3.4.69 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+5 n} \, dx\) [369]

Optimal. Leaf size=94 \[ \frac {F^{a+b (c+d x)^n} \left (24-24 b (c+d x)^n \log (F)+12 b^2 (c+d x)^{2 n} \log ^2(F)-4 b^3 (c+d x)^{3 n} \log ^3(F)+b^4 (c+d x)^{4 n} \log ^4(F)\right )}{b^5 d n \log ^5(F)} \]

[Out]

F^(a+b*(d*x+c)^n)*(24-24*b*(d*x+c)^n*ln(F)+12*b^2*(d*x+c)^(2*n)*ln(F)^2-4*b^3*(d*x+c)^(3*n)*ln(F)^3+b^4*(d*x+c
)^(4*n)*ln(F)^4)/b^5/d/n/ln(F)^5

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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2249} \begin {gather*} \frac {F^{a+b (c+d x)^n} \left (b^4 \log ^4(F) (c+d x)^{4 n}-4 b^3 \log ^3(F) (c+d x)^{3 n}+12 b^2 \log ^2(F) (c+d x)^{2 n}-24 b \log (F) (c+d x)^n+24\right )}{b^5 d n \log ^5(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 5*n),x]

[Out]

(F^(a + b*(c + d*x)^n)*(24 - 24*b*(c + d*x)^n*Log[F] + 12*b^2*(c + d*x)^(2*n)*Log[F]^2 - 4*b^3*(c + d*x)^(3*n)
*Log[F]^3 + b^4*(c + d*x)^(4*n)*Log[F]^4))/(b^5*d*n*Log[F]^5)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

Rubi steps

\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+5 n} \, dx &=\frac {F^a \Gamma \left (5,-b (c+d x)^n \log (F)\right )}{b^5 d n \log ^5(F)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.01, size = 31, normalized size = 0.33 \begin {gather*} \frac {F^a \Gamma \left (5,-b (c+d x)^n \log (F)\right )}{b^5 d n \log ^5(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 5*n),x]

[Out]

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

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Maple [A]
time = 0.06, size = 95, normalized size = 1.01

method	result	size
risch	\(\frac {F^{a +b \left (d x +c \right )^{n}} \left (24-24 b \left (d x +c \right )^{n} \ln \left (F \right )+12 b^{2} \left (d x +c \right )^{2 n} \ln \left (F \right )^{2}-4 b^{3} \left (d x +c \right )^{3 n} \ln \left (F \right )^{3}+b^{4} \left (d x +c \right )^{4 n} \ln \left (F \right )^{4}\right )}{b^{5} d n \ln \left (F \right )^{5}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x,method=_RETURNVERBOSE)

[Out]

(((d*x+c)^n)^4*b^4*ln(F)^4-4*((d*x+c)^n)^3*b^3*ln(F)^3+12*((d*x+c)^n)^2*b^2*ln(F)^2-24*b*(d*x+c)^n*ln(F)+24)/b
^5/ln(F)^5/n/d*F^(a+b*(d*x+c)^n)

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Maxima [A]
time = 0.30, size = 108, normalized size = 1.15 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{4 \, n} F^{a} b^{4} \log \left (F\right )^{4} - 4 \, {\left (d x + c\right )}^{3 \, n} F^{a} b^{3} \log \left (F\right )^{3} + 12 \, {\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \left (F\right )^{2} - 24 \, {\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) + 24 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{5} d n \log \left (F\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x, algorithm="maxima")

[Out]

((d*x + c)^(4*n)*F^a*b^4*log(F)^4 - 4*(d*x + c)^(3*n)*F^a*b^3*log(F)^3 + 12*(d*x + c)^(2*n)*F^a*b^2*log(F)^2 -
24*(d*x + c)^n*F^a*b*log(F) + 24*F^a)*F^((d*x + c)^n*b)/(b^5*d*n*log(F)^5)

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Fricas [A]
time = 0.39, size = 98, normalized size = 1.04 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{4 \, n} b^{4} \log \left (F\right )^{4} - 4 \, {\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} + 12 \, {\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} - 24 \, {\left (d x + c\right )}^{n} b \log \left (F\right ) + 24\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{5} d n \log \left (F\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x, algorithm="fricas")

[Out]

((d*x + c)^(4*n)*b^4*log(F)^4 - 4*(d*x + c)^(3*n)*b^3*log(F)^3 + 12*(d*x + c)^(2*n)*b^2*log(F)^2 - 24*(d*x + c
)^n*b*log(F) + 24)*e^((d*x + c)^n*b*log(F) + a*log(F))/(b^5*d*n*log(F)^5)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1+5*n),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5*n - 1)*F^((d*x + c)^n*b + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{5\,n-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^n)*(c + d*x)^(5*n - 1),x)

[Out]

int(F^(a + b*(c + d*x)^n)*(c + d*x)^(5*n - 1), x)

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