3.279 \(\int \frac{F^{a+b (c+d x)^2}}{(c+d x)^{12}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^{11}} \]

[Out]

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

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Rubi [A]  time = 0.0619664, 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} \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^{11}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)/(c + d*x)^12,x]

[Out]

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

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+b (c+d x)^2}}{(c+d x)^{12}} \, dx &=-\frac{F^a \Gamma \left (-\frac{11}{2},-b (c+d x)^2 \log (F)\right ) \left (-b (c+d x)^2 \log (F)\right )^{11/2}}{2 d (c+d x)^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0299479, size = 49, normalized size = 1. \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)/(c + d*x)^12,x]

[Out]

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

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Maple [A]  time = 0.182, size = 228, normalized size = 4.7 \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{11\,d \left ( dx+c \right ) ^{11}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{99\,d \left ( dx+c \right ) ^{9}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{693\,d \left ( dx+c \right ) ^{7}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{3465\,d \left ( dx+c \right ) ^{5}}}-{\frac{16\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{10395\,d \left ( dx+c \right ) ^{3}}}-{\frac{32\,{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{10395\, \left ( dx+c \right ) d}}+{\frac{32\,{b}^{6} \left ( \ln \left ( F \right ) \right ) ^{6}\sqrt{\pi }{F}^{a}}{10395\,d}{\it Erf} \left ( \sqrt{-b\ln \left ( F \right ) } \left ( dx+c \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)^12,x)

[Out]

-1/11/d/(d*x+c)^11*F^(b*(d*x+c)^2)*F^a-2/99/d*b*ln(F)/(d*x+c)^9*F^(b*(d*x+c)^2)*F^a-4/693/d*b^2*ln(F)^2/(d*x+c
)^7*F^(b*(d*x+c)^2)*F^a-8/3465/d*b^3*ln(F)^3/(d*x+c)^5*F^(b*(d*x+c)^2)*F^a-16/10395/d*b^4*ln(F)^4/(d*x+c)^3*F^
(b*(d*x+c)^2)*F^a-32/10395/d*b^5*ln(F)^5/(d*x+c)*F^(b*(d*x+c)^2)*F^a+32/10395/d*b^6*ln(F)^6*Pi^(1/2)*F^a/(-b*l
n(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^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{12}}\,{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)^12,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^12, x)

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Fricas [B]  time = 2.05855, size = 1729, normalized size = 35.29 \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)^12,x, algorithm="fricas")

[Out]

-1/10395*(32*sqrt(pi)*(b^5*d^11*x^11 + 11*b^5*c*d^10*x^10 + 55*b^5*c^2*d^9*x^9 + 165*b^5*c^3*d^8*x^8 + 330*b^5
*c^4*d^7*x^7 + 462*b^5*c^5*d^6*x^6 + 462*b^5*c^6*d^5*x^5 + 330*b^5*c^7*d^4*x^4 + 165*b^5*c^8*d^3*x^3 + 55*b^5*
c^9*d^2*x^2 + 11*b^5*c^10*d*x + b^5*c^11)*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d)*log(F)^
5 + (32*(b^5*d^11*x^10 + 10*b^5*c*d^10*x^9 + 45*b^5*c^2*d^9*x^8 + 120*b^5*c^3*d^8*x^7 + 210*b^5*c^4*d^7*x^6 +
252*b^5*c^5*d^6*x^5 + 210*b^5*c^6*d^5*x^4 + 120*b^5*c^7*d^4*x^3 + 45*b^5*c^8*d^3*x^2 + 10*b^5*c^9*d^2*x + b^5*
c^10*d)*log(F)^5 + 16*(b^4*d^9*x^8 + 8*b^4*c*d^8*x^7 + 28*b^4*c^2*d^7*x^6 + 56*b^4*c^3*d^6*x^5 + 70*b^4*c^4*d^
5*x^4 + 56*b^4*c^5*d^4*x^3 + 28*b^4*c^6*d^3*x^2 + 8*b^4*c^7*d^2*x + b^4*c^8*d)*log(F)^4 + 24*(b^3*d^7*x^6 + 6*
b^3*c*d^6*x^5 + 15*b^3*c^2*d^5*x^4 + 20*b^3*c^3*d^4*x^3 + 15*b^3*c^4*d^3*x^2 + 6*b^3*c^5*d^2*x + b^3*c^6*d)*lo
g(F)^3 + 60*(b^2*d^5*x^4 + 4*b^2*c*d^4*x^3 + 6*b^2*c^2*d^3*x^2 + 4*b^2*c^3*d^2*x + b^2*c^4*d)*log(F)^2 + 210*(
b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*log(F) + 945*d)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(d^13*x^11 + 11*c*d^
12*x^10 + 55*c^2*d^11*x^9 + 165*c^3*d^10*x^8 + 330*c^4*d^9*x^7 + 462*c^5*d^8*x^6 + 462*c^6*d^7*x^5 + 330*c^7*d
^6*x^4 + 165*c^8*d^5*x^3 + 55*c^9*d^4*x^2 + 11*c^10*d^3*x + c^11*d^2)

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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)**12,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{12}}\,{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)^12,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^12, x)