Integrand size = 15, antiderivative size = 52 \[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\frac {b (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (1+n)} \] Output:
b*(b*x+a)^(1+n)*hypergeom([2, 1+n],[2+n],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c) ^2/(1+n)
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\frac {b (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (1+n)} \] Input:
Integrate[(a + b*x)^n/(c + d*x)^2,x]
Output:
(b*(a + b*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, -((d*(a + b*x))/(b *c - a*d))])/((b*c - a*d)^2*(1 + n))
Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
\(\displaystyle \int \frac {(a+b x)^n}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {b (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{(n+1) (b c-a d)^2}\) |
Input:
Int[(a + b*x)^n/(c + d*x)^2,x]
Output:
(b*(a + b*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, -((d*(a + b*x))/(b *c - a*d))])/((b*c - a*d)^2*(1 + n))
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]
\[\int \frac {\left (b x +a \right )^{n}}{\left (x d +c \right )^{2}}d x\]
Input:
int((b*x+a)^n/(d*x+c)^2,x)
Output:
int((b*x+a)^n/(d*x+c)^2,x)
\[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((b*x+a)^n/(d*x+c)^2,x, algorithm="fricas")
Output:
integral((b*x + a)^n/(d^2*x^2 + 2*c*d*x + c^2), x)
\[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\int \frac {\left (a + b x\right )^{n}}{\left (c + d x\right )^{2}}\, dx \] Input:
integrate((b*x+a)**n/(d*x+c)**2,x)
Output:
Integral((a + b*x)**n/(c + d*x)**2, x)
\[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((b*x+a)^n/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate((b*x + a)^n/(d*x + c)^2, x)
\[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((b*x+a)^n/(d*x+c)^2,x, algorithm="giac")
Output:
integrate((b*x + a)^n/(d*x + c)^2, x)
Timed out. \[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + b*x)^n/(c + d*x)^2,x)
Output:
int((a + b*x)^n/(c + d*x)^2, x)
\[ \int \frac {(a+b x)^n}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^n/(d*x+c)^2,x)
Output:
( - (a + b*x)**n*a + int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d** 2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2*c**2 *d*n*x**2 - b**2*c*d**2*n*x**3),x)*a**2*b*c*d**2*n + int(((a + b*x)**n*x)/ (a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2* d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x* *3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a**2*b* d**3*n*x - int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3 *x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a*b**2*c**2*d*n**2 - int(((a + b*x)**n*x)/(a**2*c **2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b* *2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a*b**2*c**2*d* n - int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b* c*d**2*x**2 + a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2* c*d**2*n*x**3),x)*a*b**2*c*d**2*n**2*x - int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b* c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b**2...