3.325 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{F^{a+\frac{b}{(c+d x)^2}} \left (12 b^2 \log ^2(F) (c+d x)^4-4 b^3 \log ^3(F) (c+d x)^2+b^4 \log ^4(F)-24 b \log (F) (c+d x)^6+24 (c+d x)^8\right )}{2 b^5 d \log ^5(F) (c+d x)^8} \]

[Out]

-(F^(a + b/(c + d*x)^2)*(24*(c + d*x)^8 - 24*b*(c + d*x)^6*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 - 4*b^3*(c + d
*x)^2*Log[F]^3 + b^4*Log[F]^4))/(2*b^5*d*(c + d*x)^8*Log[F]^5)

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Rubi [C]  time = 0.0449171, antiderivative size = 31, 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} \[ -\frac{F^a \text{Gamma}\left (5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)/(c + d*x)^11,x]

[Out]

-(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^2)])/(2*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)^2}}}{(c+d x)^{11}} \, dx &=-\frac{F^a \Gamma \left (5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)}\\ \end{align*}

Mathematica [C]  time = 0.0068502, size = 31, normalized size = 0.32 \[ -\frac{F^a \text{Gamma}\left (5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^11,x]

[Out]

-(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^2)])/(2*b^5*d*Log[F]^5)

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Maple [B]  time = 0.145, size = 609, normalized size = 6.3 \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)^2)/(d*x+c)^11,x)

[Out]

(-12*d^9/ln(F)^5/b^5*x^10*exp((a+b/(d*x+c)^2)*ln(F))-c*(b^4*ln(F)^4-8*ln(F)^3*b^3*c^2+36*ln(F)^2*b^2*c^4-96*ln
(F)*b*c^6+120*c^8)/b^5/ln(F)^5*x*exp((a+b/(d*x+c)^2)*ln(F))-1/2*d*(b^4*ln(F)^4-24*ln(F)^3*b^3*c^2+180*ln(F)^2*
b^2*c^4-672*ln(F)*b*c^6+1080*c^8)/ln(F)^5/b^5*x^2*exp((a+b/(d*x+c)^2)*ln(F))+2*d^3*(ln(F)^3*b^3-45*ln(F)^2*b^2
*c^2+420*ln(F)*b*c^4-1260*c^6)/ln(F)^5/b^5*x^4*exp((a+b/(d*x+c)^2)*ln(F))-6*d^5*(ln(F)^2*b^2-56*ln(F)*b*c^2+42
0*c^4)/ln(F)^5/b^5*x^6*exp((a+b/(d*x+c)^2)*ln(F))+12*d^7*(b*ln(F)-45*c^2)/ln(F)^5/b^5*x^8*exp((a+b/(d*x+c)^2)*
ln(F))-120*d^8*c/ln(F)^5/b^5*x^9*exp((a+b/(d*x+c)^2)*ln(F))-1/2*(b^4*ln(F)^4-4*ln(F)^3*b^3*c^2+12*ln(F)^2*b^2*
c^4-24*ln(F)*b*c^6+24*c^8)*c^2/b^5/ln(F)^5/d*exp((a+b/(d*x+c)^2)*ln(F))+8*c*d^2*(ln(F)^3*b^3-15*ln(F)^2*b^2*c^
2+84*ln(F)*b*c^4-180*c^6)/ln(F)^5/b^5*x^3*exp((a+b/(d*x+c)^2)*ln(F))-12*c*d^4*(3*ln(F)^2*b^2-56*ln(F)*b*c^2+25
2*c^4)/ln(F)^5/b^5*x^5*exp((a+b/(d*x+c)^2)*ln(F))+96*c*d^6*(b*ln(F)-15*c^2)/ln(F)^5/b^5*x^7*exp((a+b/(d*x+c)^2
)*ln(F)))/(d*x+c)^10

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Maxima [B]  time = 1.08379, size = 710, normalized size = 7.4 \begin{align*} -\frac{{\left (24 \, F^{a} d^{8} x^{8} + 192 \, F^{a} c d^{7} x^{7} + 24 \, F^{a} c^{8} - 24 \, F^{a} b c^{6} \log \left (F\right ) + 12 \, F^{a} b^{2} c^{4} \log \left (F\right )^{2} - 4 \, F^{a} b^{3} c^{2} \log \left (F\right )^{3} + F^{a} b^{4} \log \left (F\right )^{4} + 24 \,{\left (28 \, F^{a} c^{2} d^{6} - F^{a} b d^{6} \log \left (F\right )\right )} x^{6} + 48 \,{\left (28 \, F^{a} c^{3} d^{5} - 3 \, F^{a} b c d^{5} \log \left (F\right )\right )} x^{5} + 12 \,{\left (140 \, F^{a} c^{4} d^{4} - 30 \, F^{a} b c^{2} d^{4} \log \left (F\right ) + F^{a} b^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 48 \,{\left (28 \, F^{a} c^{5} d^{3} - 10 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + F^{a} b^{2} c d^{3} \log \left (F\right )^{2}\right )} x^{3} + 4 \,{\left (168 \, F^{a} c^{6} d^{2} - 90 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + 18 \, F^{a} b^{2} c^{2} d^{2} \log \left (F\right )^{2} - F^{a} b^{3} d^{2} \log \left (F\right )^{3}\right )} x^{2} + 8 \,{\left (24 \, F^{a} c^{7} d - 18 \, F^{a} b c^{5} d \log \left (F\right ) + 6 \, F^{a} b^{2} c^{3} d \log \left (F\right )^{2} - F^{a} b^{3} c d \log \left (F\right )^{3}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{5} d^{9} x^{8} \log \left (F\right )^{5} + 8 \, b^{5} c d^{8} x^{7} \log \left (F\right )^{5} + 28 \, b^{5} c^{2} d^{7} x^{6} \log \left (F\right )^{5} + 56 \, b^{5} c^{3} d^{6} x^{5} \log \left (F\right )^{5} + 70 \, b^{5} c^{4} d^{5} x^{4} \log \left (F\right )^{5} + 56 \, b^{5} c^{5} d^{4} x^{3} \log \left (F\right )^{5} + 28 \, b^{5} c^{6} d^{3} x^{2} \log \left (F\right )^{5} + 8 \, b^{5} c^{7} d^{2} x \log \left (F\right )^{5} + b^{5} c^{8} d \log \left (F\right )^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^11,x, algorithm="maxima")

[Out]

-1/2*(24*F^a*d^8*x^8 + 192*F^a*c*d^7*x^7 + 24*F^a*c^8 - 24*F^a*b*c^6*log(F) + 12*F^a*b^2*c^4*log(F)^2 - 4*F^a*
b^3*c^2*log(F)^3 + F^a*b^4*log(F)^4 + 24*(28*F^a*c^2*d^6 - F^a*b*d^6*log(F))*x^6 + 48*(28*F^a*c^3*d^5 - 3*F^a*
b*c*d^5*log(F))*x^5 + 12*(140*F^a*c^4*d^4 - 30*F^a*b*c^2*d^4*log(F) + F^a*b^2*d^4*log(F)^2)*x^4 + 48*(28*F^a*c
^5*d^3 - 10*F^a*b*c^3*d^3*log(F) + F^a*b^2*c*d^3*log(F)^2)*x^3 + 4*(168*F^a*c^6*d^2 - 90*F^a*b*c^4*d^2*log(F)
+ 18*F^a*b^2*c^2*d^2*log(F)^2 - F^a*b^3*d^2*log(F)^3)*x^2 + 8*(24*F^a*c^7*d - 18*F^a*b*c^5*d*log(F) + 6*F^a*b^
2*c^3*d*log(F)^2 - F^a*b^3*c*d*log(F)^3)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^5*d^9*x^8*log(F)^5 + 8*b^5*c*d^
8*x^7*log(F)^5 + 28*b^5*c^2*d^7*x^6*log(F)^5 + 56*b^5*c^3*d^6*x^5*log(F)^5 + 70*b^5*c^4*d^5*x^4*log(F)^5 + 56*
b^5*c^5*d^4*x^3*log(F)^5 + 28*b^5*c^6*d^3*x^2*log(F)^5 + 8*b^5*c^7*d^2*x*log(F)^5 + b^5*c^8*d*log(F)^5)

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Fricas [B]  time = 1.97487, size = 907, normalized size = 9.45 \begin{align*} -\frac{{\left (24 \, d^{8} x^{8} + 192 \, c d^{7} x^{7} + 672 \, c^{2} d^{6} x^{6} + 1344 \, c^{3} d^{5} x^{5} + 1680 \, c^{4} d^{4} x^{4} + 1344 \, c^{5} d^{3} x^{3} + 672 \, c^{6} d^{2} x^{2} + 192 \, c^{7} d x + 24 \, c^{8} + b^{4} \log \left (F\right )^{4} - 4 \,{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} - 24 \,{\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{5} d^{9} x^{8} + 8 \, b^{5} c d^{8} x^{7} + 28 \, b^{5} c^{2} d^{7} x^{6} + 56 \, b^{5} c^{3} d^{6} x^{5} + 70 \, b^{5} c^{4} d^{5} x^{4} + 56 \, b^{5} c^{5} d^{4} x^{3} + 28 \, b^{5} c^{6} d^{3} x^{2} + 8 \, b^{5} c^{7} d^{2} x + b^{5} c^{8} 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)^2)/(d*x+c)^11,x, algorithm="fricas")

[Out]

-1/2*(24*d^8*x^8 + 192*c*d^7*x^7 + 672*c^2*d^6*x^6 + 1344*c^3*d^5*x^5 + 1680*c^4*d^4*x^4 + 1344*c^5*d^3*x^3 +
672*c^6*d^2*x^2 + 192*c^7*d*x + 24*c^8 + b^4*log(F)^4 - 4*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*log(F)^3 + 12*
(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(F)^2 - 24*(b*d^6*x^6 + 6*b*c
*d^5*x^5 + 15*b*c^2*d^4*x^4 + 20*b*c^3*d^3*x^3 + 15*b*c^4*d^2*x^2 + 6*b*c^5*d*x + b*c^6)*log(F))*F^((a*d^2*x^2
+ 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))/((b^5*d^9*x^8 + 8*b^5*c*d^8*x^7 + 28*b^5*c^2*d^7*x^6 + 56
*b^5*c^3*d^6*x^5 + 70*b^5*c^4*d^5*x^4 + 56*b^5*c^5*d^4*x^3 + 28*b^5*c^6*d^3*x^2 + 8*b^5*c^7*d^2*x + b^5*c^8*d)
*log(F)^5)

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Sympy [B]  time = 0.537117, size = 518, normalized size = 5.4 \begin{align*} \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}} \left (- b^{4} \log{\left (F \right )}^{4} + 4 b^{3} c^{2} \log{\left (F \right )}^{3} + 8 b^{3} c d x \log{\left (F \right )}^{3} + 4 b^{3} d^{2} x^{2} \log{\left (F \right )}^{3} - 12 b^{2} c^{4} \log{\left (F \right )}^{2} - 48 b^{2} c^{3} d x \log{\left (F \right )}^{2} - 72 b^{2} c^{2} d^{2} x^{2} \log{\left (F \right )}^{2} - 48 b^{2} c d^{3} x^{3} \log{\left (F \right )}^{2} - 12 b^{2} d^{4} x^{4} \log{\left (F \right )}^{2} + 24 b c^{6} \log{\left (F \right )} + 144 b c^{5} d x \log{\left (F \right )} + 360 b c^{4} d^{2} x^{2} \log{\left (F \right )} + 480 b c^{3} d^{3} x^{3} \log{\left (F \right )} + 360 b c^{2} d^{4} x^{4} \log{\left (F \right )} + 144 b c d^{5} x^{5} \log{\left (F \right )} + 24 b d^{6} x^{6} \log{\left (F \right )} - 24 c^{8} - 192 c^{7} d x - 672 c^{6} d^{2} x^{2} - 1344 c^{5} d^{3} x^{3} - 1680 c^{4} d^{4} x^{4} - 1344 c^{3} d^{5} x^{5} - 672 c^{2} d^{6} x^{6} - 192 c d^{7} x^{7} - 24 d^{8} x^{8}\right )}{2 b^{5} c^{8} d \log{\left (F \right )}^{5} + 16 b^{5} c^{7} d^{2} x \log{\left (F \right )}^{5} + 56 b^{5} c^{6} d^{3} x^{2} \log{\left (F \right )}^{5} + 112 b^{5} c^{5} d^{4} x^{3} \log{\left (F \right )}^{5} + 140 b^{5} c^{4} d^{5} x^{4} \log{\left (F \right )}^{5} + 112 b^{5} c^{3} d^{6} x^{5} \log{\left (F \right )}^{5} + 56 b^{5} c^{2} d^{7} x^{6} \log{\left (F \right )}^{5} + 16 b^{5} c d^{8} x^{7} \log{\left (F \right )}^{5} + 2 b^{5} d^{9} x^{8} \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)**2)/(d*x+c)**11,x)

[Out]

F**(a + b/(c + d*x)**2)*(-b**4*log(F)**4 + 4*b**3*c**2*log(F)**3 + 8*b**3*c*d*x*log(F)**3 + 4*b**3*d**2*x**2*l
og(F)**3 - 12*b**2*c**4*log(F)**2 - 48*b**2*c**3*d*x*log(F)**2 - 72*b**2*c**2*d**2*x**2*log(F)**2 - 48*b**2*c*
d**3*x**3*log(F)**2 - 12*b**2*d**4*x**4*log(F)**2 + 24*b*c**6*log(F) + 144*b*c**5*d*x*log(F) + 360*b*c**4*d**2
*x**2*log(F) + 480*b*c**3*d**3*x**3*log(F) + 360*b*c**2*d**4*x**4*log(F) + 144*b*c*d**5*x**5*log(F) + 24*b*d**
6*x**6*log(F) - 24*c**8 - 192*c**7*d*x - 672*c**6*d**2*x**2 - 1344*c**5*d**3*x**3 - 1680*c**4*d**4*x**4 - 1344
*c**3*d**5*x**5 - 672*c**2*d**6*x**6 - 192*c*d**7*x**7 - 24*d**8*x**8)/(2*b**5*c**8*d*log(F)**5 + 16*b**5*c**7
*d**2*x*log(F)**5 + 56*b**5*c**6*d**3*x**2*log(F)**5 + 112*b**5*c**5*d**4*x**3*log(F)**5 + 140*b**5*c**4*d**5*
x**4*log(F)**5 + 112*b**5*c**3*d**6*x**5*log(F)**5 + 56*b**5*c**2*d**7*x**6*log(F)**5 + 16*b**5*c*d**8*x**7*lo
g(F)**5 + 2*b**5*d**9*x**8*log(F)**5)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^11,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^11, x)