3.374 \(\int \frac {F^{a+b (c+d x)^n}}{c+d x} \, dx\)

Optimal. Leaf size=22 \[ \frac {F^a \text {Ei}\left (b (c+d x)^n \log (F)\right )}{d n} \]

[Out]

F^a*Ei(b*(d*x+c)^n*ln(F))/d/n

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2210} \[ \frac {F^a \text {Ei}\left (b (c+d x)^n \log (F)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ \frac {F^a \text {Ei}\left (b (c+d x)^n \log (F)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)

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fricas [A]  time = 0.44, size = 22, normalized size = 1.00 \[ \frac {F^{a} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \relax (F)\right )}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^a*Ei((d*x + c)^n*b*log(F))/(d*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 26, normalized size = 1.18 \[ -\frac {F^{a} \Ei \left (1, -b \left (d x +c \right )^{n} \ln \relax (F )\right )}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/n*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {F^a\,F^{b\,{\left (c+d\,x\right )}^n}}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + b \left (c + d x\right )^{n}}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)/(d*x+c),x)

[Out]

Integral(F**(a + b*(c + d*x)**n)/(c + d*x), x)

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