Integrand size = 15, antiderivative size = 54 \[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\frac {b^2 (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)} \] Output:
b^2*(b*x+a)^(1+m)*hypergeom([3, 1+m],[2+m],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b* c)^3/(1+m)
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\frac {b^2 (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)} \] Input:
Integrate[(a + b*x)^m/(c + d*x)^3,x]
Output:
(b^2*(a + b*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(a + b*x))/ (b*c - a*d))])/((b*c - a*d)^3*(1 + m))
Time = 0.16 (sec) , antiderivative size = 54, 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.
\(\displaystyle \int \frac {(a+b x)^m}{(c+d x)^3} \, dx\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {b^2 (a+b x)^{m+1} \operatorname {Hypergeometric2F1}\left (3,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3}\) |
Input:
Int[(a + b*x)^m/(c + d*x)^3,x]
Output:
(b^2*(a + b*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(a + b*x))/ (b*c - a*d))])/((b*c - a*d)^3*(1 + m))
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]
\[\int \frac {\left (b x +a \right )^{m}}{\left (x d +c \right )^{3}}d x\]
Input:
int((b*x+a)^m/(d*x+c)^3,x)
Output:
int((b*x+a)^m/(d*x+c)^3,x)
\[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:
integrate((b*x+a)^m/(d*x+c)^3,x, algorithm="fricas")
Output:
integral((b*x + a)^m/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)
\[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\int \frac {\left (a + b x\right )^{m}}{\left (c + d x\right )^{3}}\, dx \] Input:
integrate((b*x+a)**m/(d*x+c)**3,x)
Output:
Integral((a + b*x)**m/(c + d*x)**3, x)
\[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:
integrate((b*x+a)^m/(d*x+c)^3,x, algorithm="maxima")
Output:
integrate((b*x + a)^m/(d*x + c)^3, x)
\[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:
integrate((b*x+a)^m/(d*x+c)^3,x, algorithm="giac")
Output:
integrate((b*x + a)^m/(d*x + c)^3, x)
Timed out. \[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a + b*x)^m/(c + d*x)^3,x)
Output:
int((a + b*x)^m/(c + d*x)^3, x)
\[ \int \frac {(a+b x)^m}{(c+d x)^3} \, dx=\text {too large to display} \] Input:
int((b*x+a)^m/(d*x+c)^3,x)
Output:
( - (a + b*x)**m*a + 2*int(((a + b*x)**m*x)/(2*a**2*c**3*d + 6*a**2*c**2*d **2*x + 6*a**2*c*d**3*x**2 + 2*a**2*d**4*x**3 - a*b*c**4*m - 3*a*b*c**3*d* m*x + 2*a*b*c**3*d*x - 3*a*b*c**2*d**2*m*x**2 + 6*a*b*c**2*d**2*x**2 - a*b *c*d**3*m*x**3 + 6*a*b*c*d**3*x**3 + 2*a*b*d**4*x**4 - b**2*c**4*m*x - 3*b **2*c**3*d*m*x**2 - 3*b**2*c**2*d**2*m*x**3 - b**2*c*d**3*m*x**4),x)*a**2* b*c**2*d**2*m + 4*int(((a + b*x)**m*x)/(2*a**2*c**3*d + 6*a**2*c**2*d**2*x + 6*a**2*c*d**3*x**2 + 2*a**2*d**4*x**3 - a*b*c**4*m - 3*a*b*c**3*d*m*x + 2*a*b*c**3*d*x - 3*a*b*c**2*d**2*m*x**2 + 6*a*b*c**2*d**2*x**2 - a*b*c*d* *3*m*x**3 + 6*a*b*c*d**3*x**3 + 2*a*b*d**4*x**4 - b**2*c**4*m*x - 3*b**2*c **3*d*m*x**2 - 3*b**2*c**2*d**2*m*x**3 - b**2*c*d**3*m*x**4),x)*a**2*b*c*d **3*m*x + 2*int(((a + b*x)**m*x)/(2*a**2*c**3*d + 6*a**2*c**2*d**2*x + 6*a **2*c*d**3*x**2 + 2*a**2*d**4*x**3 - a*b*c**4*m - 3*a*b*c**3*d*m*x + 2*a*b *c**3*d*x - 3*a*b*c**2*d**2*m*x**2 + 6*a*b*c**2*d**2*x**2 - a*b*c*d**3*m*x **3 + 6*a*b*c*d**3*x**3 + 2*a*b*d**4*x**4 - b**2*c**4*m*x - 3*b**2*c**3*d* m*x**2 - 3*b**2*c**2*d**2*m*x**3 - b**2*c*d**3*m*x**4),x)*a**2*b*d**4*m*x* *2 - int(((a + b*x)**m*x)/(2*a**2*c**3*d + 6*a**2*c**2*d**2*x + 6*a**2*c*d **3*x**2 + 2*a**2*d**4*x**3 - a*b*c**4*m - 3*a*b*c**3*d*m*x + 2*a*b*c**3*d *x - 3*a*b*c**2*d**2*m*x**2 + 6*a*b*c**2*d**2*x**2 - a*b*c*d**3*m*x**3 + 6 *a*b*c*d**3*x**3 + 2*a*b*d**4*x**4 - b**2*c**4*m*x - 3*b**2*c**3*d*m*x**2 - 3*b**2*c**2*d**2*m*x**3 - b**2*c*d**3*m*x**4),x)*a*b**2*c**3*d*m**2 -...