3.3041 \(\int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx\)
Optimal. Leaf size=300 \[ \frac{2 b^2 (a+b x)^{1-n} (e+f x)^{n-1} (b (3 c f+d e (1-n))-a d f (4-n))}{f (1-n) (2-n) (3-n) (4-n) (b e-a f)^4}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-3} (b (3 c f+d e (1-n))-a d f (4-n))}{f (3-n) (4-n) (b e-a f)^2}+\frac{2 b (a+b x)^{1-n} (e+f x)^{n-2} (b (3 c f+d e (1-n))-a d f (4-n))}{f (2-n) (3-n) (4-n) (b e-a f)^3} \]
[Out]
-(((d*e - c*f)*(a + b*x)^(1 - n)*(e + f*x)^(-4 + n))/(f*(b*e - a*f)*(4 - n))) +
((b*(3*c*f + d*e*(1 - n)) - a*d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-3 + n))
/(f*(b*e - a*f)^2*(3 - n)*(4 - n)) + (2*b*(b*(3*c*f + d*e*(1 - n)) - a*d*f*(4 -
n))*(a + b*x)^(1 - n)*(e + f*x)^(-2 + n))/(f*(b*e - a*f)^3*(2 - n)*(3 - n)*(4 -
n)) + (2*b^2*(b*(3*c*f + d*e*(1 - n)) - a*d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*
x)^(-1 + n))/(f*(b*e - a*f)^4*(1 - n)*(2 - n)*(3 - n)*(4 - n))
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Rubi [A] time = 0.572703, antiderivative size = 297, 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
\[ \frac{2 b^2 (a+b x)^{1-n} (e+f x)^{n-1} (-a d f (4-n)+3 b c f+b d (e-e n))}{f (1-n) (2-n) (3-n) (4-n) (b e-a f)^4}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-3} (-a d f (4-n)+3 b c f+b d (e-e n))}{f (3-n) (4-n) (b e-a f)^2}+\frac{2 b (a+b x)^{1-n} (e+f x)^{n-2} (-a d f (4-n)+3 b c f+b d (e-e n))}{f (2-n) (3-n) (4-n) (b e-a f)^3} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x)*(e + f*x)^(-5 + n))/(a + b*x)^n,x]
[Out]
-(((d*e - c*f)*(a + b*x)^(1 - n)*(e + f*x)^(-4 + n))/(f*(b*e - a*f)*(4 - n))) +
((3*b*c*f - a*d*f*(4 - n) + b*d*(e - e*n))*(a + b*x)^(1 - n)*(e + f*x)^(-3 + n))
/(f*(b*e - a*f)^2*(3 - n)*(4 - n)) + (2*b*(3*b*c*f - a*d*f*(4 - n) + b*d*(e - e*
n))*(a + b*x)^(1 - n)*(e + f*x)^(-2 + n))/(f*(b*e - a*f)^3*(2 - n)*(3 - n)*(4 -
n)) + (2*b^2*(3*b*c*f - a*d*f*(4 - n) + b*d*(e - e*n))*(a + b*x)^(1 - n)*(e + f*
x)^(-1 + n))/(f*(b*e - a*f)^4*(1 - n)*(2 - n)*(3 - n)*(4 - n))
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Rubi in Sympy [A] time = 81.0724, size = 221, normalized size = 0.74 \[ - \frac{2 b^{2} \left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 1} \left (- 3 b c f + d \left (a f \left (- n + 4\right ) - b e \left (- n + 1\right )\right )\right )}{f \left (- n + 1\right ) \left (- n + 2\right ) \left (- n + 3\right ) \left (- n + 4\right ) \left (a f - b e\right )^{4}} + \frac{2 b \left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 2} \left (- 3 b c f + d \left (a f \left (- n + 4\right ) - b e \left (- n + 1\right )\right )\right )}{f \left (- n + 2\right ) \left (- n + 3\right ) \left (- n + 4\right ) \left (a f - b e\right )^{3}} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 4} \left (c f - d e\right )}{f \left (- n + 4\right ) \left (a f - b e\right )} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 3} \left (- 3 b c f + d \left (a f \left (- n + 4\right ) - b e \left (- n + 1\right )\right )\right )}{f \left (- n + 3\right ) \left (- n + 4\right ) \left (a f - b e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)*(f*x+e)**(-5+n)/((b*x+a)**n),x)
[Out]
-2*b**2*(a + b*x)**(-n + 1)*(e + f*x)**(n - 1)*(-3*b*c*f + d*(a*f*(-n + 4) - b*e
*(-n + 1)))/(f*(-n + 1)*(-n + 2)*(-n + 3)*(-n + 4)*(a*f - b*e)**4) + 2*b*(a + b*
x)**(-n + 1)*(e + f*x)**(n - 2)*(-3*b*c*f + d*(a*f*(-n + 4) - b*e*(-n + 1)))/(f*
(-n + 2)*(-n + 3)*(-n + 4)*(a*f - b*e)**3) - (a + b*x)**(-n + 1)*(e + f*x)**(n -
4)*(c*f - d*e)/(f*(-n + 4)*(a*f - b*e)) - (a + b*x)**(-n + 1)*(e + f*x)**(n - 3
)*(-3*b*c*f + d*(a*f*(-n + 4) - b*e*(-n + 1)))/(f*(-n + 3)*(-n + 4)*(a*f - b*e)*
*2)
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Mathematica [A] time = 1.18335, size = 270, normalized size = 0.9 \[ \frac{(a+b x)^{-n} (e+f x)^n \left (-\frac{2 b^3 (-a d f (n-4)-3 b c f+b d e (n-1))}{(n-4) (n-3) (n-2) (n-1) (b e-a f)^4}-\frac{2 b^2 n (a d f (n-4)+3 b c f+b d (e-e n))}{(n-1) \left (n^3-9 n^2+26 n-24\right ) (e+f x) (b e-a f)^3}+\frac{b n (a d f (n-4)+3 b c f+b d (e-e n))}{(n-2) \left (n^2-7 n+12\right ) (e+f x)^2 (b e-a f)^2}-\frac{a d f (n-4)+b c f n-2 b d e (n-2)}{(n-4) (n-3) (e+f x)^3 (b e-a f)}+\frac{c f-d e}{(n-4) (e+f x)^4}\right )}{f^2} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x)*(e + f*x)^(-5 + n))/(a + b*x)^n,x]
[Out]
((e + f*x)^n*((-2*b^3*(-3*b*c*f - a*d*f*(-4 + n) + b*d*e*(-1 + n)))/((b*e - a*f)
^4*(-4 + n)*(-3 + n)*(-2 + n)*(-1 + n)) + (-(d*e) + c*f)/((-4 + n)*(e + f*x)^4)
- (a*d*f*(-4 + n) - 2*b*d*e*(-2 + n) + b*c*f*n)/((b*e - a*f)*(-4 + n)*(-3 + n)*(
e + f*x)^3) + (b*n*(3*b*c*f + a*d*f*(-4 + n) + b*d*(e - e*n)))/((b*e - a*f)^2*(-
2 + n)*(12 - 7*n + n^2)*(e + f*x)^2) - (2*b^2*n*(3*b*c*f + a*d*f*(-4 + n) + b*d*
(e - e*n)))/((b*e - a*f)^3*(-1 + n)*(-24 + 26*n - 9*n^2 + n^3)*(e + f*x))))/(f^2
*(a + b*x)^n)
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Maple [B] time = 0.015, size = 1188, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)*(f*x+e)^(-5+n)/((b*x+a)^n),x)
[Out]
(b*x+a)*(f*x+e)^(-4+n)*(a^3*d*f^3*n^3*x-3*a^2*b*d*e*f^2*n^3*x+2*a^2*b*d*f^3*n^2*
x^2+3*a*b^2*d*e^2*f*n^3*x-4*a*b^2*d*e*f^2*n^2*x^2+2*a*b^2*d*f^3*n*x^3-b^3*d*e^3*
n^3*x+2*b^3*d*e^2*f*n^2*x^2-2*b^3*d*e*f^2*n*x^3+a^3*c*f^3*n^3-7*a^3*d*f^3*n^2*x-
3*a^2*b*c*e*f^2*n^3+3*a^2*b*c*f^3*n^2*x+22*a^2*b*d*e*f^2*n^2*x-10*a^2*b*d*f^3*n*
x^2+3*a*b^2*c*e^2*f*n^3-6*a*b^2*c*e*f^2*n^2*x+6*a*b^2*c*f^3*n*x^2-23*a*b^2*d*e^2
*f*n^2*x+20*a*b^2*d*e*f^2*n*x^2-8*a*b^2*d*f^3*x^3-b^3*c*e^3*n^3+3*b^3*c*e^2*f*n^
2*x-6*b^3*c*e*f^2*n*x^2+6*b^3*c*f^3*x^3+8*b^3*d*e^3*n^2*x-10*b^3*d*e^2*f*n*x^2+2
*b^3*d*e*f^2*x^3-6*a^3*c*f^3*n^2-a^3*d*e*f^2*n^2+14*a^3*d*f^3*n*x+21*a^2*b*c*e*f
^2*n^2-9*a^2*b*c*f^3*n*x+2*a^2*b*d*e^2*f*n^2-53*a^2*b*d*e*f^2*n*x+8*a^2*b*d*f^3*
x^2-24*a*b^2*c*e^2*f*n^2+30*a*b^2*c*e*f^2*n*x-6*a*b^2*c*f^3*x^2-a*b^2*d*e^3*n^2+
58*a*b^2*d*e^2*f*n*x-34*a*b^2*d*e*f^2*x^2+9*b^3*c*e^3*n^2-21*b^3*c*e^2*f*n*x+24*
b^3*c*e*f^2*x^2-19*b^3*d*e^3*n*x+8*b^3*d*e^2*f*x^2+11*a^3*c*f^3*n+3*a^3*d*e*f^2*
n-8*a^3*d*f^3*x-42*a^2*b*c*e*f^2*n+6*a^2*b*c*f^3*x-10*a^2*b*d*e^2*f*n+34*a^2*b*d
*e*f^2*x+57*a*b^2*c*e^2*f*n-24*a*b^2*c*e*f^2*x+7*a*b^2*d*e^3*n-56*a*b^2*d*e^2*f*
x-26*b^3*c*e^3*n+36*b^3*c*e^2*f*x+12*b^3*d*e^3*x-6*a^3*c*f^3-2*a^3*d*e*f^2+24*a^
2*b*c*e*f^2+8*a^2*b*d*e^2*f-36*a*b^2*c*e^2*f-12*a*b^2*d*e^3+24*b^3*c*e^3)/(a^4*f
^4*n^4-4*a^3*b*e*f^3*n^4+6*a^2*b^2*e^2*f^2*n^4-4*a*b^3*e^3*f*n^4+b^4*e^4*n^4-10*
a^4*f^4*n^3+40*a^3*b*e*f^3*n^3-60*a^2*b^2*e^2*f^2*n^3+40*a*b^3*e^3*f*n^3-10*b^4*
e^4*n^3+35*a^4*f^4*n^2-140*a^3*b*e*f^3*n^2+210*a^2*b^2*e^2*f^2*n^2-140*a*b^3*e^3
*f*n^2+35*b^4*e^4*n^2-50*a^4*f^4*n+200*a^3*b*e*f^3*n-300*a^2*b^2*e^2*f^2*n+200*a
*b^3*e^3*f*n-50*b^4*e^4*n+24*a^4*f^4-96*a^3*b*e*f^3+144*a^2*b^2*e^2*f^2-96*a*b^3
*e^3*f+24*b^4*e^4)/((b*x+a)^n)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}{\left (b x + a\right )}^{-n}{\left (f x + e\right )}^{n - 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 5)/(b*x + a)^n,x, algorithm="maxima")
[Out]
integrate((d*x + c)*(b*x + a)^(-n)*(f*x + e)^(n - 5), x)
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Fricas [A] time = 0.272249, size = 2349, normalized size = 7.83 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 5)/(b*x + a)^n,x, algorithm="fricas")
[Out]
-(6*a^4*c*e*f^3 - 2*(b^4*d*e*f^3 + (3*b^4*c - 4*a*b^3*d)*f^4 - (b^4*d*e*f^3 - a*
b^3*d*f^4)*n)*x^5 - 12*(2*a*b^3*c - a^2*b^2*d)*e^4 + 4*(9*a^2*b^2*c - 2*a^3*b*d)
*e^3*f - 2*(12*a^3*b*c - a^4*d)*e^2*f^2 - 2*(5*b^4*d*e^2*f^2 + 5*(3*b^4*c - 4*a*
b^3*d)*e*f^3 + (b^4*d*e^2*f^2 - 2*a*b^3*d*e*f^3 + a^2*b^2*d*f^4)*n^2 - (6*b^4*d*
e^2*f^2 + (3*b^4*c - 10*a*b^3*d)*e*f^3 - (3*a*b^3*c - 4*a^2*b^2*d)*f^4)*n)*x^4 +
(a*b^3*c*e^4 - 3*a^2*b^2*c*e^3*f + 3*a^3*b*c*e^2*f^2 - a^4*c*e*f^3)*n^3 - (20*b
^4*d*e^3*f + 20*(3*b^4*c - 4*a*b^3*d)*e^2*f^2 - (b^4*d*e^3*f - 3*a*b^3*d*e^2*f^2
+ 3*a^2*b^2*d*e*f^3 - a^3*b*d*f^4)*n^3 + (10*b^4*d*e^3*f + (3*b^4*c - 25*a*b^3*
d)*e^2*f^2 - 2*(3*a*b^3*c - 10*a^2*b^2*d)*e*f^3 + (3*a^2*b^2*c - 5*a^3*b*d)*f^4)
*n^2 - (29*b^4*d*e^3*f + 3*(9*b^4*c - 22*a*b^3*d)*e^2*f^2 - (30*a*b^3*c - 41*a^2
*b^2*d)*e*f^3 + (3*a^2*b^2*c - 4*a^3*b*d)*f^4)*n)*x^3 + (6*a^4*c*e*f^3 - (9*a*b^
3*c - a^2*b^2*d)*e^4 + 2*(12*a^2*b^2*c - a^3*b*d)*e^3*f - (21*a^3*b*c - a^4*d)*e
^2*f^2)*n^2 - (12*b^4*d*e^4 - 48*a^2*b^2*d*e^2*f^2 + 32*a^3*b*d*e*f^3 - 8*a^4*d*
f^4 + 12*(5*b^4*c - 4*a*b^3*d)*e^3*f - (b^4*d*e^4 - 3*a*b^3*c*e^2*f^2 + (b^4*c -
2*a*b^3*d)*e^3*f + (3*a^2*b^2*c + 2*a^3*b*d)*e*f^3 - (a^3*b*c + a^4*d)*f^4)*n^3
+ (8*b^4*d*e^4 + 2*(6*b^4*c - 7*a*b^3*d)*e^3*f - 3*(9*a*b^3*c + a^2*b^2*d)*e^2*
f^2 + 2*(9*a^2*b^2*c + 8*a^3*b*d)*e*f^3 - (3*a^3*b*c + 7*a^4*d)*f^4)*n^2 - (19*b
^4*d*e^4 + (47*b^4*c - 36*a*b^3*d)*e^3*f - 15*(4*a*b^3*c + a^2*b^2*d)*e^2*f^2 +
(15*a^2*b^2*c + 46*a^3*b*d)*e*f^3 - 2*(a^3*b*c + 7*a^4*d)*f^4)*n)*x^2 - (11*a^4*
c*e*f^3 - (26*a*b^3*c - 7*a^2*b^2*d)*e^4 + (57*a^2*b^2*c - 10*a^3*b*d)*e^3*f - 3
*(14*a^3*b*c - a^4*d)*e^2*f^2)*n - (24*b^4*c*e^4 - 6*a^4*c*f^4 + 12*(2*a*b^3*c -
5*a^2*b^2*d)*e^3*f - 4*(9*a^2*b^2*c - 10*a^3*b*d)*e^2*f^2 + 2*(12*a^3*b*c - 5*a
^4*d)*e*f^3 - (3*a^3*b*d*e^2*f^2 - a^4*c*f^4 + (b^4*c + a*b^3*d)*e^4 - (2*a*b^3*
c + 3*a^2*b^2*d)*e^3*f + (2*a^3*b*c - a^4*d)*e*f^3)*n^3 - (6*a^4*c*f^4 - (9*b^4*
c + 7*a*b^3*d)*e^4 + 2*(6*a*b^3*c + 11*a^2*b^2*d)*e^3*f + (9*a^2*b^2*c - 23*a^3*
b*d)*e^2*f^2 - 2*(9*a^3*b*c - 4*a^4*d)*e*f^3)*n^2 + (11*a^4*c*f^4 - 2*(13*b^4*c
+ 6*a*b^3*d)*e^4 + 5*(2*a*b^3*c + 11*a^2*b^2*d)*e^3*f + 15*(3*a^2*b^2*c - 4*a^3*
b*d)*e^2*f^2 - (40*a^3*b*c - 17*a^4*d)*e*f^3)*n)*x)*(f*x + e)^(n - 5)/((24*b^4*e
^4 - 96*a*b^3*e^3*f + 144*a^2*b^2*e^2*f^2 - 96*a^3*b*e*f^3 + 24*a^4*f^4 + (b^4*e
^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n^4 - 10*(b^4*
e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n^3 + 35*(b^4
*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n^2 - 50*(b^
4*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n)*(b*x + a
)^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((d*x+c)*(f*x+e)**(-5+n)/((b*x+a)**n),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 5}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 5)/(b*x + a)^n,x, algorithm="giac")
[Out]
integrate((d*x + c)*(f*x + e)^(n - 5)/(b*x + a)^n, x)