3.3082 \(\int (a+b x)^m (c+d x)^{-4-m} (e+f x) \, dx\)

Optimal. Leaf size=188 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b (c f (m+1)+2 d e))}{d (m+2) (m+3) (b c-a d)^2}-\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+3)-b (c f (m+1)+2 d e))}{d (m+1) (m+2) (m+3) (b c-a d)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.304577, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}+\frac{(a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+2) (m+3) (b c-a d)^2}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+1) (m+2) (m+3) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.252, size = 156, normalized size = 0.83 \[ \frac{b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- 2 b d e + f \left (a d \left (m + 3\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{3}} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 3} \left (c f - d e\right )}{d \left (m + 3\right ) \left (a d - b c\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (- 2 b d e + f \left (a d \left (m + 3\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.615199, size = 199, normalized size = 1.06 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{b^2 (-a d f (m+3)+b c f (m+1)+2 b d e)}{(m+1) (m+2) (m+3) (b c-a d)^3}+\frac{b m (-a d f (m+3)+b c f (m+1)+2 b d e)}{(m+1) \left (m^2+5 m+6\right ) (c+d x) (b c-a d)^2}+\frac{a d f (m+3)-b c f (2 m+3)+b d e m}{(m+2) (m+3) (c+d x)^2 (b c-a d)}+\frac{c f-d e}{(m+3) (c+d x)^3}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x),x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 503, normalized size = 2.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-3-m} \left ({a}^{2}{d}^{2}f{m}^{2}x-2\,abcdf{m}^{2}x-ab{d}^{2}fm{x}^{2}+{b}^{2}{c}^{2}f{m}^{2}x+{b}^{2}cdfm{x}^{2}+{a}^{2}{d}^{2}e{m}^{2}+4\,{a}^{2}{d}^{2}fmx-2\,abcde{m}^{2}-8\,abcdfmx-2\,ab{d}^{2}emx-3\,ab{d}^{2}f{x}^{2}+{b}^{2}{c}^{2}e{m}^{2}+4\,{b}^{2}{c}^{2}fmx+2\,{b}^{2}cdemx+{b}^{2}cdf{x}^{2}+2\,{b}^{2}{d}^{2}e{x}^{2}+{a}^{2}cdfm+3\,{a}^{2}{d}^{2}em+3\,{a}^{2}{d}^{2}fx-ab{c}^{2}fm-8\,abcdem-10\,abcdfx-2\,ab{d}^{2}ex+5\,{b}^{2}{c}^{2}em+3\,{b}^{2}{c}^{2}fx+6\,{b}^{2}cdex+{a}^{2}cdf+2\,{a}^{2}{d}^{2}e-3\,ab{c}^{2}f-6\,abcde+6\,{b}^{2}{c}^{2}e \right ) }{{a}^{3}{d}^{3}{m}^{3}-3\,{a}^{2}bc{d}^{2}{m}^{3}+3\,a{b}^{2}{c}^{2}d{m}^{3}-{b}^{3}{c}^{3}{m}^{3}+6\,{a}^{3}{d}^{3}{m}^{2}-18\,{a}^{2}bc{d}^{2}{m}^{2}+18\,a{b}^{2}{c}^{2}d{m}^{2}-6\,{b}^{3}{c}^{3}{m}^{2}+11\,{a}^{3}{d}^{3}m-33\,{a}^{2}bc{d}^{2}m+33\,a{b}^{2}{c}^{2}dm-11\,{b}^{3}{c}^{3}m+6\,{a}^{3}{d}^{3}-18\,{a}^{2}bc{d}^{2}+18\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e),x)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-3-m)*(a^2*d^2*f*m^2*x-2*a*b*c*d*f*m^2*x-a*b*d^2*f*m*x^2
+b^2*c^2*f*m^2*x+b^2*c*d*f*m*x^2+a^2*d^2*e*m^2+4*a^2*d^2*f*m*x-2*a*b*c*d*e*m^2-8
*a*b*c*d*f*m*x-2*a*b*d^2*e*m*x-3*a*b*d^2*f*x^2+b^2*c^2*e*m^2+4*b^2*c^2*f*m*x+2*b
^2*c*d*e*m*x+b^2*c*d*f*x^2+2*b^2*d^2*e*x^2+a^2*c*d*f*m+3*a^2*d^2*e*m+3*a^2*d^2*f
*x-a*b*c^2*f*m-8*a*b*c*d*e*m-10*a*b*c*d*f*x-2*a*b*d^2*e*x+5*b^2*c^2*e*m+3*b^2*c^
2*f*x+6*b^2*c*d*e*x+a^2*c*d*f+2*a^2*d^2*e-3*a*b*c^2*f-6*a*b*c*d*e+6*b^2*c^2*e)/(
a^3*d^3*m^3-3*a^2*b*c*d^2*m^3+3*a*b^2*c^2*d*m^3-b^3*c^3*m^3+6*a^3*d^3*m^2-18*a^2
*b*c*d^2*m^2+18*a*b^2*c^2*d*m^2-6*b^3*c^3*m^2+11*a^3*d^3*m-33*a^2*b*c*d^2*m+33*a
*b^2*c^2*d*m-11*b^3*c^3*m+6*a^3*d^3-18*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="maxima")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.238807, size = 1218, normalized size = 6.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="fricas")

[Out]

((2*b^3*d^3*e + (b^3*c*d^2 - a*b^2*d^3)*f*m + (b^3*c*d^2 - 3*a*b^2*d^3)*f)*x^4 +
(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*e*m^2 + (8*b^3*c*d^2*e + (b^3*c^2*d - 2
*a*b^2*c*d^2 + a^2*b*d^3)*f*m^2 + 4*(b^3*c^2*d - 3*a*b^2*c*d^2)*f + (2*(b^3*c*d^
2 - a*b^2*d^3)*e + (5*b^3*c^2*d - 8*a*b^2*c*d^2 + 3*a^2*b*d^3)*f)*m)*x^3 + (12*b
^3*c^2*d*e + ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e + (b^3*c^3 - a*b^2*c^2*d
- a^2*b*c*d^2 + a^3*d^3)*f)*m^2 + 3*(b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 +
a^3*d^3)*f + ((7*b^3*c^2*d - 8*a*b^2*c*d^2 + a^2*b*d^3)*e + 4*(b^3*c^3 - a*b^2*c
^2*d - a^2*b*c*d^2 + a^3*d^3)*f)*m)*x^2 + 2*(3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c
*d^2)*e - (3*a^2*b*c^3 - a^3*c^2*d)*f + ((5*a*b^2*c^3 - 8*a^2*b*c^2*d + 3*a^3*c*
d^2)*e - (a^2*b*c^3 - a^3*c^2*d)*f)*m + (((b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 +
a^3*d^3)*e + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f)*m^2 + 2*(3*b^3*c^3 + 3*
a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*e - 4*(3*a^2*b*c^2*d - a^3*c*d^2)*f + ((5
*b^3*c^3 - a*b^2*c^2*d - 7*a^2*b*c*d^2 + 3*a^3*d^3)*e + (3*a*b^2*c^3 - 8*a^2*b*c
^2*d + 5*a^3*c*d^2)*f)*m)*x)*(b*x + a)^m*(d*x + c)^(-m - 4)/(6*b^3*c^3 - 18*a*b^
2*c^2*d + 18*a^2*b*c*d^2 - 6*a^3*d^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*m^3 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*m^2 + 11*
(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*m)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4), x)