3.368 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+6 n} \, dx\)

Optimal. Leaf size=114 \[ -\frac {F^{a+b (c+d x)^n} \left (-b^5 \log ^5(F) (c+d x)^{5 n}+5 b^4 \log ^4(F) (c+d x)^{4 n}-20 b^3 \log ^3(F) (c+d x)^{3 n}+60 b^2 \log ^2(F) (c+d x)^{2 n}-120 b \log (F) (c+d x)^n+120\right )}{b^6 d n \log ^6(F)} \]

[Out]

-F^(a+b*(d*x+c)^n)*(120-120*b*(d*x+c)^n*ln(F)+60*b^2*(d*x+c)^(2*n)*ln(F)^2-20*b^3*(d*x+c)^(3*n)*ln(F)^3+5*b^4*
(d*x+c)^(4*n)*ln(F)^4-b^5*(d*x+c)^(5*n)*ln(F)^5)/b^6/d/n/ln(F)^6

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Rubi [C]  time = 0.04, antiderivative size = 32, normalized size of antiderivative = 0.28, 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 = {2218} \[ -\frac {F^a \text {Gamma}\left (6,-b \log (F) (c+d x)^n\right )}{b^6 d n \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 6*n),x]

[Out]

-((F^a*Gamma[6, -(b*(c + d*x)^n*Log[F])])/(b^6*d*n*Log[F]^6))

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+b (c+d x)^n} (c+d x)^{-1+6 n} \, dx &=-\frac {F^a \Gamma \left (6,-b (c+d x)^n \log (F)\right )}{b^6 d n \log ^6(F)}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 32, normalized size = 0.28 \[ -\frac {F^a \Gamma \left (6,-b (c+d x)^n \log (F)\right )}{b^6 d n \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 6*n),x]

[Out]

-((F^a*Gamma[6, -(b*(c + d*x)^n*Log[F])])/(b^6*d*n*Log[F]^6))

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fricas [A]  time = 0.43, size = 116, normalized size = 1.02 \[ \frac {{\left ({\left (d x + c\right )}^{5 \, n} b^{5} \log \relax (F)^{5} - 5 \, {\left (d x + c\right )}^{4 \, n} b^{4} \log \relax (F)^{4} + 20 \, {\left (d x + c\right )}^{3 \, n} b^{3} \log \relax (F)^{3} - 60 \, {\left (d x + c\right )}^{2 \, n} b^{2} \log \relax (F)^{2} + 120 \, {\left (d x + c\right )}^{n} b \log \relax (F) - 120\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \relax (F) + a \log \relax (F)\right )}}{b^{6} d n \log \relax (F)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+6*n),x, algorithm="fricas")

[Out]

((d*x + c)^(5*n)*b^5*log(F)^5 - 5*(d*x + c)^(4*n)*b^4*log(F)^4 + 20*(d*x + c)^(3*n)*b^3*log(F)^3 - 60*(d*x + c
)^(2*n)*b^2*log(F)^2 + 120*(d*x + c)^n*b*log(F) - 120)*e^((d*x + c)^n*b*log(F) + a*log(F))/(b^6*d*n*log(F)^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{6 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+6*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(6*n - 1)*F^((d*x + c)^n*b + a), x)

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maple [A]  time = 0.02, size = 113, normalized size = 0.99 \[ \frac {\left (b^{5} \left (d x +c \right )^{5 n} \ln \relax (F )^{5}-5 b^{4} \left (d x +c \right )^{4 n} \ln \relax (F )^{4}+20 b^{3} \left (d x +c \right )^{3 n} \ln \relax (F )^{3}-60 b^{2} \left (d x +c \right )^{2 n} \ln \relax (F )^{2}+120 b \left (d x +c \right )^{n} \ln \relax (F )-120\right ) F^{b \left (d x +c \right )^{n}+a}}{b^{6} d n \ln \relax (F )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*(d*x+c)^n+a)*(d*x+c)^(-1+6*n),x)

[Out]

(((d*x+c)^n)^5*b^5*ln(F)^5-5*((d*x+c)^n)^4*b^4*ln(F)^4+20*((d*x+c)^n)^3*b^3*ln(F)^3-60*((d*x+c)^n)^2*b^2*ln(F)
^2+120*b*(d*x+c)^n*ln(F)-120)/b^6/ln(F)^6/n/d*F^(b*(d*x+c)^n+a)

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maxima [A]  time = 1.05, size = 129, normalized size = 1.13 \[ \frac {{\left ({\left (d x + c\right )}^{5 \, n} F^{a} b^{5} \log \relax (F)^{5} - 5 \, {\left (d x + c\right )}^{4 \, n} F^{a} b^{4} \log \relax (F)^{4} + 20 \, {\left (d x + c\right )}^{3 \, n} F^{a} b^{3} \log \relax (F)^{3} - 60 \, {\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \relax (F)^{2} + 120 \, {\left (d x + c\right )}^{n} F^{a} b \log \relax (F) - 120 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{6} d n \log \relax (F)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+6*n),x, algorithm="maxima")

[Out]

((d*x + c)^(5*n)*F^a*b^5*log(F)^5 - 5*(d*x + c)^(4*n)*F^a*b^4*log(F)^4 + 20*(d*x + c)^(3*n)*F^a*b^3*log(F)^3 -
60*(d*x + c)^(2*n)*F^a*b^2*log(F)^2 + 120*(d*x + c)^n*F^a*b*log(F) - 120*F^a)*F^((d*x + c)^n*b)/(b^6*d*n*log(
F)^6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{6\,n-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^n)*(c + d*x)^(6*n - 1),x)

[Out]

int(F^(a + b*(c + d*x)^n)*(c + d*x)^(6*n - 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1+6*n),x)

[Out]

Timed out

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