Integrand size = 20, antiderivative size = 91 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=-\frac {b c (e x)^{1+m}}{d^2 e (1+m)}+\frac {b (e x)^{2+m}}{d e^2 (2+m)}+\frac {\left (b c^2+a d^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{c d^2 e (1+m)} \] Output:
-b*c*(e*x)^(1+m)/d^2/e/(1+m)+b*(e*x)^(2+m)/d/e^2/(2+m)+(a*d^2+b*c^2)*(e*x) ^(1+m)*hypergeom([1, 1+m],[2+m],-d*x/c)/c/d^2/e/(1+m)
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\frac {x (e x)^m \left (b c (-c (2+m)+d (1+m) x)+\left (b c^2+a d^2\right ) (2+m) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )\right )}{c d^2 (1+m) (2+m)} \] Input:
Integrate[((e*x)^m*(a + b*x^2))/(c + d*x),x]
Output:
(x*(e*x)^m*(b*c*(-(c*(2 + m)) + d*(1 + m)*x) + (b*c^2 + a*d^2)*(2 + m)*Hyp ergeometric2F1[1, 1 + m, 2 + m, -((d*x)/c)]))/(c*d^2*(1 + m)*(2 + m))
Time = 0.41 (sec) , antiderivative size = 91, 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 = {522, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) (e x)^m}{c+d x} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {(e x)^m \left (a d^2+b c^2\right )}{d^2 (c+d x)}-\frac {b c (e x)^m}{d^2}+\frac {b (e x)^{m+1}}{d e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e x)^{m+1} \left (a d^2+b c^2\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d x}{c}\right )}{c d^2 e (m+1)}-\frac {b c (e x)^{m+1}}{d^2 e (m+1)}+\frac {b (e x)^{m+2}}{d e^2 (m+2)}\) |
Input:
Int[((e*x)^m*(a + b*x^2))/(c + d*x),x]
Output:
-((b*c*(e*x)^(1 + m))/(d^2*e*(1 + m))) + (b*(e*x)^(2 + m))/(d*e^2*(2 + m)) + ((b*c^2 + a*d^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d* x)/c)])/(c*d^2*e*(1 + m))
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] && IGtQ[p, 0]
\[\int \frac {\left (e x \right )^{m} \left (b \,x^{2}+a \right )}{d x +c}d x\]
Input:
int((e*x)^m*(b*x^2+a)/(d*x+c),x)
Output:
int((e*x)^m*(b*x^2+a)/(d*x+c),x)
\[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x + c} \,d x } \] Input:
integrate((e*x)^m*(b*x^2+a)/(d*x+c),x, algorithm="fricas")
Output:
integral((b*x^2 + a)*(e*x)^m/(d*x + c), x)
Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.60 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\frac {a e^{m} m x^{m + 1} \Phi \left (\frac {d x e^{i \pi }}{c}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac {a e^{m} x^{m + 1} \Phi \left (\frac {d x e^{i \pi }}{c}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac {b e^{m} m x^{m + 3} \Phi \left (\frac {d x e^{i \pi }}{c}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{c \Gamma \left (m + 4\right )} + \frac {3 b e^{m} x^{m + 3} \Phi \left (\frac {d x e^{i \pi }}{c}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{c \Gamma \left (m + 4\right )} \] Input:
integrate((e*x)**m*(b*x**2+a)/(d*x+c),x)
Output:
a*e**m*m*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gamma(m + 1) /(c*gamma(m + 2)) + a*e**m*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gamma(m + 1)/(c*gamma(m + 2)) + b*e**m*m*x**(m + 3)*lerchphi(d*x*exp _polar(I*pi)/c, 1, m + 3)*gamma(m + 3)/(c*gamma(m + 4)) + 3*b*e**m*x**(m + 3)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 3)*gamma(m + 3)/(c*gamma(m + 4) )
\[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x + c} \,d x } \] Input:
integrate((e*x)^m*(b*x^2+a)/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)*(e*x)^m/(d*x + c), x)
\[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x + c} \,d x } \] Input:
integrate((e*x)^m*(b*x^2+a)/(d*x+c),x, algorithm="giac")
Output:
integrate((b*x^2 + a)*(e*x)^m/(d*x + c), x)
Timed out. \[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (b\,x^2+a\right )}{c+d\,x} \,d x \] Input:
int(((e*x)^m*(a + b*x^2))/(c + d*x),x)
Output:
int(((e*x)^m*(a + b*x^2))/(c + d*x), x)
\[ \int \frac {(e x)^m \left (a+b x^2\right )}{c+d x} \, dx=\frac {e^{m} \left (x^{m} a \,d^{2} m^{2}+3 x^{m} a \,d^{2} m +2 x^{m} a \,d^{2}+x^{m} b \,c^{2} m^{2}+3 x^{m} b \,c^{2} m +2 x^{m} b \,c^{2}-x^{m} b c d \,m^{2} x -2 x^{m} b c d m x +x^{m} b \,d^{2} m^{2} x^{2}+x^{m} b \,d^{2} m \,x^{2}-\left (\int \frac {x^{m}}{d \,x^{2}+c x}d x \right ) a c \,d^{2} m^{3}-3 \left (\int \frac {x^{m}}{d \,x^{2}+c x}d x \right ) a c \,d^{2} m^{2}-2 \left (\int \frac {x^{m}}{d \,x^{2}+c x}d x \right ) a c \,d^{2} m -\left (\int \frac {x^{m}}{d \,x^{2}+c x}d x \right ) b \,c^{3} m^{3}-3 \left (\int \frac {x^{m}}{d \,x^{2}+c x}d x \right ) b \,c^{3} m^{2}-2 \left (\int \frac {x^{m}}{d \,x^{2}+c x}d x \right ) b \,c^{3} m \right )}{d^{3} m \left (m^{2}+3 m +2\right )} \] Input:
int((e*x)^m*(b*x^2+a)/(d*x+c),x)
Output:
(e**m*(x**m*a*d**2*m**2 + 3*x**m*a*d**2*m + 2*x**m*a*d**2 + x**m*b*c**2*m* *2 + 3*x**m*b*c**2*m + 2*x**m*b*c**2 - x**m*b*c*d*m**2*x - 2*x**m*b*c*d*m* x + x**m*b*d**2*m**2*x**2 + x**m*b*d**2*m*x**2 - int(x**m/(c*x + d*x**2),x )*a*c*d**2*m**3 - 3*int(x**m/(c*x + d*x**2),x)*a*c*d**2*m**2 - 2*int(x**m/ (c*x + d*x**2),x)*a*c*d**2*m - int(x**m/(c*x + d*x**2),x)*b*c**3*m**3 - 3* int(x**m/(c*x + d*x**2),x)*b*c**3*m**2 - 2*int(x**m/(c*x + d*x**2),x)*b*c* *3*m))/(d**3*m*(m**2 + 3*m + 2))