Integrand size = 20, antiderivative size = 49 \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\frac {e^p \pi ^n (b x)^{1+m} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (1+m)} \] Output:
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)
Time = 0.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\frac {e^p \pi ^n x (b x)^m \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{\pi },-\frac {f x}{e}\right )}{1+m} \] Input:
Integrate[(b*x)^m*(Pi + d*x)^n*(E + f*x)^p,x]
Output:
(E^p*Pi^n*x*(b*x)^m*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x)/E) ])/(1 + m)
Time = 0.20 (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 = {150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (b x)^m (d x+\pi )^n (f x+e)^p \, dx\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\pi ^n e^p (b x)^{m+1} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (m+1)}\) |
Input:
Int[(b*x)^m*(Pi + d*x)^n*(E + f*x)^p,x]
Output:
(E^p*Pi^n*(b*x)^(1 + m)*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x )/E)])/(b*(1 + m))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> 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] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
\[\int \left (b x \right )^{m} \left (x d +\pi \right )^{n} \left (f x +{\mathrm e}\right )^{p}d x\]
Input:
int((b*x)^m*(d*x+Pi)^n*(f*x+exp(1))^p,x)
Output:
int((b*x)^m*(d*x+Pi)^n*(f*x+exp(1))^p,x)
\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int { {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + e\right )}^{p} \,d x } \] Input:
integrate((b*x)^m*(d*x+pi)^n*(f*x+exp(1))^p,x, algorithm="fricas")
Output:
integral((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)
\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int \left (b x\right )^{m} \left (d x + \pi \right )^{n} \left (f x + e\right )^{p}\, dx \] Input:
integrate((b*x)**m*(d*x+pi)**n*(f*x+exp(1))**p,x)
Output:
Integral((b*x)**m*(d*x + pi)**n*(f*x + E)**p, x)
\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int { {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + e\right )}^{p} \,d x } \] Input:
integrate((b*x)^m*(d*x+pi)^n*(f*x+exp(1))^p,x, algorithm="maxima")
Output:
integrate((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)
\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int { {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + e\right )}^{p} \,d x } \] Input:
integrate((b*x)^m*(d*x+pi)^n*(f*x+exp(1))^p,x, algorithm="giac")
Output:
integrate((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)
Timed out. \[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\int {\left (\mathrm {e}+f\,x\right )}^p\,{\left (b\,x\right )}^m\,{\left (\Pi +d\,x\right )}^n \,d x \] Input:
int((exp(1) + f*x)^p*(b*x)^m*(Pi + d*x)^n,x)
Output:
int((exp(1) + f*x)^p*(b*x)^m*(Pi + d*x)^n, x)
\[ \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx=\text {too large to display} \] Input:
int((b*x)^m*(d*x+Pi)^n*(f*x+exp(1))^p,x)
Output:
(b**m*(x**m*(e + f*x)**p*(d*x + pi)**n*d*e*m*x + x**m*(e + f*x)**p*(d*x + pi)**n*d*e*n*x + x**m*(e + f*x)**p*(d*x + pi)**n*e*n*pi + x**m*(e + f*x)** p*(d*x + pi)**n*e*p*pi + x**m*(e + f*x)**p*(d*x + pi)**n*f*m*pi*x + x**m*( e + f*x)**p*(d*x + pi)**n*f*p*pi*x + int((x**m*(e + f*x)**p*(d*x + pi)**n* x)/(d**2*e**2*m**2*x + 2*d**2*e**2*m*n*x + d**2*e**2*m*p*x + d**2*e**2*m*x + d**2*e**2*n**2*x + d**2*e**2*n*p*x + d**2*e**2*n*x + d**2*e*f*m**2*x**2 + 2*d**2*e*f*m*n*x**2 + d**2*e*f*m*p*x**2 + d**2*e*f*m*x**2 + d**2*e*f*n* *2*x**2 + d**2*e*f*n*p*x**2 + d**2*e*f*n*x**2 + d*e**2*m**2*pi + 2*d*e**2* m*n*pi + d*e**2*m*p*pi + d*e**2*m*pi + d*e**2*n**2*pi + d*e**2*n*p*pi + d* e**2*n*pi + 2*d*e*f*m**2*pi*x + 3*d*e*f*m*n*pi*x + 3*d*e*f*m*p*pi*x + 2*d* e*f*m*pi*x + d*e*f*n**2*pi*x + 2*d*e*f*n*p*pi*x + d*e*f*n*pi*x + d*e*f*p** 2*pi*x + d*e*f*p*pi*x + d*f**2*m**2*pi*x**2 + d*f**2*m*n*pi*x**2 + 2*d*f** 2*m*p*pi*x**2 + d*f**2*m*pi*x**2 + d*f**2*n*p*pi*x**2 + d*f**2*p**2*pi*x** 2 + d*f**2*p*pi*x**2 + e*f*m**2*pi**2 + e*f*m*n*pi**2 + 2*e*f*m*p*pi**2 + e*f*m*pi**2 + e*f*n*p*pi**2 + e*f*p**2*pi**2 + e*f*p*pi**2 + f**2*m**2*pi* *2*x + f**2*m*n*pi**2*x + 2*f**2*m*p*pi**2*x + f**2*m*pi**2*x + f**2*n*p*p i**2*x + f**2*p**2*pi**2*x + f**2*p*pi**2*x),x)*d**3*e**3*m**3*p + 3*int(( x**m*(e + f*x)**p*(d*x + pi)**n*x)/(d**2*e**2*m**2*x + 2*d**2*e**2*m*n*x + d**2*e**2*m*p*x + d**2*e**2*m*x + d**2*e**2*n**2*x + d**2*e**2*n*p*x + d* *2*e**2*n*x + d**2*e*f*m**2*x**2 + 2*d**2*e*f*m*n*x**2 + d**2*e*f*m*p*x...