\(\int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx\) [1776]

Optimal result

Integrand size = 26, antiderivative size = 310 \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{3-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {f (6 b d e-b c f (4-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac {(b c-a d)^3 \left (2 a b d f (5 d e-c f (3-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (20 d^2 e^2-10 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (4,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{20 (b e-a f)^6 (d e-c f)^2 (1+m)} \] Output:

-1/5*f*(b*x+a)^(1+m)*(d*x+c)^(3-m)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^5-1/20*f* 
(6*b*d*e-b*c*f*(4-m)-a*d*f*(2+m))*(b*x+a)^(1+m)*(d*x+c)^(3-m)/(-a*f+b*e)^2 
/(-c*f+d*e)^2/(f*x+e)^4-1/20*(-a*d+b*c)^3*(2*a*b*d*f*(5*d*e-c*f*(3-m))*(1+ 
m)-a^2*d^2*f^2*(m^2+3*m+2)-b^2*(20*d^2*e^2-10*c*d*e*f*(3-m)+c^2*f^2*(m^2-7 
*m+12)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)*hypergeom([4, 1+m],[2+m],(-c*f+d*e)* 
(b*x+a)/(-a*f+b*e)/(d*x+c))/(-a*f+b*e)^6/(-c*f+d*e)^2/(1+m)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (-\frac {4 f (c+d x)^4}{(e+f x)^5}-\frac {f (6 b d e+b c f (-4+m)-a d f (2+m)) (c+d x)^4}{(b e-a f) (d e-c f) (e+f x)^4}+\frac {(b c-a d)^3 \left (-2 a b d f (5 d e+c f (-3+m)) (1+m)+a^2 d^2 f^2 \left (2+3 m+m^2\right )+b^2 \left (20 d^2 e^2+10 c d e f (-3+m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (4,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^5 (d e-c f) (1+m)}\right )}{20 (b e-a f) (d e-c f)} \] Input:

Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x]
 

Output:

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*((-4*f*(c + d*x)^4)/(e + f*x)^5 - (f 
*(6*b*d*e + b*c*f*(-4 + m) - a*d*f*(2 + m))*(c + d*x)^4)/((b*e - a*f)*(d*e 
 - c*f)*(e + f*x)^4) + ((b*c - a*d)^3*(-2*a*b*d*f*(5*d*e + c*f*(-3 + m))*( 
1 + m) + a^2*d^2*f^2*(2 + 3*m + m^2) + b^2*(20*d^2*e^2 + 10*c*d*e*f*(-3 + 
m) + c^2*f^2*(12 - 7*m + m^2)))*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - 
 c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^5*(d*e - c*f)*(1 + 
 m))))/(20*(b*e - a*f)*(d*e - c*f))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {114, 25, 168, 27, 141}

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.

\(\displaystyle \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {\int -\frac {(a+b x)^m (c+d x)^{2-m} (b (5 d e-c f (4-m))-a d f (m+2)-b d f x)}{(e+f x)^5}dx}{5 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(a+b x)^m (c+d x)^{2-m} (5 b d e-b c f (4-m)-a d f (m+2)-b d f x)}{(e+f x)^5}dx}{5 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {\int \frac {\left (-\left (\left (20 d^2 e^2-10 c d f (3-m) e+c^2 f^2 \left (m^2-7 m+12\right )\right ) b^2\right )+2 a d f (5 d e-c f (3-m)) (m+1) b-a^2 d^2 f^2 \left (m^2+3 m+2\right )\right ) (a+b x)^m (c+d x)^{2-m}}{(e+f x)^4}dx}{4 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+2)-b c f (4-m)+6 b d e)}{4 (e+f x)^4 (b e-a f) (d e-c f)}}{5 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (5 d e-c f (3-m))-\left (b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )-10 c d e f (3-m)+20 d^2 e^2\right )\right )\right ) \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^4}dx}{4 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+2)-b c f (4-m)+6 b d e)}{4 (e+f x)^4 (b e-a f) (d e-c f)}}{5 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {-\frac {(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (5 d e-c f (3-m))-\left (b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )-10 c d e f (3-m)+20 d^2 e^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (4,m+1,m+2,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{4 (m+1) (b e-a f)^5 (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+2)-b c f (4-m)+6 b d e)}{4 (e+f x)^4 (b e-a f) (d e-c f)}}{5 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)}\)

Input:

Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x]
 

Output:

-1/5*(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/((b*e - a*f)*(d*e - c*f)*(e + 
 f*x)^5) + (-1/4*(f*(6*b*d*e - b*c*f*(4 - m) - a*d*f*(2 + m))*(a + b*x)^(1 
 + m)*(c + d*x)^(3 - m))/((b*e - a*f)*(d*e - c*f)*(e + f*x)^4) - ((b*c - a 
*d)^3*(2*a*b*d*f*(5*d*e - c*f*(3 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^ 
2) - b^2*(20*d^2*e^2 - 10*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)))*(a 
+ b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e 
 - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(4*(b*e - a*f)^5*(d*e - c*f)* 
(1 + m)))/(5*(b*e - a*f)*(d*e - c*f))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (x d +c \right )^{2-m}}{\left (f x +e \right )^{6}}d x\]

Input:

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x)
 

Output:

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x)
 

Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{6}} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x, algorithm="fricas")
 

Output:

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^6*x^6 + 6*e*f^5*x^5 + 15*e^2*f^ 
4*x^4 + 20*e^3*f^3*x^3 + 15*e^4*f^2*x^2 + 6*e^5*f*x + e^6), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**m*(d*x+c)**(2-m)/(f*x+e)**6,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{6}} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x, algorithm="maxima")
 

Output:

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6, x)
 

Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{6}} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x, algorithm="giac")
 

Output:

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m}}{{\left (e+f\,x\right )}^6} \,d x \] Input:

int(((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x)
 

Output:

int(((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6, x)
 

Reduce [F]

\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx=\left (\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{m} e^{6}+6 \left (d x +c \right )^{m} e^{5} f x +15 \left (d x +c \right )^{m} e^{4} f^{2} x^{2}+20 \left (d x +c \right )^{m} e^{3} f^{3} x^{3}+15 \left (d x +c \right )^{m} e^{2} f^{4} x^{4}+6 \left (d x +c \right )^{m} e \,f^{5} x^{5}+\left (d x +c \right )^{m} f^{6} x^{6}}d x \right ) c^{2}+\left (\int \frac {\left (b x +a \right )^{m} x^{2}}{\left (d x +c \right )^{m} e^{6}+6 \left (d x +c \right )^{m} e^{5} f x +15 \left (d x +c \right )^{m} e^{4} f^{2} x^{2}+20 \left (d x +c \right )^{m} e^{3} f^{3} x^{3}+15 \left (d x +c \right )^{m} e^{2} f^{4} x^{4}+6 \left (d x +c \right )^{m} e \,f^{5} x^{5}+\left (d x +c \right )^{m} f^{6} x^{6}}d x \right ) d^{2}+2 \left (\int \frac {\left (b x +a \right )^{m} x}{\left (d x +c \right )^{m} e^{6}+6 \left (d x +c \right )^{m} e^{5} f x +15 \left (d x +c \right )^{m} e^{4} f^{2} x^{2}+20 \left (d x +c \right )^{m} e^{3} f^{3} x^{3}+15 \left (d x +c \right )^{m} e^{2} f^{4} x^{4}+6 \left (d x +c \right )^{m} e \,f^{5} x^{5}+\left (d x +c \right )^{m} f^{6} x^{6}}d x \right ) c d \] Input:

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x)
 

Output:

int((a + b*x)**m/((c + d*x)**m*e**6 + 6*(c + d*x)**m*e**5*f*x + 15*(c + d* 
x)**m*e**4*f**2*x**2 + 20*(c + d*x)**m*e**3*f**3*x**3 + 15*(c + d*x)**m*e* 
*2*f**4*x**4 + 6*(c + d*x)**m*e*f**5*x**5 + (c + d*x)**m*f**6*x**6),x)*c** 
2 + int(((a + b*x)**m*x**2)/((c + d*x)**m*e**6 + 6*(c + d*x)**m*e**5*f*x + 
 15*(c + d*x)**m*e**4*f**2*x**2 + 20*(c + d*x)**m*e**3*f**3*x**3 + 15*(c + 
 d*x)**m*e**2*f**4*x**4 + 6*(c + d*x)**m*e*f**5*x**5 + (c + d*x)**m*f**6*x 
**6),x)*d**2 + 2*int(((a + b*x)**m*x)/((c + d*x)**m*e**6 + 6*(c + d*x)**m* 
e**5*f*x + 15*(c + d*x)**m*e**4*f**2*x**2 + 20*(c + d*x)**m*e**3*f**3*x**3 
 + 15*(c + d*x)**m*e**2*f**4*x**4 + 6*(c + d*x)**m*e*f**5*x**5 + (c + d*x) 
**m*f**6*x**6),x)*c*d