\(\int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx\) [1754]

Optimal result

Integrand size = 24, antiderivative size = 264 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-4-m}}{d (b c-a d) (4+m)}+\frac {(3 b d e+b c f (1+m)-a d f (4+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{d (b c-a d)^2 (3+m) (4+m)}+\frac {2 b (3 b d e+b c f (1+m)-a d f (4+m)) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {2 b^2 (3 b d e+b c f (1+m)-a d f (4+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)} \] Output:

(-c*f+d*e)*(b*x+a)^(1+m)*(d*x+c)^(-4-m)/d/(-a*d+b*c)/(4+m)+(3*b*d*e+b*c*f* 
(1+m)-a*d*f*(4+m))*(b*x+a)^(1+m)*(d*x+c)^(-3-m)/d/(-a*d+b*c)^2/(3+m)/(4+m) 
+2*b*(3*b*d*e+b*c*f*(1+m)-a*d*f*(4+m))*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/d/(-a* 
d+b*c)^3/(2+m)/(3+m)/(4+m)+2*b^2*(3*b*d*e+b*c*f*(1+m)-a*d*f*(4+m))*(b*x+a) 
^(1+m)*(d*x+c)^(-1-m)/d/(-a*d+b*c)^4/(1+m)/(2+m)/(3+m)/(4+m)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.63 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left (d e-c f+\frac {(3 b d e+b c f (1+m)-a d f (4+m)) (c+d x) \left (a^2 d^2 \left (2+3 m+m^2\right )-2 a b d (1+m) (c (3+m)+d x)+b^2 \left (c^2 \left (6+5 m+m^2\right )+2 c d (3+m) x+2 d^2 x^2\right )\right )}{(b c-a d)^3 (1+m) (2+m) (3+m)}\right )}{d (-b c+a d) (4+m)} \] Input:

Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x),x]
 

Output:

-(((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(d*e - c*f + ((3*b*d*e + b*c*f*(1 
+ m) - a*d*f*(4 + m))*(c + d*x)*(a^2*d^2*(2 + 3*m + m^2) - 2*a*b*d*(1 + m) 
*(c*(3 + m) + d*x) + b^2*(c^2*(6 + 5*m + m^2) + 2*c*d*(3 + m)*x + 2*d^2*x^ 
2)))/((b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m))))/(d*(-(b*c) + a*d)*(4 + m)))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 55, 55, 48}

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.

\(\displaystyle \int (e+f x) (a+b x)^m (c+d x)^{-m-5} \, dx\)

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x),x]
 

Output:

((d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d*(b*c - a*d)*(4 + m)) 
 + ((3*b*d*e + b*c*f*(1 + m) - a*d*f*(4 + m))*(((a + b*x)^(1 + m)*(c + d*x 
)^(-3 - m))/((b*c - a*d)*(3 + m)) + (2*b*(((a + b*x)^(1 + m)*(c + d*x)^(-2 
 - m))/((b*c - a*d)*(2 + m)) + (b*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(( 
b*c - a*d)^2*(1 + m)*(2 + m))))/((b*c - a*d)*(3 + m))))/(d*(b*c - a*d)*(4 
+ m))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 88

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)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1183\) vs. \(2(264)=528\).

Time = 0.85 (sec) , antiderivative size = 1184, normalized size of antiderivative = 4.48

method	result	size
gosper	\(\text {Expression too large to display}\)	\(1184\)
orering	\(\text {Expression too large to display}\)	\(1192\)
parallelrisch	\(\text {Expression too large to display}\)	\(4641\)

Input:

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e),x,method=_RETURNVERBOSE)
 

Output:

-(b*x+a)^(1+m)*(d*x+c)^(-4-m)/(a^4*d^4*m^4-4*a^3*b*c*d^3*m^4+6*a^2*b^2*c^2 
*d^2*m^4-4*a*b^3*c^3*d*m^4+b^4*c^4*m^4+10*a^4*d^4*m^3-40*a^3*b*c*d^3*m^3+6 
0*a^2*b^2*c^2*d^2*m^3-40*a*b^3*c^3*d*m^3+10*b^4*c^4*m^3+35*a^4*d^4*m^2-140 
*a^3*b*c*d^3*m^2+210*a^2*b^2*c^2*d^2*m^2-140*a*b^3*c^3*d*m^2+35*b^4*c^4*m^ 
2+50*a^4*d^4*m-200*a^3*b*c*d^3*m+300*a^2*b^2*c^2*d^2*m-200*a*b^3*c^3*d*m+5 
0*b^4*c^4*m+24*a^4*d^4-96*a^3*b*c*d^3+144*a^2*b^2*c^2*d^2-96*a*b^3*c^3*d+2 
4*b^4*c^4)*(a^3*d^3*f*m^3*x-3*a^2*b*c*d^2*f*m^3*x-2*a^2*b*d^3*f*m^2*x^2+3* 
a*b^2*c^2*d*f*m^3*x+4*a*b^2*c*d^2*f*m^2*x^2+2*a*b^2*d^3*f*m*x^3-b^3*c^3*f* 
m^3*x-2*b^3*c^2*d*f*m^2*x^2-2*b^3*c*d^2*f*m*x^3+a^3*d^3*e*m^3+7*a^3*d^3*f* 
m^2*x-3*a^2*b*c*d^2*e*m^3-22*a^2*b*c*d^2*f*m^2*x-3*a^2*b*d^3*e*m^2*x-10*a^ 
2*b*d^3*f*m*x^2+3*a*b^2*c^2*d*e*m^3+23*a*b^2*c^2*d*f*m^2*x+6*a*b^2*c*d^2*e 
*m^2*x+20*a*b^2*c*d^2*f*m*x^2+6*a*b^2*d^3*e*m*x^2+8*a*b^2*d^3*f*x^3-b^3*c^ 
3*e*m^3-8*b^3*c^3*f*m^2*x-3*b^3*c^2*d*e*m^2*x-10*b^3*c^2*d*f*m*x^2-6*b^3*c 
*d^2*e*m*x^2-2*b^3*c*d^2*f*x^3-6*b^3*d^3*e*x^3+a^3*c*d^2*f*m^2+6*a^3*d^3*e 
*m^2+14*a^3*d^3*f*m*x-2*a^2*b*c^2*d*f*m^2-21*a^2*b*c*d^2*e*m^2-53*a^2*b*c* 
d^2*f*m*x-9*a^2*b*d^3*e*m*x-8*a^2*b*d^3*f*x^2+a*b^2*c^3*f*m^2+24*a*b^2*c^2 
*d*e*m^2+58*a*b^2*c^2*d*f*m*x+30*a*b^2*c*d^2*e*m*x+34*a*b^2*c*d^2*f*x^2+6* 
a*b^2*d^3*e*x^2-9*b^3*c^3*e*m^2-19*b^3*c^3*f*m*x-21*b^3*c^2*d*e*m*x-8*b^3* 
c^2*d*f*x^2-24*b^3*c*d^2*e*x^2+3*a^3*c*d^2*f*m+11*a^3*d^3*e*m+8*a^3*d^3*f* 
x-10*a^2*b*c^2*d*f*m-42*a^2*b*c*d^2*e*m-34*a^2*b*c*d^2*f*x-6*a^2*b*d^3*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1777 vs. \(2 (265) = 530\).

Time = 0.19 (sec) , antiderivative size = 1777, normalized size of antiderivative = 6.73 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e),x, algorithm="fricas")
 

Output:

(2*(3*b^4*d^4*e + (b^4*c*d^3 - a*b^3*d^4)*f*m + (b^4*c*d^3 - 4*a*b^3*d^4)* 
f)*x^5 + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*e*m^3 
 + 2*(15*b^4*c*d^3*e + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f*m^2 + 
 5*(b^4*c^2*d^2 - 4*a*b^3*c*d^3)*f + (3*(b^4*c*d^3 - a*b^3*d^4)*e + 2*(3*b 
^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*f)*m)*x^4 + (60*b^4*c^2*d^2*e 
+ (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*f*m^3 + (3*( 
b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e + 5*(2*b^4*c^3*d - 5*a*b^3*c^ 
2*d^2 + 4*a^2*b^2*c*d^3 - a^3*b*d^4)*f)*m^2 + 20*(b^4*c^3*d - 4*a*b^3*c^2* 
d^2)*f + (3*(9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*e + (29*b^4*c^3 
*d - 66*a*b^3*c^2*d^2 + 41*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*f)*m)*x^3 + (3*(3* 
a*b^3*c^4 - 8*a^2*b^2*c^3*d + 7*a^3*b*c^2*d^2 - 2*a^4*c*d^3)*e - (a^2*b^2* 
c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f)*m^2 + (60*b^4*c^3*d*e + ((b^4*c^3*d 
- 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*e + (b^4*c^4 - 2*a*b^3*c^ 
3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f)*m^3 + (3*(4*b^4*c^3*d - 9*a*b^3*c^2*d^2 
+ 6*a^2*b^2*c*d^3 - a^3*b*d^4)*e + (8*b^4*c^4 - 14*a*b^3*c^3*d - 3*a^2*b^2 
*c^2*d^2 + 16*a^3*b*c*d^3 - 7*a^4*d^4)*f)*m^2 + 4*(3*b^4*c^4 - 12*a*b^3*c^ 
3*d - 12*a^2*b^2*c^2*d^2 + 8*a^3*b*c*d^3 - 2*a^4*d^4)*f + ((47*b^4*c^3*d - 
 60*a*b^3*c^2*d^2 + 15*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*e + (19*b^4*c^4 - 36*a 
*b^3*c^3*d - 15*a^2*b^2*c^2*d^2 + 46*a^3*b*c*d^3 - 14*a^4*d^4)*f)*m)*x^2 + 
 6*(4*a*b^3*c^4 - 6*a^2*b^2*c^3*d + 4*a^3*b*c^2*d^2 - a^4*c*d^3)*e - 2*...
 

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**m*(d*x+c)**(-5-m)*(f*x+e),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\int { {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e),x, algorithm="maxima")
 

Output:

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5), x)
 

Giac [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\int { {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e),x, algorithm="giac")
 

Output:

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5), x)
 

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 1658, normalized size of antiderivative = 6.28 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\text {Too large to display} \] Input:

int(((e + f*x)*(a + b*x)^m)/(c + d*x)^(m + 5),x)
 

Output:

(2*b^3*d^3*x^5*(a + b*x)^m*(b*c*f - 4*a*d*f + 3*b*d*e - a*d*f*m + b*c*f*m) 
)/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - 
(x^2*(a + b*x)^m*(8*a^4*d^4*f - 12*b^4*c^4*f + 7*a^4*d^4*f*m^2 - 8*b^4*c^4 
*f*m^2 + a^4*d^4*f*m^3 - b^4*c^4*f*m^3 - 60*b^4*c^3*d*e + 14*a^4*d^4*f*m - 
 19*b^4*c^4*f*m + 48*a*b^3*c^3*d*f - 32*a^3*b*c*d^3*f + 2*a^3*b*d^4*e*m - 
47*b^4*c^3*d*e*m + 3*a^3*b*d^4*e*m^2 + a^3*b*d^4*e*m^3 - 12*b^4*c^3*d*e*m^ 
2 - b^4*c^3*d*e*m^3 + 48*a^2*b^2*c^2*d^2*f + 27*a*b^3*c^2*d^2*e*m^2 - 18*a 
^2*b^2*c*d^3*e*m^2 + 3*a*b^3*c^2*d^2*e*m^3 - 3*a^2*b^2*c*d^3*e*m^3 + 15*a^ 
2*b^2*c^2*d^2*f*m + 36*a*b^3*c^3*d*f*m - 46*a^3*b*c*d^3*f*m + 3*a^2*b^2*c^ 
2*d^2*f*m^2 + 60*a*b^3*c^2*d^2*e*m - 15*a^2*b^2*c*d^3*e*m + 14*a*b^3*c^3*d 
*f*m^2 - 16*a^3*b*c*d^3*f*m^2 + 2*a*b^3*c^3*d*f*m^3 - 2*a^3*b*c*d^3*f*m^3) 
)/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - 
(x*(a + b*x)^m*(6*a^4*d^4*e - 24*b^4*c^4*e + 6*a^4*d^4*e*m^2 - 9*b^4*c^4*e 
*m^2 + a^4*d^4*e*m^3 - b^4*c^4*e*m^3 + 10*a^4*c*d^3*f + 11*a^4*d^4*e*m - 2 
6*b^4*c^4*e*m - 24*a*b^3*c^3*d*e - 24*a^3*b*c*d^3*e - 12*a*b^3*c^4*f*m + 1 
7*a^4*c*d^3*f*m + 60*a^2*b^2*c^3*d*f - 40*a^3*b*c^2*d^2*f - 7*a*b^3*c^4*f* 
m^2 - a*b^3*c^4*f*m^3 + 8*a^4*c*d^3*f*m^2 + a^4*c*d^3*f*m^3 + 36*a^2*b^2*c 
^2*d^2*e + 45*a^2*b^2*c^2*d^2*e*m + 22*a^2*b^2*c^3*d*f*m^2 - 23*a^3*b*c^2* 
d^2*f*m^2 + 3*a^2*b^2*c^3*d*f*m^3 - 3*a^3*b*c^2*d^2*f*m^3 + 10*a*b^3*c^3*d 
*e*m - 40*a^3*b*c*d^3*e*m + 9*a^2*b^2*c^2*d^2*e*m^2 + 12*a*b^3*c^3*d*e*...
 

Reduce [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx=\left (\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) e +\left (\int \frac {\left (b x +a \right )^{m} x}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) f \] Input:

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e),x)
 

Output:

int((a + b*x)**m/((c + d*x)**m*c**5 + 5*(c + d*x)**m*c**4*d*x + 10*(c + d* 
x)**m*c**3*d**2*x**2 + 10*(c + d*x)**m*c**2*d**3*x**3 + 5*(c + d*x)**m*c*d 
**4*x**4 + (c + d*x)**m*d**5*x**5),x)*e + int(((a + b*x)**m*x)/((c + d*x)* 
*m*c**5 + 5*(c + d*x)**m*c**4*d*x + 10*(c + d*x)**m*c**3*d**2*x**2 + 10*(c 
 + d*x)**m*c**2*d**3*x**3 + 5*(c + d*x)**m*c*d**4*x**4 + (c + d*x)**m*d**5 
*x**5),x)*f