\(\int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx\) [236]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=-\frac {B (d+e x)^{1+m}}{b e (1-m) (a+b x)^2}+\frac {e (2 b B d-A b e (1-m)-a B e (1+m)) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e)^3 (1-m) (1+m)} \] Output:

-B*(e*x+d)^(1+m)/b/e/(1-m)/(b*x+a)^2+e*(2*B*b*d-A*b*e*(1-m)-a*B*e*(1+m))*( 
e*x+d)^(1+m)*hypergeom([3, 1+m],[2+m],b*(e*x+d)/(-a*e+b*d))/b/(-a*e+b*d)^3 
/(1-m)/(1+m)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=\frac {(d+e x)^{1+m} \left (\frac {-A b+a B}{(a+b x)^2}+\frac {e (2 b B d+A b e (-1+m)-a B e (1+m)) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2 (1+m)}\right )}{2 b (b d-a e)} \] Input:

Integrate[((A + B*x)*(d + e*x)^m)/(a + b*x)^3,x]
 

Output:

((d + e*x)^(1 + m)*((-(A*b) + a*B)/(a + b*x)^2 + (e*(2*b*B*d + A*b*e*(-1 + 
 m) - a*B*e*(1 + m))*Hypergeometric2F1[2, 1 + m, 2 + m, (b*(d + e*x))/(b*d 
 - a*e)])/((b*d - a*e)^2*(1 + m))))/(2*b*(b*d - a*e))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {87, 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.

\(\displaystyle \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-a B e (m+1)-A b e (1-m)+2 b B d) \int \frac {(d+e x)^m}{(a+b x)^2}dx}{2 b (b d-a e)}-\frac {(A b-a B) (d+e x)^{m+1}}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {e (d+e x)^{m+1} (-a B e (m+1)-A b e (1-m)+2 b B d) \operatorname {Hypergeometric2F1}\left (2,m+1,m+2,\frac {b (d+e x)}{b d-a e}\right )}{2 b (m+1) (b d-a e)^3}-\frac {(A b-a B) (d+e x)^{m+1}}{2 b (a+b x)^2 (b d-a e)}\)

Input:

Int[((A + B*x)*(d + e*x)^m)/(a + b*x)^3,x]
 

Output:

-1/2*((A*b - a*B)*(d + e*x)^(1 + m))/(b*(b*d - a*e)*(a + b*x)^2) + (e*(2*b 
*B*d - A*b*e*(1 - m) - a*B*e*(1 + m))*(d + e*x)^(1 + m)*Hypergeometric2F1[ 
2, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(2*b*(b*d - a*e)^3*(1 + m))
 

Defintions of rubi rules used

rule 78

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]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [F]

\[\int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{\left (b x +a \right )^{3}}d x\]

Input:

int((B*x+A)*(e*x+d)^m/(b*x+a)^3,x)
 

Output:

int((B*x+A)*(e*x+d)^m/(b*x+a)^3,x)
 

Fricas [F]

\[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (b x + a\right )}^{3}} \,d x } \] Input:

integrate((B*x+A)*(e*x+d)^m/(b*x+a)^3,x, algorithm="fricas")
 

Output:

integral((B*x + A)*(e*x + d)^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), 
x)
 

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{m}}{\left (a + b x\right )^{3}}\, dx \] Input:

integrate((B*x+A)*(e*x+d)**m/(b*x+a)**3,x)
 

Output:

Integral((A + B*x)*(d + e*x)**m/(a + b*x)**3, x)
 

Maxima [F]

\[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (b x + a\right )}^{3}} \,d x } \] Input:

integrate((B*x+A)*(e*x+d)^m/(b*x+a)^3,x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(e*x + d)^m/(b*x + a)^3, x)
 

Giac [F]

\[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (b x + a\right )}^{3}} \,d x } \] Input:

integrate((B*x+A)*(e*x+d)^m/(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate((B*x + A)*(e*x + d)^m/(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (a+b\,x\right )}^3} \,d x \] Input:

int(((A + B*x)*(d + e*x)^m)/(a + b*x)^3,x)
 

Output:

int(((A + B*x)*(d + e*x)^m)/(a + b*x)^3, x)
 

Reduce [F]

\[ \int \frac {(A+B x) (d+e x)^m}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)^m/(b*x+a)^3,x)
 

Output:

((d + e*x)**m*d + int(((d + e*x)**m*x)/(a**3*d*e*m + a**3*e**2*m*x - a**2* 
b*d**2 + 2*a**2*b*d*e*m*x - a**2*b*d*e*x + 2*a**2*b*e**2*m*x**2 - 2*a*b**2 
*d**2*x + a*b**2*d*e*m*x**2 - 2*a*b**2*d*e*x**2 + a*b**2*e**2*m*x**3 - b** 
3*d**2*x**2 - b**3*d*e*x**3),x)*a**3*e**3*m**2 - int(((d + e*x)**m*x)/(a** 
3*d*e*m + a**3*e**2*m*x - a**2*b*d**2 + 2*a**2*b*d*e*m*x - a**2*b*d*e*x + 
2*a**2*b*e**2*m*x**2 - 2*a*b**2*d**2*x + a*b**2*d*e*m*x**2 - 2*a*b**2*d*e* 
x**2 + a*b**2*e**2*m*x**3 - b**3*d**2*x**2 - b**3*d*e*x**3),x)*a**2*b*d*e* 
*2*m**2 - int(((d + e*x)**m*x)/(a**3*d*e*m + a**3*e**2*m*x - a**2*b*d**2 + 
 2*a**2*b*d*e*m*x - a**2*b*d*e*x + 2*a**2*b*e**2*m*x**2 - 2*a*b**2*d**2*x 
+ a*b**2*d*e*m*x**2 - 2*a*b**2*d*e*x**2 + a*b**2*e**2*m*x**3 - b**3*d**2*x 
**2 - b**3*d*e*x**3),x)*a**2*b*d*e**2*m + int(((d + e*x)**m*x)/(a**3*d*e*m 
 + a**3*e**2*m*x - a**2*b*d**2 + 2*a**2*b*d*e*m*x - a**2*b*d*e*x + 2*a**2* 
b*e**2*m*x**2 - 2*a*b**2*d**2*x + a*b**2*d*e*m*x**2 - 2*a*b**2*d*e*x**2 + 
a*b**2*e**2*m*x**3 - b**3*d**2*x**2 - b**3*d*e*x**3),x)*a**2*b*e**3*m**2*x 
 + int(((d + e*x)**m*x)/(a**3*d*e*m + a**3*e**2*m*x - a**2*b*d**2 + 2*a**2 
*b*d*e*m*x - a**2*b*d*e*x + 2*a**2*b*e**2*m*x**2 - 2*a*b**2*d**2*x + a*b** 
2*d*e*m*x**2 - 2*a*b**2*d*e*x**2 + a*b**2*e**2*m*x**3 - b**3*d**2*x**2 - b 
**3*d*e*x**3),x)*a*b**2*d**2*e*m - int(((d + e*x)**m*x)/(a**3*d*e*m + a**3 
*e**2*m*x - a**2*b*d**2 + 2*a**2*b*d*e*m*x - a**2*b*d*e*x + 2*a**2*b*e**2* 
m*x**2 - 2*a*b**2*d**2*x + a*b**2*d*e*m*x**2 - 2*a*b**2*d*e*x**2 + a*b*...