (F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} - F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x + 2 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right )\right )}}{3 \, b^{3} d \log \left (F\right )^{3}} \]

[In]

integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^8,x, algorithm="maxima")

[Out]

1/3*(F^(b*c^3 + a)*b^2*d^6*x^6*log(F)^2 + 6*F^(b*c^3 + a)*b^2*c*d^5*x^5*log(F)^2 + 15*F^(b*c^3 + a)*b^2*c^2*d^
4*x^4*log(F)^2 + F^(b*c^3 + a)*b^2*c^6*log(F)^2 - 2*F^(b*c^3 + a)*b*c^3*log(F) + 2*(10*F^(b*c^3 + a)*b^2*c^3*d
^3*log(F)^2 - F^(b*c^3 + a)*b*d^3*log(F))*x^3 + 3*(5*F^(b*c^3 + a)*b^2*c^4*d^2*log(F)^2 - 2*F^(b*c^3 + a)*b*c*
d^2*log(F))*x^2 + 6*(F^(b*c^3 + a)*b^2*c^5*d*log(F)^2 - F^(b*c^3 + a)*b*c^2*d*log(F))*x + 2*F^(b*c^3 + a))*e^(
b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F))/(b^3*d*log(F)^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (90) = 180\).

Time = 0.39 (sec) , antiderivative size = 705, normalized size of antiderivative = 7.34 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=\frac {b^{2} d^{6} x^{6} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 6 \, b^{2} c d^{5} x^{5} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 15 \, b^{2} c^{2} d^{4} x^{4} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 20 \, b^{2} c^{3} d^{3} x^{3} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 15 \, b^{2} c^{4} d^{2} x^{2} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 6 \, b^{2} c^{5} d x e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + b^{2} c^{6} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} - 2 \, b d^{3} x^{3} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) - 6 \, b c d^{2} x^{2} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) - 6 \, b c^{2} d x e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) - 2 \, b c^{3} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) + 2 \, e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )}}{3 \, b^{3} d \log \left (F\right )^{3}} \]

[In]

integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^8,x, algorithm="giac")

[Out]

1/3*(b^2*d^6*x^6*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*lo
g(F)^2 + 6*b^2*c*d^5*x^5*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*lo
g(F))*log(F)^2 + 15*b^2*c^2*d^4*x^4*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*lo
g(F) + a*log(F))*log(F)^2 + 20*b^2*c^3*d^3*x^3*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F)
+ b*c^3*log(F) + a*log(F))*log(F)^2 + 15*b^2*c^4*d^2*x^2*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2
*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^2 + 6*b^2*c^5*d*x*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3
*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^2 + b^2*c^6*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3
*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^2 - 2*b*d^3*x^3*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F)
+ 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F) - 6*b*c*d^2*x^2*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*lo
g(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F) - 6*b*c^2*d*x*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*
log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F) - 2*b*c^3*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*lo
g(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F) + 2*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3
*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F)))/(b^3*d*log(F)^3)

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.04 \[ \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx=F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^2\,c^6\,{\ln \left (F\right )}^2-2\,b\,c^3\,\ln \left (F\right )+2}{3\,b^3\,d\,{\ln \left (F\right )}^3}+\frac {d^5\,x^6}{3\,b\,\ln \left (F\right )}+\frac {2\,c^2\,x\,\left (b\,c^3\,\ln \left (F\right )-1\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {2\,c\,d^4\,x^5}{b\,\ln \left (F\right )}+\frac {2\,d^2\,x^3\,\left (10\,b\,c^3\,\ln \left (F\right )-1\right )}{3\,b^2\,{\ln \left (F\right )}^2}+\frac {5\,c^2\,d^3\,x^4}{b\,\ln \left (F\right )}+\frac {c\,d\,x^2\,\left (5\,b\,c^3\,\ln \left (F\right )-2\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \]

[In]

int(F^(a + b*(c + d*x)^3)*(c + d*x)^8,x)

[Out]

F^(b*d^3*x^3)*F^(3*b*c^2*d*x)*F^a*F^(b*c^3)*F^(3*b*c*d^2*x^2)*((b^2*c^6*log(F)^2 - 2*b*c^3*log(F) + 2)/(3*b^3*
d*log(F)^3) + (d^5*x^6)/(3*b*log(F)) + (2*c^2*x*(b*c^3*log(F) - 1))/(b^2*log(F)^2) + (2*c*d^4*x^5)/(b*log(F))
+ (2*d^2*x^3*(10*b*c^3*log(F) - 1))/(3*b^2*log(F)^2) + (5*c^2*d^3*x^4)/(b*log(F)) + (c*d*x^2*(5*b*c^3*log(F) -
2))/(b^2*log(F)^2))