3.377 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1-3 n} \, dx\)

Optimal. Leaf size=139 \[ \frac{b^3 F^a \log ^3(F) \text{Ei}\left (b (c+d x)^n \log (F)\right )}{6 d n}-\frac{b^2 \log ^2(F) (c+d x)^{-n} F^{a+b (c+d x)^n}}{6 d n}-\frac{(c+d x)^{-3 n} F^{a+b (c+d x)^n}}{3 d n}-\frac{b \log (F) (c+d x)^{-2 n} F^{a+b (c+d x)^n}}{6 d n} \]

[Out]

-F^(a + b*(c + d*x)^n)/(3*d*n*(c + d*x)^(3*n)) - (b*F^(a + b*(c + d*x)^n)*Log[F])/(6*d*n*(c + d*x)^(2*n)) - (b
^2*F^(a + b*(c + d*x)^n)*Log[F]^2)/(6*d*n*(c + d*x)^n) + (b^3*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^3
)/(6*d*n)

________________________________________________________________________________________

Rubi [A]  time = 0.164464, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2215, 2210} \[ \frac{b^3 F^a \log ^3(F) \text{Ei}\left (b (c+d x)^n \log (F)\right )}{6 d n}-\frac{b^2 \log ^2(F) (c+d x)^{-n} F^{a+b (c+d x)^n}}{6 d n}-\frac{(c+d x)^{-3 n} F^{a+b (c+d x)^n}}{3 d n}-\frac{b \log (F) (c+d x)^{-2 n} F^{a+b (c+d x)^n}}{6 d n} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 3*n),x]

[Out]

-F^(a + b*(c + d*x)^n)/(3*d*n*(c + d*x)^(3*n)) - (b*F^(a + b*(c + d*x)^n)*Log[F])/(6*d*n*(c + d*x)^(2*n)) - (b
^2*F^(a + b*(c + d*x)^n)*Log[F]^2)/(6*d*n*(c + d*x)^n) + (b^3*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^3
)/(6*d*n)

Rule 2215

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^Simplify[m + n]*F^(a +
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simpl
ify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, 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 F^{a+b (c+d x)^n} (c+d x)^{-1-3 n} \, dx &=-\frac{F^{a+b (c+d x)^n} (c+d x)^{-3 n}}{3 d n}+\frac{1}{3} (b \log (F)) \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx\\ &=-\frac{F^{a+b (c+d x)^n} (c+d x)^{-3 n}}{3 d n}-\frac{b F^{a+b (c+d x)^n} (c+d x)^{-2 n} \log (F)}{6 d n}+\frac{1}{6} \left (b^2 \log ^2(F)\right ) \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx\\ &=-\frac{F^{a+b (c+d x)^n} (c+d x)^{-3 n}}{3 d n}-\frac{b F^{a+b (c+d x)^n} (c+d x)^{-2 n} \log (F)}{6 d n}-\frac{b^2 F^{a+b (c+d x)^n} (c+d x)^{-n} \log ^2(F)}{6 d n}+\frac{1}{6} \left (b^3 \log ^3(F)\right ) \int \frac{F^{a+b (c+d x)^n}}{c+d x} \, dx\\ &=-\frac{F^{a+b (c+d x)^n} (c+d x)^{-3 n}}{3 d n}-\frac{b F^{a+b (c+d x)^n} (c+d x)^{-2 n} \log (F)}{6 d n}-\frac{b^2 F^{a+b (c+d x)^n} (c+d x)^{-n} \log ^2(F)}{6 d n}+\frac{b^3 F^a \text{Ei}\left (b (c+d x)^n \log (F)\right ) \log ^3(F)}{6 d n}\\ \end{align*}

Mathematica [A]  time = 0.0068938, size = 31, normalized size = 0.22 \[ \frac{b^3 F^a \log ^3(F) \text{Gamma}\left (-3,-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 3*n),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 137, normalized size = 1. \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{3\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{3}}}-{\frac{b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{6\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{6\,dn \left ( dx+c \right ) ^{n}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{6\,dn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-3*n),x)

[Out]

-1/3/n/d*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^3-1/6/n/d*ln(F)*b*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^2-1/6/n/d*ln(F)^2*b
^2*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)-1/6/n/d*ln(F)^3*b^3*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{-3 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-3*n),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(-3*n - 1)*F^((d*x + c)^n*b + a), x)

________________________________________________________________________________________

Fricas [A]  time = 1.55027, size = 247, normalized size = 1.78 \begin{align*} \frac{{\left (d x + c\right )}^{3 \, n} F^{a} b^{3}{\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{3} -{\left ({\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} +{\left (d x + c\right )}^{n} b \log \left (F\right ) + 2\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{6 \,{\left (d x + c\right )}^{3 \, n} d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-3*n),x, algorithm="fricas")

[Out]

1/6*((d*x + c)^(3*n)*F^a*b^3*Ei((d*x + c)^n*b*log(F))*log(F)^3 - ((d*x + c)^(2*n)*b^2*log(F)^2 + (d*x + c)^n*b
*log(F) + 2)*e^((d*x + c)^n*b*log(F) + a*log(F)))/((d*x + c)^(3*n)*d*n)

________________________________________________________________________________________

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)**n)*(d*x+c)**(-1-3*n),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{-3 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-3*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(-3*n - 1)*F^((d*x + c)^n*b + a), x)