Integrand size = 35, antiderivative size = 86 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2 p,p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1-p) (d+e x)^2} \] Output:
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)*hypergeom([1, 2*p],[p],c*d*(e*x+d) /(-a*e^2+c*d^2))/(-a*e^2+c*d^2)/(1-p)/(e*x+d)^2
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\frac {c d (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((a e+c d x) (d+e x))^p \operatorname {Hypergeometric2F1}\left (2-p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{\left (c d^2-a e^2\right )^2 (1+p)} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x)^2,x]
Output:
(c*d*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^2*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.40, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1138, 80, 79}
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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1138 |
| \(\displaystyle \frac {\left (\frac {e x}{d}+1\right )^{-p} (a e+c d x)^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \int (a e+c d x)^p \left (\frac {e x}{d}+1\right )^{p-2}dx}{d^2}\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {c^2 d^2 (a e+c d x)^{-p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \int (a e+c d x)^p \left (\frac {c d^2}{c d^2-a e^2}+\frac {c e x d}{c d^2-a e^2}\right )^{p-2}dx}{\left (c d^2-a e^2\right )^2}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {c d (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (2-p,p+1,p+2,-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )^2}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x)^2,x]
Output:
(c*d*(a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hypergeometri c2F1[2 - p, 1 + p, 2 + p, -((e*(a*e + c*d*x))/(c*d^2 - a*e^2))])/((c*d^2 - a*e^2)^2*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] *(a/d + (c*x)/e)^FracPart[p])) Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[m] || GtQ[d, 0]) && !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer Q[4*p]))
\[\int \frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{p}}{\left (e x +d \right )^{2}}d x\]
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^p/(e*x+d)^2,x)
Output:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^p/(e*x+d)^2,x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p/(e*x+d)^2,x, algorithm="fric as")
Output:
integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p/(e^2*x^2 + 2*d*e*x + d^ 2), x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\text {Timed out} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p/(e*x+d)**2,x)
Output:
Timed out
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p/(e*x+d)^2,x, algorithm="maxi ma")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p/(e*x + d)^2, x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p/(e*x+d)^2,x, algorithm="giac ")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p/(e*x + d)^2, x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^2,x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^2, x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx=\text {too large to display} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p/(e*x+d)^2,x)
Output:
((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*e - int(((a*d*e + a*e**2* x + c*d**2*x + c*d*e*x**2)**p*x)/(a**2*d**2*e**3*p - a**2*d**2*e**3 + 2*a* *2*d*e**4*p*x - 2*a**2*d*e**4*x + a**2*e**5*p*x**2 - a**2*e**5*x**2 + a*c* d**4*e*p + 3*a*c*d**3*e**2*p*x - a*c*d**3*e**2*x + 3*a*c*d**2*e**3*p*x**2 - 2*a*c*d**2*e**3*x**2 + a*c*d*e**4*p*x**3 - a*c*d*e**4*x**3 + c**2*d**5*p *x + 2*c**2*d**4*e*p*x**2 + c**2*d**3*e**2*p*x**3),x)*a**2*c*d**2*e**4*p** 2 + int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*x)/(a**2*d**2*e**3* p - a**2*d**2*e**3 + 2*a**2*d*e**4*p*x - 2*a**2*d*e**4*x + a**2*e**5*p*x** 2 - a**2*e**5*x**2 + a*c*d**4*e*p + 3*a*c*d**3*e**2*p*x - a*c*d**3*e**2*x + 3*a*c*d**2*e**3*p*x**2 - 2*a*c*d**2*e**3*x**2 + a*c*d*e**4*p*x**3 - a*c* d*e**4*x**3 + c**2*d**5*p*x + 2*c**2*d**4*e*p*x**2 + c**2*d**3*e**2*p*x**3 ),x)*a**2*c*d**2*e**4*p - int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)* *p*x)/(a**2*d**2*e**3*p - a**2*d**2*e**3 + 2*a**2*d*e**4*p*x - 2*a**2*d*e* *4*x + a**2*e**5*p*x**2 - a**2*e**5*x**2 + a*c*d**4*e*p + 3*a*c*d**3*e**2* p*x - a*c*d**3*e**2*x + 3*a*c*d**2*e**3*p*x**2 - 2*a*c*d**2*e**3*x**2 + a* c*d*e**4*p*x**3 - a*c*d*e**4*x**3 + c**2*d**5*p*x + 2*c**2*d**4*e*p*x**2 + c**2*d**3*e**2*p*x**3),x)*a**2*c*d*e**5*p**2*x + int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*x)/(a**2*d**2*e**3*p - a**2*d**2*e**3 + 2*a**2* d*e**4*p*x - 2*a**2*d*e**4*x + a**2*e**5*p*x**2 - a**2*e**5*x**2 + a*c*d** 4*e*p + 3*a*c*d**3*e**2*p*x - a*c*d**3*e**2*x + 3*a*c*d**2*e**3*p*x**2 ...