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

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

[Out]

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

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Rubi [C]  time = 0.05, 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*}

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Mathematica [C]  time = 0.01, size = 29, normalized size = 0.32 \[ -\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{c+d x}\right )}{b^5 d \log ^5(F)} \]

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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fricas [B]  time = 0.41, size = 219, normalized size = 2.38 \[ -\frac {{\left (24 \, d^{4} x^{4} + b^{4} \log \relax (F)^{4} + 96 \, c d^{3} x^{3} + 144 \, c^{2} d^{2} x^{2} + 96 \, c^{3} d x + 24 \, c^{4} - 4 \, {\left (b^{3} d x + b^{3} c\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \relax (F)^{2} - 24 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{5} d^{5} x^{4} + 4 \, b^{5} c d^{4} x^{3} + 6 \, b^{5} c^{2} d^{3} x^{2} + 4 \, b^{5} c^{3} d^{2} x + b^{5} c^{4} d\right )} \log \relax (F)^{5}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{6}}\,{d x} \]

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)

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maple [B]  time = 0.06, size = 329, normalized size = 3.58 \[ \frac {-\frac {24 d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}+\frac {24 \left (b \ln \relax (F )-5 c \right ) d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {12 \left (b^{2} \ln \relax (F )^{2}-8 b c \ln \relax (F )+20 c^{2}\right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}+\frac {4 \left (b^{3} \ln \relax (F )^{3}-9 b^{2} c \ln \relax (F )^{2}+36 b \,c^{2} \ln \relax (F )-60 c^{3}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {\left (b^{4} \ln \relax (F )^{4}-8 b^{3} c \ln \relax (F )^{3}+36 b^{2} c^{2} \ln \relax (F )^{2}-96 b \,c^{3} \ln \relax (F )+120 c^{4}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {\left (b^{4} \ln \relax (F )^{4}-4 b^{3} c \ln \relax (F )^{3}+12 b^{2} c^{2} \ln \relax (F )^{2}-24 b \,c^{3} \ln \relax (F )+24 c^{4}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{5} d \ln \relax (F )^{5}}}{\left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)*b)/(d*x+c)^6,x)

[Out]

(-24*d^4/ln(F)^5/b^5*x^5*exp((a+1/(d*x+c)*b)*ln(F))-(b^4*ln(F)^4-8*ln(F)^3*b^3*c+36*ln(F)^2*b^2*c^2-96*b*c^3*l
n(F)+120*c^4)/ln(F)^5/b^5*x*exp((a+1/(d*x+c)*b)*ln(F))+4*d*(b^3*ln(F)^3-9*b^2*c*ln(F)^2+36*b*c^2*ln(F)-60*c^3)
/ln(F)^5/b^5*x^2*exp((a+1/(d*x+c)*b)*ln(F))-12*d^2*(b^2*ln(F)^2-8*b*c*ln(F)+20*c^2)/ln(F)^5/b^5*x^3*exp((a+1/(
d*x+c)*b)*ln(F))+24*d^3*(b*ln(F)-5*c)/ln(F)^5/b^5*x^4*exp((a+1/(d*x+c)*b)*ln(F))-(b^4*ln(F)^4-4*ln(F)^3*b^3*c+
12*ln(F)^2*b^2*c^2-24*b*c^3*ln(F)+24*c^4)*c/b^5/ln(F)^5/d*exp((a+1/(d*x+c)*b)*ln(F)))/(d*x+c)^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{6}}\,{d x} \]

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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mupad [B]  time = 3.88, size = 231, normalized size = 2.51 \[ -\frac {F^{a+\frac {b}{c+d\,x}}\,\left (\frac {b^4\,{\ln \relax (F)}^4-4\,b^3\,c\,{\ln \relax (F)}^3+12\,b^2\,c^2\,{\ln \relax (F)}^2-24\,b\,c^3\,\ln \relax (F)+24\,c^4}{b^5\,d^5\,{\ln \relax (F)}^5}+\frac {24\,x^4}{b^5\,d\,{\ln \relax (F)}^5}+\frac {x^2\,\left (12\,b^2\,{\ln \relax (F)}^2-72\,b\,c\,\ln \relax (F)+144\,c^2\right )}{b^5\,d^3\,{\ln \relax (F)}^5}+\frac {x^3\,\left (96\,c-24\,b\,\ln \relax (F)\right )}{b^5\,d^2\,{\ln \relax (F)}^5}-\frac {4\,x\,\left (b^3\,{\ln \relax (F)}^3-6\,b^2\,c\,{\ln \relax (F)}^2+18\,b\,c^2\,\ln \relax (F)-24\,c^3\right )}{b^5\,d^4\,{\ln \relax (F)}^5}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(F^(a + b/(c + d*x))*((b^4*log(F)^4 + 24*c^4 - 24*b*c^3*log(F) - 4*b^3*c*log(F)^3 + 12*b^2*c^2*log(F)^2)/(b^5
*d^5*log(F)^5) + (24*x^4)/(b^5*d*log(F)^5) + (x^2*(12*b^2*log(F)^2 + 144*c^2 - 72*b*c*log(F)))/(b^5*d^3*log(F)
^5) + (x^3*(96*c - 24*b*log(F)))/(b^5*d^2*log(F)^5) - (4*x*(b^3*log(F)^3 - 24*c^3 + 18*b*c^2*log(F) - 6*b^2*c*
log(F)^2))/(b^5*d^4*log(F)^5)))/(x^4 + c^4/d^4 + (4*c*x^3)/d + (4*c^3*x)/d^3 + (6*c^2*x^2)/d^2)

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sympy [B]  time = 0.35, size = 272, normalized size = 2.96 \[ \frac {F^{a + \frac {b}{c + d x}} \left (- b^{4} \log {\relax (F )}^{4} + 4 b^{3} c \log {\relax (F )}^{3} + 4 b^{3} d x \log {\relax (F )}^{3} - 12 b^{2} c^{2} \log {\relax (F )}^{2} - 24 b^{2} c d x \log {\relax (F )}^{2} - 12 b^{2} d^{2} x^{2} \log {\relax (F )}^{2} + 24 b c^{3} \log {\relax (F )} + 72 b c^{2} d x \log {\relax (F )} + 72 b c d^{2} x^{2} \log {\relax (F )} + 24 b d^{3} x^{3} \log {\relax (F )} - 24 c^{4} - 96 c^{3} d x - 144 c^{2} d^{2} x^{2} - 96 c d^{3} x^{3} - 24 d^{4} x^{4}\right )}{b^{5} c^{4} d \log {\relax (F )}^{5} + 4 b^{5} c^{3} d^{2} x \log {\relax (F )}^{5} + 6 b^{5} c^{2} d^{3} x^{2} \log {\relax (F )}^{5} + 4 b^{5} c d^{4} x^{3} \log {\relax (F )}^{5} + b^{5} d^{5} x^{4} \log {\relax (F )}^{5}} \]

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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