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 = 0.0613516, 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} \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]

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)

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)^{10}} \, dx &=-\frac{F^a \Gamma \left (-\frac{9}{2},-b (c+d x)^2 \log (F)\right ) \left (-b (c+d x)^2 \log (F)\right )^{9/2}}{2 d (c+d x)^9}\\ \end{align*}

Mathematica [A]  time = 0.0299586, size = 49, normalized size = 1. \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[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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Maple [A]  time = 0.122, size = 195, normalized size = 4. \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{9\,d \left ( dx+c \right ) ^{9}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{63\,d \left ( dx+c \right ) ^{7}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{315\,d \left ( dx+c \right ) ^{5}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{945\,d \left ( dx+c \right ) ^{3}}}-{\frac{16\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{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 ) }}}} \end{align*}

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*x+c)^2)*F^a-2/63/d*b*ln(F)/(d*x+c)^7*F^(b*(d*x+c)^2)*F^a-4/315/d*b^2*ln(F)^2/(d*x+c)^
5*F^(b*(d*x+c)^2)*F^a-8/945/d*b^3*ln(F)^3/(d*x+c)^3*F^(b*(d*x+c)^2)*F^a-16/945/d*b^4*ln(F)^4/(d*x+c)*F^(b*(d*x
+c)^2)*F^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. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{10}}\,{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)^10,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.77045, size = 1273, normalized size = 25.98 \begin{align*} -\frac{16 \, \sqrt{\pi }{\left (b^{4} d^{9} x^{9} + 9 \, b^{4} c d^{8} x^{8} + 36 \, b^{4} c^{2} d^{7} x^{7} + 84 \, b^{4} c^{3} d^{6} x^{6} + 126 \, b^{4} c^{4} d^{5} x^{5} + 126 \, b^{4} c^{5} d^{4} x^{4} + 84 \, b^{4} c^{6} d^{3} x^{3} + 36 \, b^{4} c^{7} d^{2} x^{2} + 9 \, b^{4} c^{8} d x + b^{4} c^{9}\right )} \sqrt{-b d^{2} \log \left (F\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 )^{4} +{\left (16 \,{\left (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\right )} \log \left (F\right )^{4} + 8 \,{\left (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\right )} \log \left (F\right )^{3} + 12 \,{\left (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\right )} \log \left (F\right )^{2} + 30 \,{\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \left (F\right ) + 105 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{945 \,{\left (d^{11} x^{9} + 9 \, c d^{10} x^{8} + 36 \, c^{2} d^{9} x^{7} + 84 \, c^{3} d^{8} x^{6} + 126 \, c^{4} d^{7} x^{5} + 126 \, c^{5} d^{6} x^{4} + 84 \, c^{6} d^{5} x^{3} + 36 \, c^{7} d^{4} x^{2} + 9 \, c^{8} d^{3} x + c^{9} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/945*(16*sqrt(pi)*(b^4*d^9*x^9 + 9*b^4*c*d^8*x^8 + 36*b^4*c^2*d^7*x^7 + 84*b^4*c^3*d^6*x^6 + 126*b^4*c^4*d^5
*x^5 + 126*b^4*c^5*d^4*x^4 + 84*b^4*c^6*d^3*x^3 + 36*b^4*c^7*d^2*x^2 + 9*b^4*c^8*d*x + b^4*c^9)*sqrt(-b*d^2*lo
g(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d)*log(F)^4 + (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 + 8*(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)*log(F)^3 + 12*(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 + 30*(b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*log(F) + 105*d)*F^(b*d^2*x^2 + 2*b*c*
d*x + b*c^2 + a))/(d^11*x^9 + 9*c*d^10*x^8 + 36*c^2*d^9*x^7 + 84*c^3*d^8*x^6 + 126*c^4*d^7*x^5 + 126*c^5*d^6*x
^4 + 84*c^6*d^5*x^3 + 36*c^7*d^4*x^2 + 9*c^8*d^3*x + c^9*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)**10,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 )}^{10}}\,{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)^10,x, algorithm="giac")

[Out]

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