Integrand size = 35, antiderivative size = 54 \[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m} \] Output:
-(e*x+d)^m*hypergeom([1, m],[1+m],c*d*(e*x+d)/(-a*e^2+c*d^2))/(-a*e^2+c*d^ 2)/m
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m} \] Input:
Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
Output:
-(((d + e*x)^m*Hypergeometric2F1[1, m, 1 + m, (c*d*(d + e*x))/(c*d^2 - a*e ^2)])/((c*d^2 - a*e^2)*m))
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 78}
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.
\(\displaystyle \int \frac {(d+e x)^m}{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \frac {(d+e x)^{m-1}}{a e+c d x}dx\) |
\(\Big \downarrow \) 78 |
\(\displaystyle -\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )}\) |
Input:
Int[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
Output:
-(((d + e*x)^m*Hypergeometric2F1[1, m, 1 + m, (c*d*(d + e*x))/(c*d^2 - a*e ^2)])/((c*d^2 - a*e^2)*m))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
\[\int \frac {\left (e x +d \right )^{m}}{a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}d x\]
Input:
int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e),x)
Output:
int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e),x)
\[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \,d x } \] Input:
integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas ")
Output:
integral((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)
\[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (d + e x\right ) \left (a e + c d x\right )}\, dx \] Input:
integrate((e*x+d)**m/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
Output:
Integral((d + e*x)**m/((d + e*x)*(a*e + c*d*x)), x)
\[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \,d x } \] Input:
integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima ")
Output:
integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)
\[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \,d x } \] Input:
integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")
Output:
integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)
Timed out. \[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \,d x \] Input:
int((d + e*x)^m/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)
Output:
int((d + e*x)^m/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2), x)
\[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int \frac {\left (e x +d \right )^{m}}{c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}d x \] Input:
int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
Output:
int((d + e*x)**m/(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2),x)