3.278 \(\int \frac{F^{a+b (c+d x)^2}}{(c+d x)^{10}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]

[Out]

-(F^a*Gamma[-9/2, -(b*(c + d*x)^2*Log[F])]*(-(b*(c + d*x)^2*Log[F]))^(9/2))/(2*d
*(c + d*x)^9)

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Rubi [A]  time = 0.101008, 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 \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)/(c + d*x)^10,x]

[Out]

-(F^a*Gamma[-9/2, -(b*(c + d*x)^2*Log[F])]*(-(b*(c + d*x)^2*Log[F]))^(9/2))/(2*d
*(c + d*x)^9)

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Rubi in Sympy [A]  time = 5.37808, size = 49, normalized size = 1. \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{2} \log{\left (F \right )}\right )^{\frac{9}{2}} \Gamma{\left (- \frac{9}{2},- b \left (c + d x\right )^{2} \log{\left (F \right )} \right )}}{2 d \left (c + d x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)/(d*x+c)**10,x)

[Out]

-F**a*(-b*(c + d*x)**2*log(F))**(9/2)*Gamma(-9/2, -b*(c + d*x)**2*log(F))/(2*d*(
c + d*x)**9)

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Mathematica [B]  time = 0.159481, size = 129, normalized size = 2.63 \[ \frac{F^a \left (16 \sqrt{\pi } b^{9/2} \log ^{\frac{9}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )-\frac{F^{b (c+d x)^2} \left (16 b^4 \log ^4(F) (c+d x)^8+8 b^3 \log ^3(F) (c+d x)^6+12 b^2 \log ^2(F) (c+d x)^4+30 b \log (F) (c+d x)^2+105\right )}{(c+d x)^9}\right )}{945 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)/(c + d*x)^10,x]

[Out]

(F^a*(16*b^(9/2)*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(9/2) - (F
^(b*(c + d*x)^2)*(105 + 30*b*(c + d*x)^2*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 +
8*b^3*(c + d*x)^6*Log[F]^3 + 16*b^4*(c + d*x)^8*Log[F]^4))/(c + d*x)^9))/(945*d)

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Maple [A]  time = 0.135, size = 240, normalized size = 4.9 \[ -{\frac{{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{9\,d \left ( dx+c \right ) ^{9}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{63\,d \left ( dx+c \right ) ^{7}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{315\,d \left ( dx+c \right ) ^{5}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{945\,d \left ( dx+c \right ) ^{3}}}-{\frac{16\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{945\, \left ( dx+c \right ) d}}+{\frac{16\,{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}\sqrt{\pi }{F}^{a}}{945\,d}{\it Erf} \left ( \sqrt{-b\ln \left ( F \right ) } \left ( dx+c \right ) \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)/(d*x+c)^10,x)

[Out]

-1/9/d/(d*x+c)^9*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-2/63/d*b*ln(F)/(d*x+c)^7*F^(b*d
^2*x^2+2*b*c*d*x+b*c^2+a)-4/315/d*b^2*ln(F)^2/(d*x+c)^5*F^(b*d^2*x^2+2*b*c*d*x+b
*c^2+a)-8/945/d*b^3*ln(F)^3/(d*x+c)^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-16/945/d*b
^4*ln(F)^4/(d*x+c)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+16/945/d*b^5*ln(F)^5*Pi^(1/2)
*F^a/(-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. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^10,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^10, x)

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Fricas [A]  time = 0.256721, size = 805, normalized size = 16.43 \[ \frac{16 \, \sqrt{\pi }{\left (b^{5} d^{10} x^{9} + 9 \, b^{5} c d^{9} x^{8} + 36 \, b^{5} c^{2} d^{8} x^{7} + 84 \, b^{5} c^{3} d^{7} x^{6} + 126 \, b^{5} c^{4} d^{6} x^{5} + 126 \, b^{5} c^{5} d^{5} x^{4} + 84 \, b^{5} c^{6} d^{4} x^{3} + 36 \, b^{5} c^{7} d^{3} x^{2} + 9 \, b^{5} c^{8} d^{2} x + b^{5} c^{9} d\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) \log \left (F\right )^{5} -{\left (16 \,{\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} + 8 \,{\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 30 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 105\right )} \sqrt{-b d^{2} \log \left (F\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{945 \,{\left (d^{10} x^{9} + 9 \, c d^{9} x^{8} + 36 \, c^{2} d^{8} x^{7} + 84 \, c^{3} d^{7} x^{6} + 126 \, c^{4} d^{6} x^{5} + 126 \, c^{5} d^{5} x^{4} + 84 \, c^{6} d^{4} x^{3} + 36 \, c^{7} d^{3} x^{2} + 9 \, c^{8} d^{2} x + c^{9} d\right )} \sqrt{-b d^{2} \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^10,x, algorithm="fricas")

[Out]

1/945*(16*sqrt(pi)*(b^5*d^10*x^9 + 9*b^5*c*d^9*x^8 + 36*b^5*c^2*d^8*x^7 + 84*b^5
*c^3*d^7*x^6 + 126*b^5*c^4*d^6*x^5 + 126*b^5*c^5*d^5*x^4 + 84*b^5*c^6*d^4*x^3 +
36*b^5*c^7*d^3*x^2 + 9*b^5*c^8*d^2*x + b^5*c^9*d)*F^a*erf(sqrt(-b*d^2*log(F))*(d
*x + c)/d)*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 + 8*(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 + 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 + 30*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) + 105)*sqrt(
-b*d^2*log(F))*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/((d^10*x^9 + 9*c*d^9*x^8 +
36*c^2*d^8*x^7 + 84*c^3*d^7*x^6 + 126*c^4*d^6*x^5 + 126*c^5*d^5*x^4 + 84*c^6*d^
4*x^3 + 36*c^7*d^3*x^2 + 9*c^8*d^2*x + c^9*d)*sqrt(-b*d^2*log(F)))

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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)**10,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(d*x + c)^10,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^10, x)