Integrand size = 22, antiderivative size = 148 \[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\frac {(e x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{2 a e (1+m)}+\frac {(e x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{2 a e (1+m)} \] Output:
1/2*(e*x)^(1+m)*(d*x+c)^n*AppellF1(1+m,1,-n,2+m,-b^(1/2)*x/(-a)^(1/2),-d*x /c)/a/e/(1+m)/((1+d*x/c)^n)+1/2*(e*x)^(1+m)*(d*x+c)^n*AppellF1(1+m,1,-n,2+ m,b^(1/2)*x/(-a)^(1/2),-d*x/c)/a/e/(1+m)/((1+d*x/c)^n)
\[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx \] Input:
Integrate[((e*x)^m*(c + d*x)^n)/(a + b*x^2),x]
Output:
Integrate[((e*x)^m*(c + d*x)^n)/(a + b*x^2), x]
Time = 0.57 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {615, 2009}
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.
\(\displaystyle \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \int \left (\frac {\sqrt {-a} (e x)^m (c+d x)^n}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} (e x)^m (c+d x)^n}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{2 a e (m+1)}+\frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{2 a e (m+1)}\) |
Input:
Int[((e*x)^m*(c + d*x)^n)/(a + b*x^2),x]
Output:
((e*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((d*x)/c), -((Sq rt[b]*x)/Sqrt[-a])])/(2*a*e*(1 + m)*(1 + (d*x)/c)^n) + ((e*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((d*x)/c), (Sqrt[b]*x)/Sqrt[-a]])/( 2*a*e*(1 + m)*(1 + (d*x)/c)^n)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )^{n}}{b \,x^{2}+a}d x\]
Input:
int((e*x)^m*(d*x+c)^n/(b*x^2+a),x)
Output:
int((e*x)^m*(d*x+c)^n/(b*x^2+a),x)
\[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} \left (e x\right )^{m}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^n/(b*x^2+a),x, algorithm="fricas")
Output:
integral((d*x + c)^n*(e*x)^m/(b*x^2 + a), x)
Exception generated. \[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((e*x)**m*(d*x+c)**n/(b*x**2+a),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} \left (e x\right )^{m}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^n/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^n*(e*x)^m/(b*x^2 + a), x)
\[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} \left (e x\right )^{m}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^n/(b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x + c)^n*(e*x)^m/(b*x^2 + a), x)
Timed out. \[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (c+d\,x\right )}^n}{b\,x^2+a} \,d x \] Input:
int(((e*x)^m*(c + d*x)^n)/(a + b*x^2),x)
Output:
int(((e*x)^m*(c + d*x)^n)/(a + b*x^2), x)
\[ \int \frac {(e x)^m (c+d x)^n}{a+b x^2} \, dx=e^{m} \left (\int \frac {x^{m} \left (d x +c \right )^{n}}{b \,x^{2}+a}d x \right ) \] Input:
int((e*x)^m*(d*x+c)^n/(b*x^2+a),x)
Output:
e**m*int((x**m*(c + d*x)**n)/(a + b*x**2),x)