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

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

[Out]

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

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Rubi [A]  time = 0.0484642, 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} \[ \frac{F^a (c+d x)^{11} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 F^{a+\frac{b}{(c+d x)^2}} (c+d x)^{10} \, dx &=\frac{F^a (c+d x)^{11} \Gamma \left (-\frac{11}{2},-\frac{b \log (F)}{(c+d x)^2}\right ) \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{11/2}}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0293848, size = 49, normalized size = 1. \[ \frac{F^a (c+d x)^{11} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.14, size = 1173, normalized size = 23.9 \begin{align*} \text{result too large to display} \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]

2/11*d^7*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c*x^8+8/11*d^6*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^2*x^7+56/33*d^5*F^a*b*ln(F)*
F^(b/(d*x+c)^2)*c^3*x^6+30*d^3*F^a*F^(b/(d*x+c)^2)*c^7*x^4+15*d^2*F^a*F^(b/(d*x+c)^2)*c^8*x^3+5*d*F^a*F^(b/(d*
x+c)^2)*c^9*x^2+28/11*d^4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^5+28/11*d^3*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^5*x^4+56
/33*d^2*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^6*x^3+8/11*d*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^7*x^2+4/99*d^5*F^a*b^2*ln(F)^
2*F^(b/(d*x+c)^2)*c*x^6+4/33*d^4*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^2*x^5+20/99*d^3*F^a*b^2*ln(F)^2*F^(b/(d*x+c
)^2)*c^3*x^4+20/99*d^2*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^4*x^3+4/33*d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^5*x^2-
32/10395/d*F^a*b^6*ln(F)^6*Pi^(1/2)/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)/(d*x+c))+8/693*d^3*F^a*b^3*ln(F)^3*F
^(b/(d*x+c)^2)*c*x^4+16/693*d^2*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2*x^3+16/693*d*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^
2)*c^3*x^2+16/3465*d*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c*x^2+2/11*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^8*x+4/99*F^a*b^2
*ln(F)^2*F^(b/(d*x+c)^2)*c^6*x+8/693*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^4*x+16/3465*F^a*b^4*ln(F)^4*F^(b/(d*x+c
)^2)*c^2*x+2/99/d*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^9+1/11/d*F^a*F^(b/(d*x+c)^2)*c^11+32/10395*F^a*b^5*ln(F)^5*F^(
b/(d*x+c)^2)*x+d^9*F^a*F^(b/(d*x+c)^2)*c*x^10+5*d^8*F^a*F^(b/(d*x+c)^2)*c^2*x^9+15*d^7*F^a*F^(b/(d*x+c)^2)*c^3
*x^8+30*d^6*F^a*F^(b/(d*x+c)^2)*c^4*x^7+42*d^5*F^a*F^(b/(d*x+c)^2)*c^5*x^6+42*d^4*F^a*F^(b/(d*x+c)^2)*c^6*x^5+
1/11*d^10*F^a*F^(b/(d*x+c)^2)*x^11+F^a*F^(b/(d*x+c)^2)*c^10*x+4/693/d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^7+8/34
65/d*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^5+16/10395/d*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c^3+32/10395/d*F^a*b^5*ln(
F)^5*F^(b/(d*x+c)^2)*c+4/693*d^6*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^7+8/3465*d^4*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2
)*x^5+16/10395*d^2*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*x^3+2/99*d^8*F^a*b*ln(F)*F^(b/(d*x+c)^2)*x^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^10,x, algorithm="maxima")

[Out]

1/10395*(945*F^a*d^10*x^11 + 10395*F^a*c*d^9*x^10 + 105*(495*F^a*c^2*d^8 + 2*F^a*b*d^8*log(F))*x^9 + 945*(165*
F^a*c^3*d^7 + 2*F^a*b*c*d^7*log(F))*x^8 + 30*(10395*F^a*c^4*d^6 + 252*F^a*b*c^2*d^6*log(F) + 2*F^a*b^2*d^6*log
(F)^2)*x^7 + 210*(2079*F^a*c^5*d^5 + 84*F^a*b*c^3*d^5*log(F) + 2*F^a*b^2*c*d^5*log(F)^2)*x^6 + 6*(72765*F^a*c^
6*d^4 + 4410*F^a*b*c^4*d^4*log(F) + 210*F^a*b^2*c^2*d^4*log(F)^2 + 4*F^a*b^3*d^4*log(F)^3)*x^5 + 30*(10395*F^a
*c^7*d^3 + 882*F^a*b*c^5*d^3*log(F) + 70*F^a*b^2*c^3*d^3*log(F)^2 + 4*F^a*b^3*c*d^3*log(F)^3)*x^4 + (155925*F^
a*c^8*d^2 + 17640*F^a*b*c^6*d^2*log(F) + 2100*F^a*b^2*c^4*d^2*log(F)^2 + 240*F^a*b^3*c^2*d^2*log(F)^3 + 16*F^a
*b^4*d^2*log(F)^4)*x^3 + 3*(17325*F^a*c^9*d + 2520*F^a*b*c^7*d*log(F) + 420*F^a*b^2*c^5*d*log(F)^2 + 80*F^a*b^
3*c^3*d*log(F)^3 + 16*F^a*b^4*c*d*log(F)^4)*x^2 + (10395*F^a*c^10 + 1890*F^a*b*c^8*log(F) + 420*F^a*b^2*c^6*lo
g(F)^2 + 120*F^a*b^3*c^4*log(F)^3 + 48*F^a*b^4*c^2*log(F)^4 + 32*F^a*b^5*log(F)^5)*x)*F^(b/(d^2*x^2 + 2*c*d*x
+ c^2)) + integrate(2/10395*(32*F^a*b^6*d*x*log(F)^6 - 945*F^a*b*c^11*log(F) - 210*F^a*b^2*c^9*log(F)^2 - 60*F
^a*b^3*c^7*log(F)^3 - 24*F^a*b^4*c^5*log(F)^4 - 16*F^a*b^5*c^3*log(F)^5)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*
x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

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Fricas [B]  time = 1.68314, size = 1285, normalized size = 26.22 \begin{align*} \frac{32 \, \sqrt{\pi } F^{a} b^{5} d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) \log \left (F\right )^{5} +{\left (945 \, d^{11} x^{11} + 10395 \, c d^{10} x^{10} + 51975 \, c^{2} d^{9} x^{9} + 155925 \, c^{3} d^{8} x^{8} + 311850 \, c^{4} d^{7} x^{7} + 436590 \, c^{5} d^{6} x^{6} + 436590 \, c^{6} d^{5} x^{5} + 311850 \, c^{7} d^{4} x^{4} + 155925 \, c^{8} d^{3} x^{3} + 51975 \, c^{9} d^{2} x^{2} + 10395 \, c^{10} d x + 945 \, c^{11} + 32 \,{\left (b^{5} d x + b^{5} c\right )} \log \left (F\right )^{5} + 16 \,{\left (b^{4} d^{3} x^{3} + 3 \, b^{4} c d^{2} x^{2} + 3 \, b^{4} c^{2} d x + b^{4} c^{3}\right )} \log \left (F\right )^{4} + 24 \,{\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \left (F\right )^{3} + 60 \,{\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + 35 \, b^{2} c^{3} d^{4} x^{4} + 35 \, b^{2} c^{4} d^{3} x^{3} + 21 \, b^{2} c^{5} d^{2} x^{2} + 7 \, b^{2} c^{6} d x + b^{2} c^{7}\right )} \log \left (F\right )^{2} + 210 \,{\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{10395 \, d} \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]

1/10395*(32*sqrt(pi)*F^a*b^5*d*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c))*log(F)^5 + (945*d^11*x
^11 + 10395*c*d^10*x^10 + 51975*c^2*d^9*x^9 + 155925*c^3*d^8*x^8 + 311850*c^4*d^7*x^7 + 436590*c^5*d^6*x^6 + 4
36590*c^6*d^5*x^5 + 311850*c^7*d^4*x^4 + 155925*c^8*d^3*x^3 + 51975*c^9*d^2*x^2 + 10395*c^10*d*x + 945*c^11 +
32*(b^5*d*x + b^5*c)*log(F)^5 + 16*(b^4*d^3*x^3 + 3*b^4*c*d^2*x^2 + 3*b^4*c^2*d*x + b^4*c^3)*log(F)^4 + 24*(b^
3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*c^4*d*x + b^3*c^5)*log(F)^3 + 60
*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + 35*b^2*c^3*d^4*x^4 + 35*b^2*c^4*d^3*x^3 + 21*b^2*c^5*d^
2*x^2 + 7*b^2*c^6*d*x + b^2*c^7)*log(F)^2 + 210*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d^7*x^7 + 84*b*c^3*d^6*x
^6 + 126*b*c^4*d^5*x^5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 36*b*c^7*d^2*x^2 + 9*b*c^8*d*x + b*c^9)*log(F)
)*F^((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)))/d

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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{\left (d x + c\right )}^{10} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{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((d*x + c)^10*F^(a + b/(d*x + c)^2), x)