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

Optimal. Leaf size=108 \[ \frac{F^{a+\frac{b}{c+d x}} \left (60 b^2 \log ^2(F) (c+d x)^3-20 b^3 \log ^3(F) (c+d x)^2+5 b^4 \log ^4(F) (c+d x)-b^5 \log ^5(F)-120 b \log (F) (c+d x)^4+120 (c+d x)^5\right )}{b^6 d \log ^6(F) (c+d x)^5} \]

[Out]

(F^(a + b/(c + d*x))*(120*(c + d*x)^5 - 120*b*(c + d*x)^4*Log[F] + 60*b^2*(c + d*x)^3*Log[F]^2 - 20*b^3*(c + d
*x)^2*Log[F]^3 + 5*b^4*(c + d*x)*Log[F]^4 - b^5*Log[F]^5))/(b^6*d*(c + d*x)^5*Log[F]^6)

________________________________________________________________________________________

Rubi [C]  time = 0.0533513, antiderivative size = 28, normalized size of antiderivative = 0.26, 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} \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^7} \, dx &=\frac{F^a \Gamma \left (6,-\frac{b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)}\\ \end{align*}

Mathematica [C]  time = 0.0060418, size = 28, normalized size = 0.26 \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 427, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c))/(d*x+c)^7,x)

[Out]

(120*d^5/ln(F)^6/b^6*x^6*exp((a+b/(d*x+c))*ln(F))-(b^5*ln(F)^5-10*ln(F)^4*b^4*c+60*ln(F)^3*b^3*c^2-240*ln(F)^2
*b^2*c^3+600*ln(F)*b*c^4-720*c^5)/ln(F)^6/b^6*x*exp((a+b/(d*x+c))*ln(F))+5*d*(b^4*ln(F)^4-12*ln(F)^3*b^3*c+72*
ln(F)^2*b^2*c^2-240*ln(F)*b*c^3+360*c^4)/b^6/ln(F)^6*x^2*exp((a+b/(d*x+c))*ln(F))-20*d^2*(ln(F)^3*b^3-12*ln(F)
^2*b^2*c+60*ln(F)*b*c^2-120*c^3)/ln(F)^6/b^6*x^3*exp((a+b/(d*x+c))*ln(F))+60*d^3*(ln(F)^2*b^2-10*b*c*ln(F)+30*
c^2)/ln(F)^6/b^6*x^4*exp((a+b/(d*x+c))*ln(F))-120*d^4*(b*ln(F)-6*c)/ln(F)^6/b^6*x^5*exp((a+b/(d*x+c))*ln(F))-(
b^5*ln(F)^5-5*ln(F)^4*b^4*c+20*ln(F)^3*b^3*c^2-60*ln(F)^2*b^2*c^3+120*ln(F)*b*c^4-120*c^5)*c/b^6/ln(F)^6/d*exp
((a+b/(d*x+c))*ln(F)))/(d*x+c)^6

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^(a + b/(d*x + c))/(d*x + c)^7, x)

________________________________________________________________________________________

Fricas [B]  time = 1.65352, size = 663, normalized size = 6.14 \begin{align*} \frac{{\left (120 \, d^{5} x^{5} - b^{5} \log \left (F\right )^{5} + 600 \, c d^{4} x^{4} + 1200 \, c^{2} d^{3} x^{3} + 1200 \, c^{3} d^{2} x^{2} + 600 \, c^{4} d x + 120 \, c^{5} + 5 \,{\left (b^{4} d x + b^{4} c\right )} \log \left (F\right )^{4} - 20 \,{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 60 \,{\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} - 120 \,{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{{\left (b^{6} d^{6} x^{5} + 5 \, b^{6} c d^{5} x^{4} + 10 \, b^{6} c^{2} d^{4} x^{3} + 10 \, b^{6} c^{3} d^{3} x^{2} + 5 \, b^{6} c^{4} d^{2} x + b^{6} c^{5} d\right )} \log \left (F\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(120*d^5*x^5 - b^5*log(F)^5 + 600*c*d^4*x^4 + 1200*c^2*d^3*x^3 + 1200*c^3*d^2*x^2 + 600*c^4*d*x + 120*c^5 + 5*
(b^4*d*x + b^4*c)*log(F)^4 - 20*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*log(F)^3 + 60*(b^2*d^3*x^3 + 3*b^2*c*d^2
*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*log(F)^2 - 120*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*
c^4)*log(F))*F^((a*d*x + a*c + b)/(d*x + c))/((b^6*d^6*x^5 + 5*b^6*c*d^5*x^4 + 10*b^6*c^2*d^4*x^3 + 10*b^6*c^3
*d^3*x^2 + 5*b^6*c^4*d^2*x + b^6*c^5*d)*log(F)^6)

________________________________________________________________________________________

Sympy [B]  time = 0.361202, size = 388, normalized size = 3.59 \begin{align*} \frac{F^{a + \frac{b}{c + d x}} \left (- b^{5} \log{\left (F \right )}^{5} + 5 b^{4} c \log{\left (F \right )}^{4} + 5 b^{4} d x \log{\left (F \right )}^{4} - 20 b^{3} c^{2} \log{\left (F \right )}^{3} - 40 b^{3} c d x \log{\left (F \right )}^{3} - 20 b^{3} d^{2} x^{2} \log{\left (F \right )}^{3} + 60 b^{2} c^{3} \log{\left (F \right )}^{2} + 180 b^{2} c^{2} d x \log{\left (F \right )}^{2} + 180 b^{2} c d^{2} x^{2} \log{\left (F \right )}^{2} + 60 b^{2} d^{3} x^{3} \log{\left (F \right )}^{2} - 120 b c^{4} \log{\left (F \right )} - 480 b c^{3} d x \log{\left (F \right )} - 720 b c^{2} d^{2} x^{2} \log{\left (F \right )} - 480 b c d^{3} x^{3} \log{\left (F \right )} - 120 b d^{4} x^{4} \log{\left (F \right )} + 120 c^{5} + 600 c^{4} d x + 1200 c^{3} d^{2} x^{2} + 1200 c^{2} d^{3} x^{3} + 600 c d^{4} x^{4} + 120 d^{5} x^{5}\right )}{b^{6} c^{5} d \log{\left (F \right )}^{6} + 5 b^{6} c^{4} d^{2} x \log{\left (F \right )}^{6} + 10 b^{6} c^{3} d^{3} x^{2} \log{\left (F \right )}^{6} + 10 b^{6} c^{2} d^{4} x^{3} \log{\left (F \right )}^{6} + 5 b^{6} c d^{5} x^{4} \log{\left (F \right )}^{6} + b^{6} d^{6} x^{5} \log{\left (F \right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c))/(d*x+c)**7,x)

[Out]

F**(a + b/(c + d*x))*(-b**5*log(F)**5 + 5*b**4*c*log(F)**4 + 5*b**4*d*x*log(F)**4 - 20*b**3*c**2*log(F)**3 - 4
0*b**3*c*d*x*log(F)**3 - 20*b**3*d**2*x**2*log(F)**3 + 60*b**2*c**3*log(F)**2 + 180*b**2*c**2*d*x*log(F)**2 +
180*b**2*c*d**2*x**2*log(F)**2 + 60*b**2*d**3*x**3*log(F)**2 - 120*b*c**4*log(F) - 480*b*c**3*d*x*log(F) - 720
*b*c**2*d**2*x**2*log(F) - 480*b*c*d**3*x**3*log(F) - 120*b*d**4*x**4*log(F) + 120*c**5 + 600*c**4*d*x + 1200*
c**3*d**2*x**2 + 1200*c**2*d**3*x**3 + 600*c*d**4*x**4 + 120*d**5*x**5)/(b**6*c**5*d*log(F)**6 + 5*b**6*c**4*d
**2*x*log(F)**6 + 10*b**6*c**3*d**3*x**2*log(F)**6 + 10*b**6*c**2*d**4*x**3*log(F)**6 + 5*b**6*c*d**5*x**4*log
(F)**6 + b**6*d**6*x**5*log(F)**6)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^(a + b/(d*x + c))/(d*x + c)^7, x)