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

Optimal. Leaf size=31 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^2\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0991752, antiderivative size = 31, 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 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^2\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.55515, size = 31, normalized size = 1. \[ \frac{F^{a} b^{5} \Gamma{\left (-5,- b \left (c + d x\right )^{2} \log{\left (F \right )} \right )} \log{\left (F \right )}^{5}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

F**a*b**5*Gamma(-5, -b*(c + d*x)**2*log(F))*log(F)**5/(2*d)

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Mathematica [B]  time = 0.122616, size = 111, normalized size = 3.58 \[ \frac{F^a \left (b^5 \log ^5(F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^2\right )-\frac{F^{b (c+d x)^2} \left (b^4 \log ^4(F) (c+d x)^8+b^3 \log ^3(F) (c+d x)^6+2 b^2 \log ^2(F) (c+d x)^4+6 b \log (F) (c+d x)^2+24\right )}{(c+d x)^{10}}\right )}{240 d} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(b^5*ExpIntegralEi[b*(c + d*x)^2*Log[F]]*Log[F]^5 - (F^(b*(c + d*x)^2)*(24
+ 6*b*(c + d*x)^2*Log[F] + 2*b^2*(c + d*x)^4*Log[F]^2 + b^3*(c + d*x)^6*Log[F]^3
+ b^4*(c + d*x)^8*Log[F]^4))/(c + d*x)^10))/(240*d)

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Maple [B]  time = 0.139, size = 230, normalized size = 7.4 \[ -{\frac{{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{10\,d \left ( dx+c \right ) ^{10}}}-{\frac{b\ln \left ( F \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{40\,d \left ( dx+c \right ) ^{8}}}-{\frac{{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{120\,d \left ( dx+c \right ) ^{6}}}-{\frac{{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{240\,d \left ( dx+c \right ) ^{4}}}-{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{240\,d \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{2}\ln \left ( F \right ) \right ) }{240\,d}} \]

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]

-1/10/d/(d*x+c)^10*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/40/d*b*ln(F)/(d*x+c)^8*F^(b
*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/120/d*b^2*ln(F)^2/(d*x+c)^6*F^(b*d^2*x^2+2*b*c*d*x
+b*c^2+a)-1/240/d*b^3*ln(F)^3/(d*x+c)^4*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/240/d*
b^4*ln(F)^4/(d*x+c)^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/240/d*b^5*ln(F)^5*F^a*Ei
(1,-b*(d*x+c)^2*ln(F))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.251526, size = 805, normalized size = 25.97 \[ \frac{{\left (b^{5} d^{10} x^{10} + 10 \, b^{5} c d^{9} x^{9} + 45 \, b^{5} c^{2} d^{8} x^{8} + 120 \, b^{5} c^{3} d^{7} x^{7} + 210 \, b^{5} c^{4} d^{6} x^{6} + 252 \, b^{5} c^{5} d^{5} x^{5} + 210 \, b^{5} c^{6} d^{4} x^{4} + 120 \, b^{5} c^{7} d^{3} x^{3} + 45 \, b^{5} c^{8} d^{2} x^{2} + 10 \, b^{5} c^{9} d x + b^{5} c^{10}\right )} F^{a}{\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right ) \log \left (F\right )^{5} -{\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} +{\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 2 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 6 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{240 \,{\left (d^{11} x^{10} + 10 \, c d^{10} x^{9} + 45 \, c^{2} d^{9} x^{8} + 120 \, c^{3} d^{8} x^{7} + 210 \, c^{4} d^{7} x^{6} + 252 \, c^{5} d^{6} x^{5} + 210 \, c^{6} d^{5} x^{4} + 120 \, c^{7} d^{4} x^{3} + 45 \, c^{8} d^{3} x^{2} + 10 \, c^{9} d^{2} x + c^{10} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^11,x, algorithm="fricas")

[Out]

1/240*((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)*F^a*Ei((b*d^2*x^2
+ 2*b*c*d*x + b*c^2)*log(F))*log(F)^5 - ((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 + (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 + 2*(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 + 6*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) +
24)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(d^11*x^10 + 10*c*d^10*x^9 + 45*c^2*
d^9*x^8 + 120*c^3*d^8*x^7 + 210*c^4*d^7*x^6 + 252*c^5*d^6*x^5 + 210*c^6*d^5*x^4
+ 120*c^7*d^4*x^3 + 45*c^8*d^3*x^2 + 10*c^9*d^2*x + c^10*d)

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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)**11,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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