Integrand size = 15, antiderivative size = 51 \[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\frac {d (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^2 (1+n)} \] Output:
d*(d*x+c)^(1+n)*hypergeom([2, 1+n],[2+n],b*(d*x+c)/(-a*d+b*c))/(-a*d+b*c)^ 2/(1+n)
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\frac {d (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,-\frac {b (c+d x)}{-b c+a d}\right )}{(-b c+a d)^2 (1+n)} \] Input:
Integrate[(c + d*x)^n/(a + b*x)^2,x]
Output:
(d*(c + d*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, -((b*(c + d*x))/(- (b*c) + a*d))])/((-(b*c) + a*d)^2*(1 + n))
Time = 0.16 (sec) , antiderivative size = 51, 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.
\(\displaystyle \int \frac {(c+d x)^n}{(a+b x)^2} \, dx\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {d (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^2}\) |
Input:
Int[(c + d*x)^n/(a + b*x)^2,x]
Output:
(d*(c + d*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (b*(c + d*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]
\[\int \frac {\left (x d +c \right )^{n}}{\left (b x +a \right )^{2}}d x\]
Input:
int((d*x+c)^n/(b*x+a)^2,x)
Output:
int((d*x+c)^n/(b*x+a)^2,x)
\[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}} \,d x } \] Input:
integrate((d*x+c)^n/(b*x+a)^2,x, algorithm="fricas")
Output:
integral((d*x + c)^n/(b^2*x^2 + 2*a*b*x + a^2), x)
\[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{n}}{\left (a + b x\right )^{2}}\, dx \] Input:
integrate((d*x+c)**n/(b*x+a)**2,x)
Output:
Integral((c + d*x)**n/(a + b*x)**2, x)
\[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}} \,d x } \] Input:
integrate((d*x+c)^n/(b*x+a)^2,x, algorithm="maxima")
Output:
integrate((d*x + c)^n/(b*x + a)^2, x)
\[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}} \,d x } \] Input:
integrate((d*x+c)^n/(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*x + c)^n/(b*x + a)^2, x)
Timed out. \[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx=\int \frac {{\left (c+d\,x\right )}^n}{{\left (a+b\,x\right )}^2} \,d x \] Input:
int((c + d*x)^n/(a + b*x)^2,x)
Output:
int((c + d*x)^n/(a + b*x)^2, x)
\[ \int \frac {(c+d x)^n}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^n/(b*x+a)^2,x)
Output:
((c + d*x)**n*c + int(((c + d*x)**n*x)/(a**3*c*d*n + a**3*d**2*n*x - a**2* b*c**2 + 2*a**2*b*c*d*n*x - a**2*b*c*d*x + 2*a**2*b*d**2*n*x**2 - 2*a*b**2 *c**2*x + a*b**2*c*d*n*x**2 - 2*a*b**2*c*d*x**2 + a*b**2*d**2*n*x**3 - b** 3*c**2*x**2 - b**3*c*d*x**3),x)*a**3*d**3*n**2 - int(((c + d*x)**n*x)/(a** 3*c*d*n + a**3*d**2*n*x - a**2*b*c**2 + 2*a**2*b*c*d*n*x - a**2*b*c*d*x + 2*a**2*b*d**2*n*x**2 - 2*a*b**2*c**2*x + a*b**2*c*d*n*x**2 - 2*a*b**2*c*d* x**2 + a*b**2*d**2*n*x**3 - b**3*c**2*x**2 - b**3*c*d*x**3),x)*a**2*b*c*d* *2*n**2 - int(((c + d*x)**n*x)/(a**3*c*d*n + a**3*d**2*n*x - a**2*b*c**2 + 2*a**2*b*c*d*n*x - a**2*b*c*d*x + 2*a**2*b*d**2*n*x**2 - 2*a*b**2*c**2*x + a*b**2*c*d*n*x**2 - 2*a*b**2*c*d*x**2 + a*b**2*d**2*n*x**3 - b**3*c**2*x **2 - b**3*c*d*x**3),x)*a**2*b*c*d**2*n + int(((c + d*x)**n*x)/(a**3*c*d*n + a**3*d**2*n*x - a**2*b*c**2 + 2*a**2*b*c*d*n*x - a**2*b*c*d*x + 2*a**2* b*d**2*n*x**2 - 2*a*b**2*c**2*x + a*b**2*c*d*n*x**2 - 2*a*b**2*c*d*x**2 + a*b**2*d**2*n*x**3 - b**3*c**2*x**2 - b**3*c*d*x**3),x)*a**2*b*d**3*n**2*x + int(((c + d*x)**n*x)/(a**3*c*d*n + a**3*d**2*n*x - a**2*b*c**2 + 2*a**2 *b*c*d*n*x - a**2*b*c*d*x + 2*a**2*b*d**2*n*x**2 - 2*a*b**2*c**2*x + a*b** 2*c*d*n*x**2 - 2*a*b**2*c*d*x**2 + a*b**2*d**2*n*x**3 - b**3*c**2*x**2 - b **3*c*d*x**3),x)*a*b**2*c**2*d*n - int(((c + d*x)**n*x)/(a**3*c*d*n + a**3 *d**2*n*x - a**2*b*c**2 + 2*a**2*b*c*d*n*x - a**2*b*c*d*x + 2*a**2*b*d**2* n*x**2 - 2*a*b**2*c**2*x + a*b**2*c*d*n*x**2 - 2*a*b**2*c*d*x**2 + a*b*...