3.3091 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx\)
Optimal. Leaf size=268 \[ -\frac{2 b^2 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f (m+1)+3 d e))}{d (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4}}{d (m+4) (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b (c f (m+1)+3 d e))}{d (m+3) (m+4) (b c-a d)^2}-\frac{2 b (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+4)-b (c f (m+1)+3 d e))}{d (m+2) (m+3) (m+4) (b c-a d)^3} \]
[Out]
((d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d*(b*c - a*d)*(4 + m)) - ((a
*d*f*(4 + m) - b*(3*d*e + c*f*(1 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d
*(b*c - a*d)^2*(3 + m)*(4 + m)) - (2*b*(a*d*f*(4 + m) - b*(3*d*e + c*f*(1 + m)))
*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m))
- (2*b^2*(a*d*f*(4 + m) - b*(3*d*e + c*f*(1 + 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))
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Rubi [A] time = 0.487891, antiderivative size = 264, normalized size of antiderivative =
0.99, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125
\[ \frac{2 b^2 (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4}}{d (m+4) (b c-a d)}+\frac{(a+b x)^{m+1} (c+d x)^{-m-3} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+3) (m+4) (b c-a d)^2}+\frac{2 b (a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+2) (m+3) (m+4) (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x),x]
[Out]
((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))/(d
*(b*c - a*d)^2*(3 + m)*(4 + m)) + (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))
+ (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))
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Rubi in Sympy [A] time = 77.9362, size = 228, normalized size = 0.85 \[ - \frac{2 b^{2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- 3 b d e + f \left (a d \left (m + 4\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right ) \left (a d - b c\right )^{4}} + \frac{2 b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (- 3 b d e + f \left (a d \left (m + 4\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right ) \left (a d - b c\right )^{3}} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 4} \left (c f - d e\right )}{d \left (m + 4\right ) \left (a d - b c\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 3} \left (- 3 b d e + f \left (a d \left (m + 4\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 3\right ) \left (m + 4\right ) \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-5-m)*(f*x+e),x)
[Out]
-2*b**2*(a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(-3*b*d*e + f*(a*d*(m + 4) - b*c*
(m + 1)))/(d*(m + 1)*(m + 2)*(m + 3)*(m + 4)*(a*d - b*c)**4) + 2*b*(a + b*x)**(m
+ 1)*(c + d*x)**(-m - 2)*(-3*b*d*e + f*(a*d*(m + 4) - b*c*(m + 1)))/(d*(m + 2)*
(m + 3)*(m + 4)*(a*d - b*c)**3) + (a + b*x)**(m + 1)*(c + d*x)**(-m - 4)*(c*f -
d*e)/(d*(m + 4)*(a*d - b*c)) - (a + b*x)**(m + 1)*(c + d*x)**(-m - 3)*(-3*b*d*e
+ f*(a*d*(m + 4) - b*c*(m + 1)))/(d*(m + 3)*(m + 4)*(a*d - b*c)**2)
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Mathematica [A] time = 0.919652, size = 267, normalized size = 1. \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{2 b^3 (-a d f (m+4)+b c f (m+1)+3 b d e)}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{2 b^2 m (-a d f (m+4)+b c f (m+1)+3 b d e)}{(m+1) \left (m^3+9 m^2+26 m+24\right ) (c+d x) (b c-a d)^3}+\frac{b m (-a d f (m+4)+b c f (m+1)+3 b d e)}{(m+2) \left (m^2+7 m+12\right ) (c+d x)^2 (b c-a d)^2}+\frac{a d f (m+4)-2 b c f (m+2)+b d e m}{(m+3) (m+4) (c+d x)^3 (b c-a d)}+\frac{c f-d e}{(m+4) (c+d x)^4}\right )}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x),x]
[Out]
((a + b*x)^m*((2*b^3*(3*b*d*e + b*c*f*(1 + m) - a*d*f*(4 + m)))/((b*c - a*d)^4*(
1 + m)*(2 + m)*(3 + m)*(4 + m)) + (-(d*e) + c*f)/((4 + m)*(c + d*x)^4) + (b*d*e*
m - 2*b*c*f*(2 + m) + a*d*f*(4 + m))/((b*c - a*d)*(3 + m)*(4 + m)*(c + d*x)^3) +
(b*m*(3*b*d*e + b*c*f*(1 + m) - a*d*f*(4 + m)))/((b*c - a*d)^2*(2 + m)*(12 + 7*
m + m^2)*(c + d*x)^2) + (2*b^2*m*(3*b*d*e + b*c*f*(1 + m) - a*d*f*(4 + m)))/((b*
c - a*d)^3*(1 + m)*(24 + 26*m + 9*m^2 + m^3)*(c + d*x))))/(d^2*(c + d*x)^m)
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Maple [B] time = 0.013, size = 1184, normalized size = 4.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e),x)
[Out]
-(b*x+a)^(1+m)*(d*x+c)^(-4-m)*(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*e*x+7*a*b^2*c^3*f*m+57*a*b^2*c^2*d*e*m+56*a*b^2*c^2*d*f*x+24*a*b^2*c
*d^2*e*x-26*b^3*c^3*e*m-12*b^3*c^3*f*x-36*b^3*c^2*d*e*x+2*a^3*c*d^2*f+6*a^3*d^3*
e-8*a^2*b*c^2*d*f-24*a^2*b*c*d^2*e+12*a*b^2*c^3*f+36*a*b^2*c^2*d*e-24*b^3*c^3*e)
/(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+60*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+50*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^2-9
6*a*b^3*c^3*d+24*b^4*c^4)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="maxima")
[Out]
integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5), x)
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Fricas [A] time = 0.24947, size = 2399, normalized size = 8.95 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="fricas")
[Out]
(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*(6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^
2)*f + ((26*a*b^3*c^4 - 57*a^2*b^2*c^3*d + 42*a^3*b*c^2*d^2 - 11*a^4*c*d^3)*e -
(7*a^2*b^2*c^4 - 10*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f)*m + (((b^4*c^4 - 2*a*b^3*c^3
*d + 2*a^3*b*c*d^3 - a^4*d^4)*e + (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)*f)*m^3 + (3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 6*a^3
*b*c*d^3 - 2*a^4*d^4)*e + (7*a*b^3*c^4 - 22*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 8
*a^4*c*d^3)*f)*m^2 + 6*(4*b^4*c^4 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*
c*d^3 - a^4*d^4)*e - 10*(6*a^2*b^2*c^3*d - 4*a^3*b*c^2*d^2 + a^4*c*d^3)*f + ((26
*b^4*c^4 - 10*a*b^3*c^3*d - 45*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 - 11*a^4*d^4)*e
+ (12*a*b^3*c^4 - 55*a^2*b^2*c^3*d + 60*a^3*b*c^2*d^2 - 17*a^4*c*d^3)*f)*m)*x)*(
b*x + a)^m*(d*x + c)^(-m - 5)/(24*b^4*c^4 - 96*a*b^3*c^3*d + 144*a^2*b^2*c^2*d^2
- 96*a^3*b*c*d^3 + 24*a^4*d^4 + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 -
4*a^3*b*c*d^3 + a^4*d^4)*m^4 + 10*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 -
4*a^3*b*c*d^3 + a^4*d^4)*m^3 + 35*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2
- 4*a^3*b*c*d^3 + a^4*d^4)*m^2 + 50*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2
- 4*a^3*b*c*d^3 + a^4*d^4)*m)
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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)**(-5-m)*(f*x+e),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="giac")
[Out]
integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 5), x)