Integrand size = 26, antiderivative size = 53 \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^5 (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (6,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^6 (1+m)} \] Output:
e^5*(e*x+d)^(1+m)*hypergeom([6, 1+m],[2+m],b*(e*x+d)/(-a*e+b*d))/(-a*e+b*d )^6/(1+m)
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^5 (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (6,1+m,2+m,-\frac {b (d+e x)}{-b d+a e}\right )}{(-b d+a e)^6 (1+m)} \] Input:
Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(e^5*(d + e*x)^(1 + m)*Hypergeometric2F1[6, 1 + m, 2 + m, -((b*(d + e*x))/ (-(b*d) + a*e))])/((-(b*d) + a*e)^6*(1 + m))
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1098, 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.
| \(\displaystyle \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle b^6 \int \frac {(d+e x)^m}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^m}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {e^5 (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (6,m+1,m+2,\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^6}\) |
Input:
Int[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(e^5*(d + e*x)^(1 + m)*Hypergeometric2F1[6, 1 + m, 2 + m, (b*(d + e*x))/(b *d - a*e)])/((b*d - a*e)^6*(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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
\[\int \frac {\left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}d x\]
Input:
int((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
int((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x)
\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}} \,d x } \] Input:
integrate((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
integral((e*x + d)^m/(b^6*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3* x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6), x)
\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (a + b x\right )^{6}}\, dx \] Input:
integrate((e*x+d)**m/(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
Integral((d + e*x)**m/(a + b*x)**6, x)
\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}} \,d x } \] Input:
integrate((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^3, x)
\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}} \,d x } \] Input:
integrate((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^3, x)
Timed out. \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^3} \,d x \] Input:
int((d + e*x)^m/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
Output:
int((d + e*x)^m/(a^2 + b^2*x^2 + 2*a*b*x)^3, x)
\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {too large to display} \] Input:
int((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
((d + e*x)**m*d + int(((d + e*x)**m*x)/(a**7*d*e*m + a**7*e**2*m*x - 5*a** 6*b*d**2 + 6*a**6*b*d*e*m*x - 5*a**6*b*d*e*x + 6*a**6*b*e**2*m*x**2 - 30*a **5*b**2*d**2*x + 15*a**5*b**2*d*e*m*x**2 - 30*a**5*b**2*d*e*x**2 + 15*a** 5*b**2*e**2*m*x**3 - 75*a**4*b**3*d**2*x**2 + 20*a**4*b**3*d*e*m*x**3 - 75 *a**4*b**3*d*e*x**3 + 20*a**4*b**3*e**2*m*x**4 - 100*a**3*b**4*d**2*x**3 + 15*a**3*b**4*d*e*m*x**4 - 100*a**3*b**4*d*e*x**4 + 15*a**3*b**4*e**2*m*x* *5 - 75*a**2*b**5*d**2*x**4 + 6*a**2*b**5*d*e*m*x**5 - 75*a**2*b**5*d*e*x* *5 + 6*a**2*b**5*e**2*m*x**6 - 30*a*b**6*d**2*x**5 + a*b**6*d*e*m*x**6 - 3 0*a*b**6*d*e*x**6 + a*b**6*e**2*m*x**7 - 5*b**7*d**2*x**6 - 5*b**7*d*e*x** 7),x)*a**7*e**3*m**2 - int(((d + e*x)**m*x)/(a**7*d*e*m + a**7*e**2*m*x - 5*a**6*b*d**2 + 6*a**6*b*d*e*m*x - 5*a**6*b*d*e*x + 6*a**6*b*e**2*m*x**2 - 30*a**5*b**2*d**2*x + 15*a**5*b**2*d*e*m*x**2 - 30*a**5*b**2*d*e*x**2 + 1 5*a**5*b**2*e**2*m*x**3 - 75*a**4*b**3*d**2*x**2 + 20*a**4*b**3*d*e*m*x**3 - 75*a**4*b**3*d*e*x**3 + 20*a**4*b**3*e**2*m*x**4 - 100*a**3*b**4*d**2*x **3 + 15*a**3*b**4*d*e*m*x**4 - 100*a**3*b**4*d*e*x**4 + 15*a**3*b**4*e**2 *m*x**5 - 75*a**2*b**5*d**2*x**4 + 6*a**2*b**5*d*e*m*x**5 - 75*a**2*b**5*d *e*x**5 + 6*a**2*b**5*e**2*m*x**6 - 30*a*b**6*d**2*x**5 + a*b**6*d*e*m*x** 6 - 30*a*b**6*d*e*x**6 + a*b**6*e**2*m*x**7 - 5*b**7*d**2*x**6 - 5*b**7*d* e*x**7),x)*a**6*b*d*e**2*m**2 - 5*int(((d + e*x)**m*x)/(a**7*d*e*m + a**7* e**2*m*x - 5*a**6*b*d**2 + 6*a**6*b*d*e*m*x - 5*a**6*b*d*e*x + 6*a**6*b...