3.341 \(\int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^{14} \, dx\)

Optimal. Leaf size=31 \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

-(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^3)]*Log[F]^5)/(3*d)

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Rubi [A]  time = 0.0463526, 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} \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)*(c + d*x)^14,x]

[Out]

-(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^3)]*Log[F]^5)/(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)^{14} \, dx &=-\frac{b^5 F^a \Gamma \left (-5,-\frac{b \log (F)}{(c+d x)^3}\right ) \log ^5(F)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0099077, size = 31, normalized size = 1. \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^14,x]

[Out]

-(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^3)]*Log[F]^5)/(3*d)

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Maple [F]  time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{14}\, 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)^14,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^14,x)

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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)^3)*(d*x+c)^14,x, algorithm="maxima")

[Out]

1/360*(24*F^a*d^14*x^15 + 360*F^a*c*d^13*x^14 + 2520*F^a*c^2*d^12*x^13 + 6*(1820*F^a*c^3*d^11 + F^a*b*d^11*log
(F))*x^12 + 72*(455*F^a*c^4*d^10 + F^a*b*c*d^10*log(F))*x^11 + 396*(182*F^a*c^5*d^9 + F^a*b*c^2*d^9*log(F))*x^
10 + 2*(60060*F^a*c^6*d^8 + 660*F^a*b*c^3*d^8*log(F) + F^a*b^2*d^8*log(F)^2)*x^9 + 18*(8580*F^a*c^7*d^7 + 165*
F^a*b*c^4*d^7*log(F) + F^a*b^2*c*d^7*log(F)^2)*x^8 + 72*(2145*F^a*c^8*d^6 + 66*F^a*b*c^5*d^6*log(F) + F^a*b^2*
c^2*d^6*log(F)^2)*x^7 + (120120*F^a*c^9*d^5 + 5544*F^a*b*c^6*d^5*log(F) + 168*F^a*b^2*c^3*d^5*log(F)^2 + F^a*b
^3*d^5*log(F)^3)*x^6 + 6*(12012*F^a*c^10*d^4 + 792*F^a*b*c^7*d^4*log(F) + 42*F^a*b^2*c^4*d^4*log(F)^2 + F^a*b^
3*c*d^4*log(F)^3)*x^5 + 3*(10920*F^a*c^11*d^3 + 990*F^a*b*c^8*d^3*log(F) + 84*F^a*b^2*c^5*d^3*log(F)^2 + 5*F^a
*b^3*c^2*d^3*log(F)^3)*x^4 + (10920*F^a*c^12*d^2 + 1320*F^a*b*c^9*d^2*log(F) + 168*F^a*b^2*c^6*d^2*log(F)^2 +
20*F^a*b^3*c^3*d^2*log(F)^3 + F^a*b^4*d^2*log(F)^4)*x^3 + 3*(840*F^a*c^13*d + 132*F^a*b*c^10*d*log(F) + 24*F^a
*b^2*c^7*d*log(F)^2 + 5*F^a*b^3*c^4*d*log(F)^3 + F^a*b^4*c*d*log(F)^4)*x^2 + 3*(120*F^a*c^14 + 24*F^a*b*c^11*l
og(F) + 6*F^a*b^2*c^8*log(F)^2 + 2*F^a*b^3*c^5*log(F)^3 + F^a*b^4*c^2*log(F)^4)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2
+ 3*c^2*d*x + c^3)) + integrate(-1/120*(24*F^a*b*c^15*log(F) + 6*F^a*b^2*c^12*log(F)^2 - F^a*b^5*d^3*x^3*log(
F)^5 + 2*F^a*b^3*c^9*log(F)^3 - 3*F^a*b^5*c*d^2*x^2*log(F)^5 + F^a*b^4*c^6*log(F)^4 - 3*F^a*b^5*c^2*d*x*log(F)
^5)*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)

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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)^14,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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)**3)*(d*x+c)**14,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 )}^{14} 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)^14,x, algorithm="giac")

[Out]

integrate((d*x + c)^14*F^(a + b/(d*x + c)^3), x)