3.4.13 \(\int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^7} \, dx\) [313]

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

[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 [A]
time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, 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 = {2249} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[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)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

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] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.01, size = 28, normalized size = 0.26 \begin {gather*} \frac {F^a \Gamma \left (6,-\frac {b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs. \(2(108)=216\).
time = 0.08, size = 271, normalized size = 2.51

method	result	size
risch	\(-\frac {\left (b^{5} \ln \left (F \right )^{5}-5 \ln \left (F \right )^{4} b^{4} d x +20 \ln \left (F \right )^{3} b^{3} d^{2} x^{2}-60 \ln \left (F \right )^{2} b^{2} d^{3} x^{3}+120 \ln \left (F \right ) b \,d^{4} x^{4}-120 d^{5} x^{5}-5 \ln \left (F \right )^{4} b^{4} c +40 \ln \left (F \right )^{3} b^{3} c d x -180 \ln \left (F \right )^{2} b^{2} c \,d^{2} x^{2}+480 \ln \left (F \right ) b c \,d^{3} x^{3}-600 c \,d^{4} x^{4}+20 \ln \left (F \right )^{3} b^{3} c^{2}-180 \ln \left (F \right )^{2} b^{2} c^{2} d x +720 \ln \left (F \right ) b \,c^{2} d^{2} x^{2}-1200 c^{2} d^{3} x^{3}-60 \ln \left (F \right )^{2} b^{2} c^{3}+480 \ln \left (F \right ) b \,c^{3} d x -1200 c^{3} d^{2} x^{2}+120 \ln \left (F \right ) b \,c^{4}-600 c^{4} d x -120 c^{5}\right ) F^{\frac {x a d +c a +b}{d x +c}}}{b^{6} \ln \left (F \right )^{6} d \left (d x +c \right )^{5}}\)	\(271\)
norman	\(\frac {\frac {120 d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6}}-\frac {\left (b^{5} \ln \left (F \right )^{5}-10 \ln \left (F \right )^{4} b^{4} c +60 \ln \left (F \right )^{3} b^{3} c^{2}-240 \ln \left (F \right )^{2} b^{2} c^{3}+600 \ln \left (F \right ) b \,c^{4}-720 c^{5}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6}}+\frac {5 d \left (b^{4} \ln \left (F \right )^{4}-12 \ln \left (F \right )^{3} b^{3} c +72 \ln \left (F \right )^{2} b^{2} c^{2}-240 \ln \left (F \right ) b \,c^{3}+360 c^{4}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{b^{6} \ln \left (F \right )^{6}}-\frac {20 d^{2} \left (\ln \left (F \right )^{3} b^{3}-12 \ln \left (F \right )^{2} b^{2} c +60 \ln \left (F \right ) b \,c^{2}-120 c^{3}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6}}+\frac {60 d^{3} \left (\ln \left (F \right )^{2} b^{2}-10 c b \ln \left (F \right )+30 c^{2}\right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6}}-\frac {120 d^{4} \left (b \ln \left (F \right )-6 c \right ) x^{5} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6}}-\frac {\left (b^{5} \ln \left (F \right )^{5}-5 \ln \left (F \right )^{4} b^{4} c +20 \ln \left (F \right )^{3} b^{3} c^{2}-60 \ln \left (F \right )^{2} b^{2} c^{3}+120 \ln \left (F \right ) b \,c^{4}-120 c^{5}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{b^{6} \ln \left (F \right )^{6} d}}{\left (d x +c \right )^{6}}\)	\(427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^5*ln(F)^5-5*ln(F)^4*b^4*d*x+20*ln(F)^3*b^3*d^2*x^2-60*ln(F)^2*b^2*d^3*x^3+120*ln(F)*b*d^4*x^4-120*d^5*x^5-
5*ln(F)^4*b^4*c+40*ln(F)^3*b^3*c*d*x-180*ln(F)^2*b^2*c*d^2*x^2+480*ln(F)*b*c*d^3*x^3-600*c*d^4*x^4+20*ln(F)^3*
b^3*c^2-180*ln(F)^2*b^2*c^2*d*x+720*ln(F)*b*c^2*d^2*x^2-1200*c^2*d^3*x^3-60*ln(F)^2*b^2*c^3+480*ln(F)*b*c^3*d*
x-1200*c^3*d^2*x^2+120*ln(F)*b*c^4-600*c^4*d*x-120*c^5)/b^6/ln(F)^6/d/(d*x+c)^5*F^((a*d*x+a*c+b)/(d*x+c))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (108) = 216\).
time = 0.39, size = 302, normalized size = 2.80 \begin {gather*} \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 {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (105) = 210\).
time = 0.18, size = 388, normalized size = 3.59 \begin {gather*} \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 {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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