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

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

[Out]

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

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Rubi [C]  time = 0.05, 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*}

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Mathematica [C]  time = 0.01, size = 28, normalized size = 0.26 \[ \frac {F^a \Gamma \left (6,-\frac {b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]

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)

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fricas [B]  time = 0.43, size = 302, normalized size = 2.80 \[ \frac {{\left (120 \, d^{5} x^{5} - b^{5} \log \relax (F)^{5} + 600 \, c d^{4} x^{4} + 1200 \, c^{2} d^{3} x^{3} + 1200 \, c^{3} d^{2} x^{2} + 600 \, c^{4} d x + 120 \, c^{5} + 5 \, {\left (b^{4} d x + b^{4} c\right )} \log \relax (F)^{4} - 20 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \relax (F)^{3} + 60 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} - 120 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{6} d^{6} x^{5} + 5 \, b^{6} c d^{5} x^{4} + 10 \, b^{6} c^{2} d^{4} x^{3} + 10 \, b^{6} c^{3} d^{3} x^{2} + 5 \, b^{6} c^{4} d^{2} x + b^{6} c^{5} d\right )} \log \relax (F)^{6}} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \]

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)

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maple [B]  time = 0.08, size = 427, normalized size = 3.95 \[ \frac {\frac {120 d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {120 \left (b \ln \relax (F )-6 c \right ) d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}+\frac {60 \left (b^{2} \ln \relax (F )^{2}-10 b c \ln \relax (F )+30 c^{2}\right ) d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {20 \left (b^{3} \ln \relax (F )^{3}-12 b^{2} c \ln \relax (F )^{2}+60 b \,c^{2} \ln \relax (F )-120 c^{3}\right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}+\frac {5 \left (b^{4} \ln \relax (F )^{4}-12 b^{3} c \ln \relax (F )^{3}+72 b^{2} c^{2} \ln \relax (F )^{2}-240 b \,c^{3} \ln \relax (F )+360 c^{4}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {\left (b^{5} \ln \relax (F )^{5}-10 b^{4} c \ln \relax (F )^{4}+60 b^{3} c^{2} \ln \relax (F )^{3}-240 b^{2} c^{3} \ln \relax (F )^{2}+600 b \,c^{4} \ln \relax (F )-720 c^{5}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {\left (b^{5} \ln \relax (F )^{5}-5 b^{4} c \ln \relax (F )^{4}+20 b^{3} c^{2} \ln \relax (F )^{3}-60 b^{2} c^{3} \ln \relax (F )^{2}+120 b \,c^{4} \ln \relax (F )-120 c^{5}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} d \ln \relax (F )^{6}}}{\left (d x +c \right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)*b)/(d*x+c)^7,x)

[Out]

(120*d^5/ln(F)^6/b^6*x^6*exp((a+1/(d*x+c)*b)*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+1/(d*x+c)*b)*ln(F))+5*d*(b^4*ln(F)^4-12*b^3*c*ln(F)^3
+72*b^2*c^2*ln(F)^2-240*b*c^3*ln(F)+360*c^4)/b^6/ln(F)^6*x^2*exp((a+1/(d*x+c)*b)*ln(F))-20*d^2*(b^3*ln(F)^3-12
*b^2*c*ln(F)^2+60*b*c^2*ln(F)-120*c^3)/ln(F)^6/b^6*x^3*exp((a+1/(d*x+c)*b)*ln(F))+60*d^3*(b^2*ln(F)^2-10*b*c*l
n(F)+30*c^2)/ln(F)^6/b^6*x^4*exp((a+1/(d*x+c)*b)*ln(F))-120*d^4*(b*ln(F)-6*c)/ln(F)^6/b^6*x^5*exp((a+1/(d*x+c)
*b)*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/l
n(F)^6/d*exp((a+1/(d*x+c)*b)*ln(F)))/(d*x+c)^6

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \]

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)

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mupad [B]  time = 3.94, size = 315, normalized size = 2.92 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,\left (\frac {120\,x^5}{b^6\,d\,{\ln \relax (F)}^6}-\frac {b^5\,{\ln \relax (F)}^5-5\,b^4\,c\,{\ln \relax (F)}^4+20\,b^3\,c^2\,{\ln \relax (F)}^3-60\,b^2\,c^3\,{\ln \relax (F)}^2+120\,b\,c^4\,\ln \relax (F)-120\,c^5}{b^6\,d^6\,{\ln \relax (F)}^6}-\frac {20\,x^2\,\left (b^3\,{\ln \relax (F)}^3-9\,b^2\,c\,{\ln \relax (F)}^2+36\,b\,c^2\,\ln \relax (F)-60\,c^3\right )}{b^6\,d^4\,{\ln \relax (F)}^6}+\frac {60\,x^3\,\left (b^2\,{\ln \relax (F)}^2-8\,b\,c\,\ln \relax (F)+20\,c^2\right )}{b^6\,d^3\,{\ln \relax (F)}^6}+\frac {120\,x^4\,\left (5\,c-b\,\ln \relax (F)\right )}{b^6\,d^2\,{\ln \relax (F)}^6}+\frac {5\,x\,\left (b^4\,{\ln \relax (F)}^4-8\,b^3\,c\,{\ln \relax (F)}^3+36\,b^2\,c^2\,{\ln \relax (F)}^2-96\,b\,c^3\,\ln \relax (F)+120\,c^4\right )}{b^6\,d^5\,{\ln \relax (F)}^6}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*F^(b/(c + d*x))*((120*x^5)/(b^6*d*log(F)^6) - (b^5*log(F)^5 - 120*c^5 + 120*b*c^4*log(F) - 5*b^4*c*log(F)
^4 - 60*b^2*c^3*log(F)^2 + 20*b^3*c^2*log(F)^3)/(b^6*d^6*log(F)^6) - (20*x^2*(b^3*log(F)^3 - 60*c^3 + 36*b*c^2
*log(F) - 9*b^2*c*log(F)^2))/(b^6*d^4*log(F)^6) + (60*x^3*(b^2*log(F)^2 + 20*c^2 - 8*b*c*log(F)))/(b^6*d^3*log
(F)^6) + (120*x^4*(5*c - b*log(F)))/(b^6*d^2*log(F)^6) + (5*x*(b^4*log(F)^4 + 120*c^4 - 96*b*c^3*log(F) - 8*b^
3*c*log(F)^3 + 36*b^2*c^2*log(F)^2))/(b^6*d^5*log(F)^6)))/(x^5 + c^5/d^5 + (5*c*x^4)/d + (5*c^4*x)/d^4 + (10*c
^2*x^3)/d^2 + (10*c^3*x^2)/d^3)

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sympy [B]  time = 0.40, size = 388, normalized size = 3.59 \[ \frac {F^{a + \frac {b}{c + d x}} \left (- b^{5} \log {\relax (F )}^{5} + 5 b^{4} c \log {\relax (F )}^{4} + 5 b^{4} d x \log {\relax (F )}^{4} - 20 b^{3} c^{2} \log {\relax (F )}^{3} - 40 b^{3} c d x \log {\relax (F )}^{3} - 20 b^{3} d^{2} x^{2} \log {\relax (F )}^{3} + 60 b^{2} c^{3} \log {\relax (F )}^{2} + 180 b^{2} c^{2} d x \log {\relax (F )}^{2} + 180 b^{2} c d^{2} x^{2} \log {\relax (F )}^{2} + 60 b^{2} d^{3} x^{3} \log {\relax (F )}^{2} - 120 b c^{4} \log {\relax (F )} - 480 b c^{3} d x \log {\relax (F )} - 720 b c^{2} d^{2} x^{2} \log {\relax (F )} - 480 b c d^{3} x^{3} \log {\relax (F )} - 120 b d^{4} x^{4} \log {\relax (F )} + 120 c^{5} + 600 c^{4} d x + 1200 c^{3} d^{2} x^{2} + 1200 c^{2} d^{3} x^{3} + 600 c d^{4} x^{4} + 120 d^{5} x^{5}\right )}{b^{6} c^{5} d \log {\relax (F )}^{6} + 5 b^{6} c^{4} d^{2} x \log {\relax (F )}^{6} + 10 b^{6} c^{3} d^{3} x^{2} \log {\relax (F )}^{6} + 10 b^{6} c^{2} d^{4} x^{3} \log {\relax (F )}^{6} + 5 b^{6} c d^{5} x^{4} \log {\relax (F )}^{6} + b^{6} d^{6} x^{5} \log {\relax (F )}^{6}} \]

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)

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