Integrand size = 31, antiderivative size = 54 \[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {e^2 (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m)} \] Output:
-e^2*(e*x+d)^(1+m)*hypergeom([3, 1+m],[2+m],b*(e*x+d)/(-a*e+b*d))/(-a*e+b* d)^3/(1+m)
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^2 (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,-\frac {b (d+e x)}{-b d+a e}\right )}{(-b d+a e)^3 (1+m)} \] Input:
Integrate[((a + b*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(e^2*(d + e*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((b*(d + e*x))/ (-(b*d) + a*e))])/((-(b*d) + a*e)^3*(1 + m))
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1184, 27, 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.
\(\displaystyle \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^4 \int \frac {(d+e x)^m}{b^4 (a+b x)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(d+e x)^m}{(a+b x)^3}dx\) |
\(\Big \downarrow \) 78 |
\(\displaystyle -\frac {e^2 (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (3,m+1,m+2,\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^3}\) |
Input:
Int[((a + b*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
-((e^2*(d + e*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, (b*(d + e*x))/ (b*d - a*e)])/((b*d - a*e)^3*(1 + m)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
\[\int \frac {\left (b x +a \right ) \left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}d x\]
Input:
int((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
int((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^2,x)
\[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\int { \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}} \,d x } \] Input:
integrate((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
integral((e*x + d)^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)
\[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (a + b x\right )^{3}}\, dx \] Input:
integrate((b*x+a)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
Integral((d + e*x)**m/(a + b*x)**3, x)
\[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\int { \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}} \,d x } \] Input:
integrate((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
integrate((b*x + a)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^2, x)
\[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\int { \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}} \,d x } \] Input:
integrate((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
integrate((b*x + a)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^2, x)
Timed out. \[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^2} \,d x \] Input:
int(((a + b*x)*(d + e*x)^m)/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
int(((a + b*x)*(d + e*x)^m)/(a^2 + b^2*x^2 + 2*a*b*x)^2, x)
\[ \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {too large to display} \] Input:
int((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
((d + e*x)**m*d + int(((d + e*x)**m*x)/(a**4*d*e*m + a**4*e**2*m*x - 2*a** 3*b*d**2 + 3*a**3*b*d*e*m*x - 2*a**3*b*d*e*x + 3*a**3*b*e**2*m*x**2 - 6*a* *2*b**2*d**2*x + 3*a**2*b**2*d*e*m*x**2 - 6*a**2*b**2*d*e*x**2 + 3*a**2*b* *2*e**2*m*x**3 - 6*a*b**3*d**2*x**2 + a*b**3*d*e*m*x**3 - 6*a*b**3*d*e*x** 3 + a*b**3*e**2*m*x**4 - 2*b**4*d**2*x**3 - 2*b**4*d*e*x**4),x)*a**4*e**3* m**2 - int(((d + e*x)**m*x)/(a**4*d*e*m + a**4*e**2*m*x - 2*a**3*b*d**2 + 3*a**3*b*d*e*m*x - 2*a**3*b*d*e*x + 3*a**3*b*e**2*m*x**2 - 6*a**2*b**2*d** 2*x + 3*a**2*b**2*d*e*m*x**2 - 6*a**2*b**2*d*e*x**2 + 3*a**2*b**2*e**2*m*x **3 - 6*a*b**3*d**2*x**2 + a*b**3*d*e*m*x**3 - 6*a*b**3*d*e*x**3 + a*b**3* e**2*m*x**4 - 2*b**4*d**2*x**3 - 2*b**4*d*e*x**4),x)*a**3*b*d*e**2*m**2 - 2*int(((d + e*x)**m*x)/(a**4*d*e*m + a**4*e**2*m*x - 2*a**3*b*d**2 + 3*a** 3*b*d*e*m*x - 2*a**3*b*d*e*x + 3*a**3*b*e**2*m*x**2 - 6*a**2*b**2*d**2*x + 3*a**2*b**2*d*e*m*x**2 - 6*a**2*b**2*d*e*x**2 + 3*a**2*b**2*e**2*m*x**3 - 6*a*b**3*d**2*x**2 + a*b**3*d*e*m*x**3 - 6*a*b**3*d*e*x**3 + a*b**3*e**2* m*x**4 - 2*b**4*d**2*x**3 - 2*b**4*d*e*x**4),x)*a**3*b*d*e**2*m + 2*int((( d + e*x)**m*x)/(a**4*d*e*m + a**4*e**2*m*x - 2*a**3*b*d**2 + 3*a**3*b*d*e* m*x - 2*a**3*b*d*e*x + 3*a**3*b*e**2*m*x**2 - 6*a**2*b**2*d**2*x + 3*a**2* b**2*d*e*m*x**2 - 6*a**2*b**2*d*e*x**2 + 3*a**2*b**2*e**2*m*x**3 - 6*a*b** 3*d**2*x**2 + a*b**3*d*e*m*x**3 - 6*a*b**3*d*e*x**3 + a*b**3*e**2*m*x**4 - 2*b**4*d**2*x**3 - 2*b**4*d*e*x**4),x)*a**3*b*e**3*m**2*x + 2*int(((d ...