\(\int (b x)^m (\pi +d x)^n (e+f x)^p \, dx\) [966]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\frac {e^p \pi ^n (b x)^{1+m} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (1+m)} \]

[Out]

exp(p)*Pi^n*(b*x)^(1+m)*AppellF1(1+m,-n,-p,2+m,-d*x/Pi,-f*x/exp(1))/b/(1+m)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {138} \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\frac {\pi ^n e^p (b x)^{m+1} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (m+1)} \]

[In]

Int[(b*x)^m*(Pi + d*x)^n*(E + f*x)^p,x]

[Out]

(E^p*Pi^n*(b*x)^(1 + m)*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x)/E)])/(b*(1 + m))

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {e^p \pi ^n (b x)^{1+m} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\frac {e^p \pi ^n x (b x)^m \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{\pi },-\frac {f x}{e}\right )}{1+m} \]

[In]

Integrate[(b*x)^m*(Pi + d*x)^n*(E + f*x)^p,x]

[Out]

(E^p*Pi^n*x*(b*x)^m*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x)/E)])/(1 + m)

Maple [F]

\[\int \left (b x \right )^{m} \left (d x +\pi \right )^{n} \left ({\mathrm e}+f x \right )^{p}d x\]

[In]

int((b*x)^m*(d*x+Pi)^n*(exp(1)+f*x)^p,x)

[Out]

int((b*x)^m*(d*x+Pi)^n*(exp(1)+f*x)^p,x)

Fricas [F]

\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int { {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + e\right )}^{p} \,d x } \]

[In]

integrate((b*x)^m*(d*x+pi)^n*(exp(1)+f*x)^p,x, algorithm="fricas")

[Out]

integral((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)

Sympy [F]

\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int \left (b x\right )^{m} \left (d x + \pi \right )^{n} \left (f x + e\right )^{p}\, dx \]

[In]

integrate((b*x)**m*(d*x+pi)**n*(exp(1)+f*x)**p,x)

[Out]

Integral((b*x)**m*(d*x + pi)**n*(f*x + E)**p, x)

Maxima [F]

\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int { {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + e\right )}^{p} \,d x } \]

[In]

integrate((b*x)^m*(d*x+pi)^n*(exp(1)+f*x)^p,x, algorithm="maxima")

[Out]

integrate((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)

Giac [F]

\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int { {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + e\right )}^{p} \,d x } \]

[In]

integrate((b*x)^m*(d*x+pi)^n*(exp(1)+f*x)^p,x, algorithm="giac")

[Out]

integrate((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)

Mupad [F(-1)]

Timed out. \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int {\left (\mathrm {e}+f\,x\right )}^p\,{\left (b\,x\right )}^m\,{\left (\Pi +d\,x\right )}^n \,d x \]

[In]

int((exp(1) + f*x)^p*(b*x)^m*(Pi + d*x)^n,x)

[Out]

int((exp(1) + f*x)^p*(b*x)^m*(Pi + d*x)^n, x)