Integrand size = 24, antiderivative size = 91 \[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\frac {(e-x)^{\sqrt {2}} (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (1-\frac {x}{e}\right )^{-\sqrt {2}} \operatorname {AppellF1}\left (1+m,-n,-\sqrt {2},2+m,-\frac {d x}{c},\frac {x}{e}\right )}{b (1+m)} \] Output:
(e-x)^(2^(1/2))*(b*x)^(1+m)*(d*x+c)^n*AppellF1(1+m,-2^(1/2),-n,2+m,x/e,-d* x/c)/b/(1+m)/((1+d*x/c)^n)/((1-x/e)^(2^(1/2)))
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\frac {(e-x)^{\sqrt {2}} \left (\frac {e-x}{e}\right )^{-\sqrt {2}} x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-\sqrt {2},-n,2+m,\frac {x}{e},-\frac {d x}{c}\right )}{1+m} \] Input:
Integrate[(e - x)^Sqrt[2]*(b*x)^m*(c + d*x)^n,x]
Output:
((e - x)^Sqrt[2]*x*(b*x)^m*(c + d*x)^n*AppellF1[1 + m, -Sqrt[2], -n, 2 + m , x/e, -((d*x)/c)])/((1 + m)*((e - x)/e)^Sqrt[2]*((c + d*x)/c)^n)
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {152, 152, 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.
\(\displaystyle \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx\) |
\(\Big \downarrow \) 152 |
\(\displaystyle (e-x)^{\sqrt {2}} \left (1-\frac {x}{e}\right )^{-\sqrt {2}} \int (b x)^m (c+d x)^n \left (1-\frac {x}{e}\right )^{\sqrt {2}}dx\) |
\(\Big \downarrow \) 152 |
\(\displaystyle (e-x)^{\sqrt {2}} \left (1-\frac {x}{e}\right )^{-\sqrt {2}} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \int (b x)^m \left (\frac {d x}{c}+1\right )^n \left (1-\frac {x}{e}\right )^{\sqrt {2}}dx\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {(e-x)^{\sqrt {2}} \left (1-\frac {x}{e}\right )^{-\sqrt {2}} (b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,-\sqrt {2},m+2,-\frac {d x}{c},\frac {x}{e}\right )}{b (m+1)}\) |
Input:
Int[(e - x)^Sqrt[2]*(b*x)^m*(c + d*x)^n,x]
Output:
((e - x)^Sqrt[2]*(b*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, -Sqrt[2], 2 + m, -((d*x)/c), x/e])/(b*(1 + m)*(1 + (d*x)/c)^n*(1 - x/e)^Sqrt[2])
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[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0]
\[\int \left (e -x \right )^{\sqrt {2}} \left (b x \right )^{m} \left (x d +c \right )^{n}d x\]
Input:
int((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x)
Output:
int((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x)
\[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\int { \left (b x\right )^{m} {\left (d x + c\right )}^{n} {\left (e - x\right )}^{\left (\sqrt {2}\right )} \,d x } \] Input:
integrate((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x, algorithm="fricas")
Output:
integral((b*x)^m*(d*x + c)^n*(e - x)^sqrt(2), x)
Timed out. \[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\text {Timed out} \] Input:
integrate((e-x)**(2**(1/2))*(b*x)**m*(d*x+c)**n,x)
Output:
Timed out
\[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\int { \left (b x\right )^{m} {\left (d x + c\right )}^{n} {\left (e - x\right )}^{\left (\sqrt {2}\right )} \,d x } \] Input:
integrate((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x, algorithm="maxima")
Output:
integrate((b*x)^m*(d*x + c)^n*(e - x)^sqrt(2), x)
\[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\int { \left (b x\right )^{m} {\left (d x + c\right )}^{n} {\left (e - x\right )}^{\left (\sqrt {2}\right )} \,d x } \] Input:
integrate((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x, algorithm="giac")
Output:
integrate((b*x)^m*(d*x + c)^n*(e - x)^sqrt(2), x)
Timed out. \[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\int {\left (b\,x\right )}^m\,{\left (e-x\right )}^{\sqrt {2}}\,{\left (c+d\,x\right )}^n \,d x \] Input:
int((b*x)^m*(e - x)^(2^(1/2))*(c + d*x)^n,x)
Output:
int((b*x)^m*(e - x)^(2^(1/2))*(c + d*x)^n, x)
\[ \int (e-x)^{\sqrt {2}} (b x)^m (c+d x)^n \, dx=\int \left (e -x \right )^{\sqrt {2}} \left (b x \right )^{m} \left (d x +c \right )^{n}d x \] Input:
int((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x)
Output:
int((e-x)^(2^(1/2))*(b*x)^m*(d*x+c)^n,x)