3.339 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^{14}} \, dx\)

Optimal. Leaf size=49 \[ \frac{F^a \text{Gamma}\left (\frac{13}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{13} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{13/2}} \]

[Out]

(F^a*Gamma[13/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^13*(-((b*Log[F])/(c + d*x)^2))^(13/2))

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Rubi [A]  time = 0.0453299, antiderivative size = 49, normalized size of antiderivative = 1., 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} \[ \frac{F^a \text{Gamma}\left (\frac{13}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{13} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)/(c + d*x)^14,x]

[Out]

(F^a*Gamma[13/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^13*(-((b*Log[F])/(c + d*x)^2))^(13/2))

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)^{14}} \, dx &=\frac{F^a \Gamma \left (\frac{13}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{13} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.026041, size = 49, normalized size = 1. \[ \frac{F^a \text{Gamma}\left (\frac{13}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{13} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^14,x]

[Out]

(F^a*Gamma[13/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^13*(-((b*Log[F])/(c + d*x)^2))^(13/2))

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Maple [A]  time = 0.341, size = 241, normalized size = 4.9 \begin{align*} -{\frac{{F}^{a}}{2\,d \left ( dx+c \right ) ^{11}b\ln \left ( F \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{11\,{F}^{a}}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d \left ( dx+c \right ) ^{9}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{99\,{F}^{a}}{8\,d{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3} \left ( dx+c \right ) ^{7}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{693\,{F}^{a}}{16\,d{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4} \left ( dx+c \right ) ^{5}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{3465\,{F}^{a}}{32\,d \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5} \left ( dx+c \right ) ^{3}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{10395\,{F}^{a}}{64\,d \left ( \ln \left ( F \right ) \right ) ^{6}{b}^{6} \left ( dx+c \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{10395\,{F}^{a}\sqrt{\pi }}{128\,d \left ( \ln \left ( F \right ) \right ) ^{6}{b}^{6}}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^2)/(d*x+c)^14,x)

[Out]

-1/2/d*F^a*F^(b/(d*x+c)^2)/(d*x+c)^11/b/ln(F)+11/4/d*F^a/b^2/ln(F)^2*F^(b/(d*x+c)^2)/(d*x+c)^9-99/8/d*F^a/b^3/
ln(F)^3*F^(b/(d*x+c)^2)/(d*x+c)^7+693/16/d*F^a/b^4/ln(F)^4*F^(b/(d*x+c)^2)/(d*x+c)^5-3465/32/d*F^a/b^5/ln(F)^5
*F^(b/(d*x+c)^2)/(d*x+c)^3+10395/64/d*F^a/b^6/ln(F)^6*F^(b/(d*x+c)^2)/(d*x+c)-10395/128/d*F^a/b^6/ln(F)^6*Pi^(
1/2)/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)/(d*x+c))

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Maxima [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 )}^{14}}\,{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)^14,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^14, x)

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Fricas [A]  time = 2.0817, size = 1750, normalized size = 35.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^14,x, algorithm="fricas")

[Out]

1/128*(10395*sqrt(pi)*(d^12*x^11 + 11*c*d^11*x^10 + 55*c^2*d^10*x^9 + 165*c^3*d^9*x^8 + 330*c^4*d^8*x^7 + 462*
c^5*d^7*x^6 + 462*c^6*d^6*x^5 + 330*c^7*d^5*x^4 + 165*c^8*d^4*x^3 + 55*c^9*d^3*x^2 + 11*c^10*d^2*x + c^11*d)*F
^a*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c)) - 2*(32*b^6*log(F)^6 - 176*(b^5*d^2*x^2 + 2*b^5*c*
d*x + b^5*c^2)*log(F)^5 + 792*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*lo
g(F)^4 - 2772*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + 15*b^3*c^2*d^4*x^4 + 20*b^3*c^3*d^3*x^3 + 15*b^3*c^4*d^2*x^2 +
6*b^3*c^5*d*x + b^3*c^6)*log(F)^3 + 6930*(b^2*d^8*x^8 + 8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 56*b^2*c^3*d^5*
x^5 + 70*b^2*c^4*d^4*x^4 + 56*b^2*c^5*d^3*x^3 + 28*b^2*c^6*d^2*x^2 + 8*b^2*c^7*d*x + b^2*c^8)*log(F)^2 - 10395
*(b*d^10*x^10 + 10*b*c*d^9*x^9 + 45*b*c^2*d^8*x^8 + 120*b*c^3*d^7*x^7 + 210*b*c^4*d^6*x^6 + 252*b*c^5*d^5*x^5
+ 210*b*c^6*d^4*x^4 + 120*b*c^7*d^3*x^3 + 45*b*c^8*d^2*x^2 + 10*b*c^9*d*x + b*c^10)*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^7*d^12*x^11 + 11*b^7*c*d^11*x^10 + 55*b^7*c^2*d^10*x^9 +
165*b^7*c^3*d^9*x^8 + 330*b^7*c^4*d^8*x^7 + 462*b^7*c^5*d^7*x^6 + 462*b^7*c^6*d^6*x^5 + 330*b^7*c^7*d^5*x^4 +
165*b^7*c^8*d^4*x^3 + 55*b^7*c^9*d^3*x^2 + 11*b^7*c^10*d^2*x + b^7*c^11*d)*log(F)^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**14,x)

[Out]

Timed out

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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 )}^{14}}\,{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)^14,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^14, x)