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)

_______________________________________________________________________________________

Rubi [A]  time = 0.103438, 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 \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^{11}} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.38391, size = 49, normalized size = 1. \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{2} \log{\left (F \right )}\right )^{\frac{11}{2}} \Gamma{\left (- \frac{11}{2},- b \left (c + d x\right )^{2} \log{\left (F \right )} \right )}}{2 d \left (c + d x\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F**a*(-b*(c + d*x)**2*log(F))**(11/2)*Gamma(-11/2, -b*(c + d*x)**2*log(F))/(2*d
*(c + d*x)**11)

_______________________________________________________________________________________

Mathematica [B]  time = 0.1571, size = 152, normalized size = 3.1 \[ \frac{F^a \left (32 \sqrt{\pi } b^{11/2} \log ^{\frac{11}{2}}(F) (c+d x)^{11} \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )-F^{b (c+d x)^2} \left (32 b^5 \log ^5(F) (c+d x)^{10}+16 b^4 \log ^4(F) (c+d x)^8+24 b^3 \log ^3(F) (c+d x)^6+60 b^2 \log ^2(F) (c+d x)^4+210 b \log (F) (c+d x)^2+945\right )\right )}{10395 d (c+d x)^{11}} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(32*b^(11/2)*Sqrt[Pi]*(c + d*x)^11*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log
[F]^(11/2) - F^(b*(c + d*x)^2)*(945 + 210*b*(c + d*x)^2*Log[F] + 60*b^2*(c + d*x
)^4*Log[F]^2 + 24*b^3*(c + d*x)^6*Log[F]^3 + 16*b^4*(c + d*x)^8*Log[F]^4 + 32*b^
5*(c + d*x)^10*Log[F]^5)))/(10395*d*(c + d*x)^11)

_______________________________________________________________________________________

Maple [A]  time = 0.171, size = 282, normalized size = 5.8 \[ -{\frac{{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{11\,d \left ( dx+c \right ) ^{11}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{99\,d \left ( dx+c \right ) ^{9}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{693\,d \left ( dx+c \right ) ^{7}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{3465\,d \left ( dx+c \right ) ^{5}}}-{\frac{16\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{10395\,d \left ( dx+c \right ) ^{3}}}-{\frac{32\,{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{10395\, \left ( dx+c \right ) d}}+{\frac{32\,{b}^{6} \left ( \ln \left ( F \right ) \right ) ^{6}\sqrt{\pi }{F}^{a}}{10395\,d}{\it Erf} \left ( \sqrt{-b\ln \left ( F \right ) } \left ( dx+c \right ) \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

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^2*x^2+2*b*c*d*x+b*c^2+a)-2/99/d*b*ln(F)/(d*x+c)^9*F^(b
*d^2*x^2+2*b*c*d*x+b*c^2+a)-4/693/d*b^2*ln(F)^2/(d*x+c)^7*F^(b*d^2*x^2+2*b*c*d*x
+b*c^2+a)-8/3465/d*b^3*ln(F)^3/(d*x+c)^5*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-16/1039
5/d*b^4*ln(F)^4/(d*x+c)^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-32/10395/d*b^5*ln(F)^5
/(d*x+c)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+32/10395/d*b^6*ln(F)^6*Pi^(1/2)*F^a/(-b
*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)*(d*x+c))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^12,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.286857, size = 1067, normalized size = 21.78 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^12,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^12,x, algorithm="giac")

[Out]

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