3.3035 \(\int (a+b x) (c+d x)^{-5+n} (e+f x)^{-n} \, dx\)
Optimal. Leaf size=299 \[ \frac{2 f^2 (c+d x)^{n-1} (e+f x)^{1-n} (3 a d f+b (c f (1-n)-d e (4-n)))}{d (1-n) (2-n) (3-n) (4-n) (d e-c f)^4}+\frac{(b c-a d) (c+d x)^{n-4} (e+f x)^{1-n}}{d (4-n) (d e-c f)}+\frac{(c+d x)^{n-3} (e+f x)^{1-n} (3 a d f+b (c f (1-n)-d e (4-n)))}{d (3-n) (4-n) (d e-c f)^2}-\frac{2 f (c+d x)^{n-2} (e+f x)^{1-n} (3 a d f+b (c f (1-n)-d e (4-n)))}{d (2-n) (3-n) (4-n) (d e-c f)^3} \]
[Out]
((b*c - a*d)*(c + d*x)^(-4 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)*(4 - n)) + ((3
*a*d*f + b*(c*f*(1 - n) - d*e*(4 - n)))*(c + d*x)^(-3 + n)*(e + f*x)^(1 - n))/(d
*(d*e - c*f)^2*(3 - n)*(4 - n)) - (2*f*(3*a*d*f + b*(c*f*(1 - n) - d*e*(4 - n)))
*(c + d*x)^(-2 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)^3*(2 - n)*(3 - n)*(4 - n))
+ (2*f^2*(3*a*d*f + b*(c*f*(1 - n) - d*e*(4 - n)))*(c + d*x)^(-1 + n)*(e + f*x)
^(1 - n))/(d*(d*e - c*f)^4*(1 - n)*(2 - n)*(3 - n)*(4 - n))
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Rubi [A] time = 0.590912, antiderivative size = 296, 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
\[ \frac{2 f^2 (c+d x)^{n-1} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (1-n) (2-n) (3-n) (4-n) (d e-c f)^4}+\frac{(b c-a d) (c+d x)^{n-4} (e+f x)^{1-n}}{d (4-n) (d e-c f)}+\frac{(c+d x)^{n-3} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (3-n) (4-n) (d e-c f)^2}-\frac{2 f (c+d x)^{n-2} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (2-n) (3-n) (4-n) (d e-c f)^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(c + d*x)^(-5 + n))/(e + f*x)^n,x]
[Out]
((b*c - a*d)*(c + d*x)^(-4 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)*(4 - n)) + ((3
*a*d*f + b*c*f*(1 - n) - b*d*e*(4 - n))*(c + d*x)^(-3 + n)*(e + f*x)^(1 - n))/(d
*(d*e - c*f)^2*(3 - n)*(4 - n)) - (2*f*(3*a*d*f + b*c*f*(1 - n) - b*d*e*(4 - n))
*(c + d*x)^(-2 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)^3*(2 - n)*(3 - n)*(4 - n))
+ (2*f^2*(3*a*d*f + b*c*f*(1 - n) - b*d*e*(4 - n))*(c + d*x)^(-1 + n)*(e + f*x)
^(1 - n))/(d*(d*e - c*f)^4*(1 - n)*(2 - n)*(3 - n)*(4 - n))
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Rubi in Sympy [A] time = 82.1127, size = 221, normalized size = 0.74 \[ \frac{2 f^{2} \left (c + d x\right )^{n - 1} \left (e + f x\right )^{- n + 1} \left (3 a d f + b \left (c f \left (- n + 1\right ) - d e \left (- n + 4\right )\right )\right )}{d \left (- n + 1\right ) \left (- n + 2\right ) \left (- n + 3\right ) \left (- n + 4\right ) \left (c f - d e\right )^{4}} + \frac{2 f \left (c + d x\right )^{n - 2} \left (e + f x\right )^{- n + 1} \left (3 a d f + b \left (c f \left (- n + 1\right ) - d e \left (- n + 4\right )\right )\right )}{d \left (- n + 2\right ) \left (- n + 3\right ) \left (- n + 4\right ) \left (c f - d e\right )^{3}} + \frac{\left (c + d x\right )^{n - 4} \left (e + f x\right )^{- n + 1} \left (a d - b c\right )}{d \left (- n + 4\right ) \left (c f - d e\right )} + \frac{\left (c + d x\right )^{n - 3} \left (e + f x\right )^{- n + 1} \left (3 a d f + b \left (c f \left (- n + 1\right ) - d e \left (- n + 4\right )\right )\right )}{d \left (- n + 3\right ) \left (- n + 4\right ) \left (c f - d e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(d*x+c)**(-5+n)/((f*x+e)**n),x)
[Out]
2*f**2*(c + d*x)**(n - 1)*(e + f*x)**(-n + 1)*(3*a*d*f + b*(c*f*(-n + 1) - d*e*(
-n + 4)))/(d*(-n + 1)*(-n + 2)*(-n + 3)*(-n + 4)*(c*f - d*e)**4) + 2*f*(c + d*x)
**(n - 2)*(e + f*x)**(-n + 1)*(3*a*d*f + b*(c*f*(-n + 1) - d*e*(-n + 4)))/(d*(-n
+ 2)*(-n + 3)*(-n + 4)*(c*f - d*e)**3) + (c + d*x)**(n - 4)*(e + f*x)**(-n + 1)
*(a*d - b*c)/(d*(-n + 4)*(c*f - d*e)) + (c + d*x)**(n - 3)*(e + f*x)**(-n + 1)*(
3*a*d*f + b*(c*f*(-n + 1) - d*e*(-n + 4)))/(d*(-n + 3)*(-n + 4)*(c*f - d*e)**2)
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Mathematica [A] time = 0.855908, size = 267, normalized size = 0.89 \[ \frac{(c+d x)^n (e+f x)^{-n} \left (\frac{2 f^3 (3 a d f-b c f (n-1)+b d e (n-4))}{(n-4) (n-3) (n-2) (n-1) (d e-c f)^4}+\frac{2 f^2 n (3 a d f-b c f (n-1)+b d e (n-4))}{(n-1) \left (n^3-9 n^2+26 n-24\right ) (c+d x) (d e-c f)^3}+\frac{f n (3 a d f-b c f (n-1)+b d e (n-4))}{(n-2) \left (n^2-7 n+12\right ) (c+d x)^2 (d e-c f)^2}+\frac{a d f n-2 b c f (n-2)+b d e (n-4)}{(n-4) (n-3) (c+d x)^3 (d e-c f)}+\frac{a d-b c}{(n-4) (c+d x)^4}\right )}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(c + d*x)^(-5 + n))/(e + f*x)^n,x]
[Out]
((c + d*x)^n*((2*f^3*(3*a*d*f + b*d*e*(-4 + n) - b*c*f*(-1 + n)))/((d*e - c*f)^4
*(-4 + n)*(-3 + n)*(-2 + n)*(-1 + n)) + (-(b*c) + a*d)/((-4 + n)*(c + d*x)^4) +
(b*d*e*(-4 + n) - 2*b*c*f*(-2 + n) + a*d*f*n)/((d*e - c*f)*(-4 + n)*(-3 + n)*(c
+ d*x)^3) + (f*(3*a*d*f + b*d*e*(-4 + n) - b*c*f*(-1 + n))*n)/((d*e - c*f)^2*(-2
+ n)*(12 - 7*n + n^2)*(c + d*x)^2) + (2*f^2*(3*a*d*f + b*d*e*(-4 + n) - b*c*f*(
-1 + n))*n)/((d*e - c*f)^3*(-1 + n)*(-24 + 26*n - 9*n^2 + n^3)*(c + d*x))))/(d^2
*(e + f*x)^n)
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Maple [B] time = 0.013, size = 1187, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(d*x+c)^(-5+n)/((f*x+e)^n),x)
[Out]
-(d*x+c)^(-4+n)*(f*x+e)*(b*c^3*f^3*n^3*x-3*b*c^2*d*e*f^2*n^3*x-2*b*c^2*d*f^3*n^2
*x^2+3*b*c*d^2*e^2*f*n^3*x+4*b*c*d^2*e*f^2*n^2*x^2+2*b*c*d^2*f^3*n*x^3-b*d^3*e^3
*n^3*x-2*b*d^3*e^2*f*n^2*x^2-2*b*d^3*e*f^2*n*x^3+a*c^3*f^3*n^3-3*a*c^2*d*e*f^2*n
^3-3*a*c^2*d*f^3*n^2*x+3*a*c*d^2*e^2*f*n^3+6*a*c*d^2*e*f^2*n^2*x+6*a*c*d^2*f^3*n
*x^2-a*d^3*e^3*n^3-3*a*d^3*e^2*f*n^2*x-6*a*d^3*e*f^2*n*x^2-6*a*d^3*f^3*x^3-8*b*c
^3*f^3*n^2*x+23*b*c^2*d*e*f^2*n^2*x+10*b*c^2*d*f^3*n*x^2-22*b*c*d^2*e^2*f*n^2*x-
20*b*c*d^2*e*f^2*n*x^2-2*b*c*d^2*f^3*x^3+7*b*d^3*e^3*n^2*x+10*b*d^3*e^2*f*n*x^2+
8*b*d^3*e*f^2*x^3-9*a*c^3*f^3*n^2+24*a*c^2*d*e*f^2*n^2+21*a*c^2*d*f^3*n*x-21*a*c
*d^2*e^2*f*n^2-30*a*c*d^2*e*f^2*n*x-24*a*c*d^2*f^3*x^2+6*a*d^3*e^3*n^2+9*a*d^3*e
^2*f*n*x+6*a*d^3*e*f^2*x^2+b*c^3*e*f^2*n^2+19*b*c^3*f^3*n*x-2*b*c^2*d*e^2*f*n^2-
58*b*c^2*d*e*f^2*n*x-8*b*c^2*d*f^3*x^2+b*c*d^2*e^3*n^2+53*b*c*d^2*e^2*f*n*x+34*b
*c*d^2*e*f^2*x^2-14*b*d^3*e^3*n*x-8*b*d^3*e^2*f*x^2+26*a*c^3*f^3*n-57*a*c^2*d*e*
f^2*n-36*a*c^2*d*f^3*x+42*a*c*d^2*e^2*f*n+24*a*c*d^2*e*f^2*x-11*a*d^3*e^3*n-6*a*
d^3*e^2*f*x-7*b*c^3*e*f^2*n-12*b*c^3*f^3*x+10*b*c^2*d*e^2*f*n+56*b*c^2*d*e*f^2*x
-3*b*c*d^2*e^3*n-34*b*c*d^2*e^2*f*x+8*b*d^3*e^3*x-24*a*c^3*f^3+36*a*c^2*d*e*f^2-
24*a*c*d^2*e^2*f+6*a*d^3*e^3+12*b*c^3*e*f^2-8*b*c^2*d*e^2*f+2*b*c*d^2*e^3)/(c^4*
f^4*n^4-4*c^3*d*e*f^3*n^4+6*c^2*d^2*e^2*f^2*n^4-4*c*d^3*e^3*f*n^4+d^4*e^4*n^4-10
*c^4*f^4*n^3+40*c^3*d*e*f^3*n^3-60*c^2*d^2*e^2*f^2*n^3+40*c*d^3*e^3*f*n^3-10*d^4
*e^4*n^3+35*c^4*f^4*n^2-140*c^3*d*e*f^3*n^2+210*c^2*d^2*e^2*f^2*n^2-140*c*d^3*e^
3*f*n^2+35*d^4*e^4*n^2-50*c^4*f^4*n+200*c^3*d*e*f^3*n-300*c^2*d^2*e^2*f^2*n+200*
c*d^3*e^3*f*n-50*d^4*e^4*n+24*c^4*f^4-96*c^3*d*e*f^3+144*c^2*d^2*e^2*f^2-96*c*d^
3*e^3*f+24*d^4*e^4)/((f*x+e)^n)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (d x + c\right )}^{n - 5}{\left (f x + e\right )}^{-n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^(n - 5)/(f*x + e)^n,x, algorithm="maxima")
[Out]
integrate((b*x + a)*(d*x + c)^(n - 5)*(f*x + e)^(-n), x)
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Fricas [A] time = 0.267873, size = 2350, normalized size = 7.86 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^(n - 5)/(f*x + e)^n,x, algorithm="fricas")
[Out]
(24*a*c^4*e*f^3 - 2*(4*b*d^4*e*f^3 - (b*c*d^3 + 3*a*d^4)*f^4 - (b*d^4*e*f^3 - b*
c*d^3*f^4)*n)*x^5 - 2*(b*c^2*d^2 + 3*a*c*d^3)*e^4 + 8*(b*c^3*d + 3*a*c^2*d^2)*e^
3*f - 12*(b*c^4 + 3*a*c^3*d)*e^2*f^2 - 2*(20*b*c*d^3*e*f^3 - 5*(b*c^2*d^2 + 3*a*
c*d^3)*f^4 - (b*d^4*e^2*f^2 - 2*b*c*d^3*e*f^3 + b*c^2*d^2*f^4)*n^2 + (4*b*d^4*e^
2*f^2 - (10*b*c*d^3 + 3*a*d^4)*e*f^3 + 3*(2*b*c^2*d^2 + a*c*d^3)*f^4)*n)*x^4 + (
a*c*d^3*e^4 - 3*a*c^2*d^2*e^3*f + 3*a*c^3*d*e^2*f^2 - a*c^4*e*f^3)*n^3 - (80*b*c
^2*d^2*e*f^3 - 20*(b*c^3*d + 3*a*c^2*d^2)*f^4 - (b*d^4*e^3*f - 3*b*c*d^3*e^2*f^2
+ 3*b*c^2*d^2*e*f^3 - b*c^3*d*f^4)*n^3 + (5*b*d^4*e^3*f - (20*b*c*d^3 + 3*a*d^4
)*e^2*f^2 + (25*b*c^2*d^2 + 6*a*c*d^3)*e*f^3 - (10*b*c^3*d + 3*a*c^2*d^2)*f^4)*n
^2 - (4*b*d^4*e^3*f - (41*b*c*d^3 + 3*a*d^4)*e^2*f^2 + 6*(11*b*c^2*d^2 + 5*a*c*d
^3)*e*f^3 - (29*b*c^3*d + 27*a*c^2*d^2)*f^4)*n)*x^3 + (9*a*c^4*e*f^3 - (b*c^2*d^
2 + 6*a*c*d^3)*e^4 + (2*b*c^3*d + 21*a*c^2*d^2)*e^3*f - (b*c^4 + 24*a*c^3*d)*e^2
*f^2)*n^2 - (8*b*d^4*e^4 - 32*b*c*d^3*e^3*f + 48*b*c^2*d^2*e^2*f^2 + 48*b*c^3*d*
e*f^3 - 12*(b*c^4 + 5*a*c^3*d)*f^4 - (b*d^4*e^4 - 3*a*c*d^3*e^2*f^2 - (2*b*c*d^3
- a*d^4)*e^3*f + (2*b*c^3*d + 3*a*c^2*d^2)*e*f^3 - (b*c^4 + a*c^3*d)*f^4)*n^3 +
(7*b*d^4*e^4 - (16*b*c*d^3 - 3*a*d^4)*e^3*f + 3*(b*c^2*d^2 - 6*a*c*d^3)*e^2*f^2
+ (14*b*c^3*d + 27*a*c^2*d^2)*e*f^3 - 4*(2*b*c^4 + 3*a*c^3*d)*f^4)*n^2 - (14*b*
d^4*e^4 - 2*(23*b*c*d^3 - a*d^4)*e^3*f + 15*(b*c^2*d^2 - a*c*d^3)*e^2*f^2 + 12*(
3*b*c^3*d + 5*a*c^2*d^2)*e*f^3 - (19*b*c^4 + 47*a*c^3*d)*f^4)*n)*x^2 - (26*a*c^4
*e*f^3 - (3*b*c^2*d^2 + 11*a*c*d^3)*e^4 + 2*(5*b*c^3*d + 21*a*c^2*d^2)*e^3*f - (
7*b*c^4 + 57*a*c^3*d)*e^2*f^2)*n + (24*a*c^3*d*e*f^3 + 24*a*c^4*f^4 - 2*(5*b*c*d
^3 + 3*a*d^4)*e^4 + 8*(5*b*c^2*d^2 + 3*a*c*d^3)*e^3*f - 12*(5*b*c^3*d + 3*a*c^2*
d^2)*e^2*f^2 + (3*b*c^3*d*e^2*f^2 - a*c^4*f^4 + (b*c*d^3 + a*d^4)*e^4 - (3*b*c^2
*d^2 + 2*a*c*d^3)*e^3*f - (b*c^4 - 2*a*c^3*d)*e*f^3)*n^3 + (9*a*c^4*f^4 - 2*(4*b
*c*d^3 + 3*a*d^4)*e^4 + (23*b*c^2*d^2 + 18*a*c*d^3)*e^3*f - (22*b*c^3*d + 9*a*c^
2*d^2)*e^2*f^2 + (7*b*c^4 - 12*a*c^3*d)*e*f^3)*n^2 - (26*a*c^4*f^4 - (17*b*c*d^3
+ 11*a*d^4)*e^4 + 20*(3*b*c^2*d^2 + 2*a*c*d^3)*e^3*f - 5*(11*b*c^3*d + 9*a*c^2*
d^2)*e^2*f^2 + 2*(6*b*c^4 - 5*a*c^3*d)*e*f^3)*n)*x)*(d*x + c)^(n - 5)/((24*d^4*e
^4 - 96*c*d^3*e^3*f + 144*c^2*d^2*e^2*f^2 - 96*c^3*d*e*f^3 + 24*c^4*f^4 + (d^4*e
^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n^4 - 10*(d^4*
e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n^3 + 35*(d^4
*e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n^2 - 50*(d^
4*e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n)*(f*x + e
)^n)
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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)*(d*x+c)**(-5+n)/((f*x+e)**n),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (d x + c\right )}^{n - 5}}{{\left (f x + e\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^(n - 5)/(f*x + e)^n,x, algorithm="giac")
[Out]
integrate((b*x + a)*(d*x + c)^(n - 5)/(f*x + e)^n, x)