3.342 \(\int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^{11} \, dx\)

Optimal. Leaf size=31 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0463236, 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, Rules used = {2218} \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)*(c + d*x)^11,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*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)^3}} (c+d x)^{11} \, dx &=\frac{b^4 F^a \Gamma \left (-4,-\frac{b \log (F)}{(c+d x)^3}\right ) \log ^4(F)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0092728, size = 31, normalized size = 1. \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^11,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)

________________________________________________________________________________________

Maple [F]  time = 0.114, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{11}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{72} \,{\left (6 \, F^{a} d^{11} x^{12} + 72 \, F^{a} c d^{10} x^{11} + 396 \, F^{a} c^{2} d^{9} x^{10} + 2 \,{\left (660 \, F^{a} c^{3} d^{8} + F^{a} b d^{8} \log \left (F\right )\right )} x^{9} + 18 \,{\left (165 \, F^{a} c^{4} d^{7} + F^{a} b c d^{7} \log \left (F\right )\right )} x^{8} + 72 \,{\left (66 \, F^{a} c^{5} d^{6} + F^{a} b c^{2} d^{6} \log \left (F\right )\right )} x^{7} +{\left (5544 \, F^{a} c^{6} d^{5} + 168 \, F^{a} b c^{3} d^{5} \log \left (F\right ) + F^{a} b^{2} d^{5} \log \left (F\right )^{2}\right )} x^{6} + 6 \,{\left (792 \, F^{a} c^{7} d^{4} + 42 \, F^{a} b c^{4} d^{4} \log \left (F\right ) + F^{a} b^{2} c d^{4} \log \left (F\right )^{2}\right )} x^{5} + 3 \,{\left (990 \, F^{a} c^{8} d^{3} + 84 \, F^{a} b c^{5} d^{3} \log \left (F\right ) + 5 \, F^{a} b^{2} c^{2} d^{3} \log \left (F\right )^{2}\right )} x^{4} +{\left (1320 \, F^{a} c^{9} d^{2} + 168 \, F^{a} b c^{6} d^{2} \log \left (F\right ) + 20 \, F^{a} b^{2} c^{3} d^{2} \log \left (F\right )^{2} + F^{a} b^{3} d^{2} \log \left (F\right )^{3}\right )} x^{3} + 3 \,{\left (132 \, F^{a} c^{10} d + 24 \, F^{a} b c^{7} d \log \left (F\right ) + 5 \, F^{a} b^{2} c^{4} d \log \left (F\right )^{2} + F^{a} b^{3} c d \log \left (F\right )^{3}\right )} x^{2} + 3 \,{\left (24 \, F^{a} c^{11} + 6 \, F^{a} b c^{8} \log \left (F\right ) + 2 \, F^{a} b^{2} c^{5} \log \left (F\right )^{2} + F^{a} b^{3} c^{2} \log \left (F\right )^{3}\right )} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int -\frac{{\left (6 \, F^{a} b c^{12} \log \left (F\right ) - F^{a} b^{4} d^{3} x^{3} \log \left (F\right )^{4} + 2 \, F^{a} b^{2} c^{9} \log \left (F\right )^{2} - 3 \, F^{a} b^{4} c d^{2} x^{2} \log \left (F\right )^{4} + F^{a} b^{3} c^{6} \log \left (F\right )^{3} - 3 \, F^{a} b^{4} c^{2} d x \log \left (F\right )^{4}\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{24 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x, algorithm="maxima")

[Out]

1/72*(6*F^a*d^11*x^12 + 72*F^a*c*d^10*x^11 + 396*F^a*c^2*d^9*x^10 + 2*(660*F^a*c^3*d^8 + F^a*b*d^8*log(F))*x^9
+ 18*(165*F^a*c^4*d^7 + F^a*b*c*d^7*log(F))*x^8 + 72*(66*F^a*c^5*d^6 + F^a*b*c^2*d^6*log(F))*x^7 + (5544*F^a*
c^6*d^5 + 168*F^a*b*c^3*d^5*log(F) + F^a*b^2*d^5*log(F)^2)*x^6 + 6*(792*F^a*c^7*d^4 + 42*F^a*b*c^4*d^4*log(F)
+ F^a*b^2*c*d^4*log(F)^2)*x^5 + 3*(990*F^a*c^8*d^3 + 84*F^a*b*c^5*d^3*log(F) + 5*F^a*b^2*c^2*d^3*log(F)^2)*x^4
+ (1320*F^a*c^9*d^2 + 168*F^a*b*c^6*d^2*log(F) + 20*F^a*b^2*c^3*d^2*log(F)^2 + F^a*b^3*d^2*log(F)^3)*x^3 + 3*
(132*F^a*c^10*d + 24*F^a*b*c^7*d*log(F) + 5*F^a*b^2*c^4*d*log(F)^2 + F^a*b^3*c*d*log(F)^3)*x^2 + 3*(24*F^a*c^1
1 + 6*F^a*b*c^8*log(F) + 2*F^a*b^2*c^5*log(F)^2 + F^a*b^3*c^2*log(F)^3)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2
*d*x + c^3)) + integrate(-1/24*(6*F^a*b*c^12*log(F) - F^a*b^4*d^3*x^3*log(F)^4 + 2*F^a*b^2*c^9*log(F)^2 - 3*F^
a*b^4*c*d^2*x^2*log(F)^4 + F^a*b^3*c^6*log(F)^3 - 3*F^a*b^4*c^2*d*x*log(F)^4)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*
c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), x)

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x, algorithm="fricas")

[Out]

Exception raised: TypeError

________________________________________________________________________________________

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)**3)*(d*x+c)**11,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{11} F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x, algorithm="giac")

[Out]

integrate((d*x + c)^11*F^(a + b/(d*x + c)^3), x)