Integrand size = 20, antiderivative size = 63 \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{b e (1+m)} \] Output:
(b*x)^(1+m)*(d*x+c)^n*AppellF1(1+m,-n,1,2+m,-d*x/c,-f*x/e)/b/e/(1+m)/((1+d *x/c)^n)
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{e (1+m)} \] Input:
Integrate[((b*x)^m*(c + d*x)^n)/(e + f*x),x]
Output:
(x*(b*x)^m*(c + d*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((d*x)/c), -((f*x)/e )])/(e*(1 + m)*((c + d*x)/c)^n)
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
\(\displaystyle \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx\) |
\(\Big \downarrow \) 152 |
\(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \int \frac {(b x)^m \left (\frac {d x}{c}+1\right )^n}{e+f x}dx\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {d x}{c},-\frac {f x}{e}\right )}{b e (m+1)}\) |
Input:
Int[((b*x)^m*(c + d*x)^n)/(e + f*x),x]
Output:
((b*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((d*x)/c), -((f* x)/e)])/(b*e*(1 + m)*(1 + (d*x)/c)^n)
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]
\[\int \frac {\left (b x \right )^{m} \left (x d +c \right )^{n}}{f x +e}d x\]
Input:
int((b*x)^m*(d*x+c)^n/(f*x+e),x)
Output:
int((b*x)^m*(d*x+c)^n/(f*x+e),x)
\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f x + e} \,d x } \] Input:
integrate((b*x)^m*(d*x+c)^n/(f*x+e),x, algorithm="fricas")
Output:
integral((b*x)^m*(d*x + c)^n/(f*x + e), x)
Exception generated. \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x)**m*(d*x+c)**n/(f*x+e),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f x + e} \,d x } \] Input:
integrate((b*x)^m*(d*x+c)^n/(f*x+e),x, algorithm="maxima")
Output:
integrate((b*x)^m*(d*x + c)^n/(f*x + e), x)
\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f x + e} \,d x } \] Input:
integrate((b*x)^m*(d*x+c)^n/(f*x+e),x, algorithm="giac")
Output:
integrate((b*x)^m*(d*x + c)^n/(f*x + e), x)
Timed out. \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int \frac {{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{e+f\,x} \,d x \] Input:
int(((b*x)^m*(c + d*x)^n)/(e + f*x),x)
Output:
int(((b*x)^m*(c + d*x)^n)/(e + f*x), x)
\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx =\text {Too large to display} \] Input:
int((b*x)^m*(d*x+c)^n/(f*x+e),x)
Output:
(b**m*(x**m*(c + d*x)**n*c - int((x**m*(c + d*x)**n*x)/(c**2*e*f*m + c**2* f**2*m*x + c*d*e**2*m + c*d*e**2*n + 2*c*d*e*f*m*x + c*d*e*f*n*x + c*d*f** 2*m*x**2 + d**2*e**2*m*x + d**2*e**2*n*x + d**2*e*f*m*x**2 + d**2*e*f*n*x* *2),x)*c**2*d*f**2*m*n + int((x**m*(c + d*x)**n*x)/(c**2*e*f*m + c**2*f**2 *m*x + c*d*e**2*m + c*d*e**2*n + 2*c*d*e*f*m*x + c*d*e*f*n*x + c*d*f**2*m* x**2 + d**2*e**2*m*x + d**2*e**2*n*x + d**2*e*f*m*x**2 + d**2*e*f*n*x**2), x)*c*d**2*e*f*m**2 - int((x**m*(c + d*x)**n*x)/(c**2*e*f*m + c**2*f**2*m*x + c*d*e**2*m + c*d*e**2*n + 2*c*d*e*f*m*x + c*d*e*f*n*x + c*d*f**2*m*x**2 + d**2*e**2*m*x + d**2*e**2*n*x + d**2*e*f*m*x**2 + d**2*e*f*n*x**2),x)*c *d**2*e*f*n**2 + int((x**m*(c + d*x)**n*x)/(c**2*e*f*m + c**2*f**2*m*x + c *d*e**2*m + c*d*e**2*n + 2*c*d*e*f*m*x + c*d*e*f*n*x + c*d*f**2*m*x**2 + d **2*e**2*m*x + d**2*e**2*n*x + d**2*e*f*m*x**2 + d**2*e*f*n*x**2),x)*d**3* e**2*m**2 + 2*int((x**m*(c + d*x)**n*x)/(c**2*e*f*m + c**2*f**2*m*x + c*d* e**2*m + c*d*e**2*n + 2*c*d*e*f*m*x + c*d*e*f*n*x + c*d*f**2*m*x**2 + d**2 *e**2*m*x + d**2*e**2*n*x + d**2*e*f*m*x**2 + d**2*e*f*n*x**2),x)*d**3*e** 2*m*n + int((x**m*(c + d*x)**n*x)/(c**2*e*f*m + c**2*f**2*m*x + c*d*e**2*m + c*d*e**2*n + 2*c*d*e*f*m*x + c*d*e*f*n*x + c*d*f**2*m*x**2 + d**2*e**2* m*x + d**2*e**2*n*x + d**2*e*f*m*x**2 + d**2*e*f*n*x**2),x)*d**3*e**2*n**2 - int((x**m*(c + d*x)**n)/(c**2*e*f*m*x + c**2*f**2*m*x**2 + c*d*e**2*m*x + c*d*e**2*n*x + 2*c*d*e*f*m*x**2 + c*d*e*f*n*x**2 + c*d*f**2*m*x**3 +...