3.312 \(\int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^6} \, dx\)

Optimal. Leaf size=92 \[ -\frac{F^{a+\frac{b}{c+d x}} \left (12 b^2 \log ^2(F) (c+d x)^2-4 b^3 \log ^3(F) (c+d x)+b^4 \log ^4(F)-24 b \log (F) (c+d x)^3+24 (c+d x)^4\right )}{b^5 d \log ^5(F) (c+d x)^4} \]

[Out]

-((F^(a + b/(c + d*x))*(24*(c + d*x)^4 - 24*b*(c + d*x)^3*Log[F] + 12*b^2*(c + d*x)^2*Log[F]^2 - 4*b^3*(c + d*
x)*Log[F]^3 + b^4*Log[F]^4))/(b^5*d*(c + d*x)^4*Log[F]^5))

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Rubi [C]  time = 0.0485709, antiderivative size = 29, normalized size of antiderivative = 0.32, 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} \[ -\frac{F^a \text{Gamma}\left (5,-\frac{b \log (F)}{c+d x}\right )}{b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^a*Gamma[5, -((b*Log[F])/(c + d*x))])/(b^5*d*Log[F]^5))

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)^6} \, dx &=-\frac{F^a \Gamma \left (5,-\frac{b \log (F)}{c+d x}\right )}{b^5 d \log ^5(F)}\\ \end{align*}

Mathematica [C]  time = 0.006196, size = 29, normalized size = 0.32 \[ -\frac{F^a \text{Gamma}\left (5,-\frac{b \log (F)}{c+d x}\right )}{b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^a*Gamma[5, -((b*Log[F])/(c + d*x))])/(b^5*d*Log[F]^5))

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Maple [B]  time = 0.048, size = 329, normalized size = 3.6 \begin{align*}{\frac{1}{ \left ( dx+c \right ) ^{5}} \left ( -24\,{\frac{{d}^{4}{x}^{5}}{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ({b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}-8\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}c+36\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}-96\,\ln \left ( F \right ) b{c}^{3}+120\,{c}^{4} \right ) x}{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}+4\,{\frac{d \left ( \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}-9\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}c+36\,\ln \left ( F \right ) b{c}^{2}-60\,{c}^{3} \right ){x}^{2}}{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-12\,{\frac{{d}^{2} \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-8\,bc\ln \left ( F \right ) +20\,{c}^{2} \right ){x}^{3}}{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}+24\,{\frac{{d}^{3} \left ( b\ln \left ( F \right ) -5\,c \right ){x}^{4}}{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ({b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}c+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}-24\,\ln \left ( F \right ) b{c}^{3}+24\,{c}^{4} \right ) c}{d \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c))/(d*x+c)^6,x)

[Out]

(-24*d^4/ln(F)^5/b^5*x^5*exp((a+b/(d*x+c))*ln(F))-(b^4*ln(F)^4-8*ln(F)^3*b^3*c+36*ln(F)^2*b^2*c^2-96*ln(F)*b*c
^3+120*c^4)/ln(F)^5/b^5*x*exp((a+b/(d*x+c))*ln(F))+4*d*(ln(F)^3*b^3-9*ln(F)^2*b^2*c+36*ln(F)*b*c^2-60*c^3)/ln(
F)^5/b^5*x^2*exp((a+b/(d*x+c))*ln(F))-12*d^2*(ln(F)^2*b^2-8*b*c*ln(F)+20*c^2)/ln(F)^5/b^5*x^3*exp((a+b/(d*x+c)
)*ln(F))+24*d^3*(b*ln(F)-5*c)/ln(F)^5/b^5*x^4*exp((a+b/(d*x+c))*ln(F))-(b^4*ln(F)^4-4*ln(F)^3*b^3*c+12*ln(F)^2
*b^2*c^2-24*ln(F)*b*c^3+24*c^4)*c/b^5/ln(F)^5/d*exp((a+b/(d*x+c))*ln(F)))/(d*x+c)^5

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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 )}^{6}}\,{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)^6,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c))/(d*x + c)^6, x)

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Fricas [B]  time = 1.66196, size = 479, normalized size = 5.21 \begin{align*} -\frac{{\left (24 \, d^{4} x^{4} + b^{4} \log \left (F\right )^{4} + 96 \, c d^{3} x^{3} + 144 \, c^{2} d^{2} x^{2} + 96 \, c^{3} d x + 24 \, c^{4} - 4 \,{\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 24 \,{\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 )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{{\left (b^{5} d^{5} x^{4} + 4 \, b^{5} c d^{4} x^{3} + 6 \, b^{5} c^{2} d^{3} x^{2} + 4 \, b^{5} c^{3} d^{2} x + b^{5} c^{4} d\right )} \log \left (F\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))/(d*x+c)^6,x, algorithm="fricas")

[Out]

-(24*d^4*x^4 + b^4*log(F)^4 + 96*c*d^3*x^3 + 144*c^2*d^2*x^2 + 96*c^3*d*x + 24*c^4 - 4*(b^3*d*x + b^3*c)*log(F
)^3 + 12*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(F)^2 - 24*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)
*log(F))*F^((a*d*x + a*c + b)/(d*x + c))/((b^5*d^5*x^4 + 4*b^5*c*d^4*x^3 + 6*b^5*c^2*d^3*x^2 + 4*b^5*c^3*d^2*x
+ b^5*c^4*d)*log(F)^5)

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Sympy [B]  time = 0.318188, size = 272, normalized size = 2.96 \begin{align*} \frac{F^{a + \frac{b}{c + d x}} \left (- b^{4} \log{\left (F \right )}^{4} + 4 b^{3} c \log{\left (F \right )}^{3} + 4 b^{3} d x \log{\left (F \right )}^{3} - 12 b^{2} c^{2} \log{\left (F \right )}^{2} - 24 b^{2} c d x \log{\left (F \right )}^{2} - 12 b^{2} d^{2} x^{2} \log{\left (F \right )}^{2} + 24 b c^{3} \log{\left (F \right )} + 72 b c^{2} d x \log{\left (F \right )} + 72 b c d^{2} x^{2} \log{\left (F \right )} + 24 b d^{3} x^{3} \log{\left (F \right )} - 24 c^{4} - 96 c^{3} d x - 144 c^{2} d^{2} x^{2} - 96 c d^{3} x^{3} - 24 d^{4} x^{4}\right )}{b^{5} c^{4} d \log{\left (F \right )}^{5} + 4 b^{5} c^{3} d^{2} x \log{\left (F \right )}^{5} + 6 b^{5} c^{2} d^{3} x^{2} \log{\left (F \right )}^{5} + 4 b^{5} c d^{4} x^{3} \log{\left (F \right )}^{5} + b^{5} d^{5} x^{4} \log{\left (F \right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c))/(d*x+c)**6,x)

[Out]

F**(a + b/(c + d*x))*(-b**4*log(F)**4 + 4*b**3*c*log(F)**3 + 4*b**3*d*x*log(F)**3 - 12*b**2*c**2*log(F)**2 - 2
4*b**2*c*d*x*log(F)**2 - 12*b**2*d**2*x**2*log(F)**2 + 24*b*c**3*log(F) + 72*b*c**2*d*x*log(F) + 72*b*c*d**2*x
**2*log(F) + 24*b*d**3*x**3*log(F) - 24*c**4 - 96*c**3*d*x - 144*c**2*d**2*x**2 - 96*c*d**3*x**3 - 24*d**4*x**
4)/(b**5*c**4*d*log(F)**5 + 4*b**5*c**3*d**2*x*log(F)**5 + 6*b**5*c**2*d**3*x**2*log(F)**5 + 4*b**5*c*d**4*x**
3*log(F)**5 + b**5*d**5*x**4*log(F)**5)

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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 )}^{6}}\,{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)^6,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c))/(d*x + c)^6, x)