3.3040 \(\int (a+b x)^{-n} (c+d x) (e+f x)^{-4+n} \, dx\)

Optimal. Leaf size=207 \[ -\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-3}}{f (3-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-2} (b (2 c f+d e (1-n))-a d f (3-n))}{f (2-n) (3-n) (b e-a f)^2}+\frac{b (a+b x)^{1-n} (e+f x)^{n-1} (b (2 c f+d e (1-n))-a d f (3-n))}{f (1-n) (2-n) (3-n) (b e-a f)^3} \]

[Out]

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

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Rubi [A]  time = 0.373809, antiderivative size = 205, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-3}}{f (3-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-2} (-a d f (3-n)+2 b c f+b d (e-e n))}{f (2-n) (3-n) (b e-a f)^2}+\frac{b (a+b x)^{1-n} (e+f x)^{n-1} (-a d f (3-n)+2 b c f+b d (e-e n))}{f (1-n) (2-n) (3-n) (b e-a f)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.8952, size = 151, normalized size = 0.73 \[ \frac{b \left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 1} \left (- 2 b c f + d \left (a f \left (- n + 3\right ) - b e \left (- n + 1\right )\right )\right )}{f \left (- n + 1\right ) \left (- n + 2\right ) \left (- n + 3\right ) \left (a f - b e\right )^{3}} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 3} \left (c f - d e\right )}{f \left (- n + 3\right ) \left (a f - b e\right )} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 2} \left (- 2 b c f + d \left (a f \left (- n + 3\right ) - b e \left (- n + 1\right )\right )\right )}{f \left (- n + 2\right ) \left (- n + 3\right ) \left (a f - b e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.618447, size = 201, normalized size = 0.97 \[ \frac{(a+b x)^{-n} (e+f x)^n \left (\frac{b^2 (-a d f (n-3)-2 b c f+b d e (n-1))}{(n-3) (n-2) (n-1) (b e-a f)^3}+\frac{b n (a d f (n-3)+2 b c f+b d (e-e n))}{(n-1) \left (n^2-5 n+6\right ) (e+f x) (b e-a f)^2}+\frac{-a d f (n-3)-b c f n+b d e (2 n-3)}{(n-3) (n-2) (e+f x)^2 (b e-a f)}+\frac{c f-d e}{(n-3) (e+f x)^3}\right )}{f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 505, normalized size = 2.4 \[{\frac{ \left ( bx+a \right ) \left ( fx+e \right ) ^{-3+n} \left ({a}^{2}d{f}^{2}{n}^{2}x-2\,abdef{n}^{2}x+abd{f}^{2}n{x}^{2}+{b}^{2}d{e}^{2}{n}^{2}x-{b}^{2}defn{x}^{2}+{a}^{2}c{f}^{2}{n}^{2}-4\,{a}^{2}d{f}^{2}nx-2\,abcef{n}^{2}+2\,abc{f}^{2}nx+8\,abdefnx-3\,abd{f}^{2}{x}^{2}+{b}^{2}c{e}^{2}{n}^{2}-2\,{b}^{2}cefnx+2\,{b}^{2}c{f}^{2}{x}^{2}-4\,{b}^{2}d{e}^{2}nx+{b}^{2}def{x}^{2}-3\,{a}^{2}c{f}^{2}n-{a}^{2}defn+3\,{a}^{2}d{f}^{2}x+8\,abcefn-2\,abc{f}^{2}x+abd{e}^{2}n-10\,abdefx-5\,{b}^{2}c{e}^{2}n+6\,{b}^{2}cefx+3\,{b}^{2}d{e}^{2}x+2\,{a}^{2}c{f}^{2}+{a}^{2}def-6\,abcef-3\,abd{e}^{2}+6\,{b}^{2}c{e}^{2} \right ) }{ \left ({a}^{3}{f}^{3}{n}^{3}-3\,{a}^{2}be{f}^{2}{n}^{3}+3\,a{b}^{2}{e}^{2}f{n}^{3}-{b}^{3}{e}^{3}{n}^{3}-6\,{a}^{3}{f}^{3}{n}^{2}+18\,{a}^{2}be{f}^{2}{n}^{2}-18\,a{b}^{2}{e}^{2}f{n}^{2}+6\,{b}^{3}{e}^{3}{n}^{2}+11\,{a}^{3}{f}^{3}n-33\,{a}^{2}be{f}^{2}n+33\,a{b}^{2}{e}^{2}fn-11\,{b}^{3}{e}^{3}n-6\,{a}^{3}{f}^{3}+18\,{a}^{2}be{f}^{2}-18\,a{b}^{2}{e}^{2}f+6\,{b}^{3}{e}^{3} \right ) \left ( bx+a \right ) ^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.258467, size = 1193, normalized size = 5.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a^3*c*e*f^2 + (b^3*d*e*f^2 + (2*b^3*c - 3*a*b^2*d)*f^3 - (b^3*d*e*f^2 - a*b^2
*d*f^3)*n)*x^4 + 3*(2*a*b^2*c - a^2*b*d)*e^3 - (6*a^2*b*c - a^3*d)*e^2*f + (4*b^
3*d*e^2*f + 4*(2*b^3*c - 3*a*b^2*d)*e*f^2 + (b^3*d*e^2*f - 2*a*b^2*d*e*f^2 + a^2
*b*d*f^3)*n^2 - (5*b^3*d*e^2*f + 2*(b^3*c - 4*a*b^2*d)*e*f^2 - (2*a*b^2*c - 3*a^
2*b*d)*f^3)*n)*x^3 + (a*b^2*c*e^3 - 2*a^2*b*c*e^2*f + a^3*c*e*f^2)*n^2 + (3*b^3*
d*e^3 - 9*a^2*b*d*e*f^2 + 3*a^3*d*f^3 + 3*(4*b^3*c - 3*a*b^2*d)*e^2*f + (b^3*d*e
^3 + (b^3*c - a*b^2*d)*e^2*f - (2*a*b^2*c + a^2*b*d)*e*f^2 + (a^2*b*c + a^3*d)*f
^3)*n^2 - (4*b^3*d*e^3 + (7*b^3*c - 4*a*b^2*d)*e^2*f - 4*(2*a*b^2*c + a^2*b*d)*e
*f^2 + (a^2*b*c + 4*a^3*d)*f^3)*n)*x^2 - (3*a^3*c*e*f^2 + (5*a*b^2*c - a^2*b*d)*
e^3 - (8*a^2*b*c - a^3*d)*e^2*f)*n + (6*b^3*c*e^3 + 2*a^3*c*f^3 + 6*(a*b^2*c - 2
*a^2*b*d)*e^2*f - 2*(3*a^2*b*c - 2*a^3*d)*e*f^2 + (a^3*c*f^3 + (b^3*c + a*b^2*d)
*e^3 - (a*b^2*c + 2*a^2*b*d)*e^2*f - (a^2*b*c - a^3*d)*e*f^2)*n^2 - (3*a^3*c*f^3
+ (5*b^3*c + 3*a*b^2*d)*e^3 - (a*b^2*c + 8*a^2*b*d)*e^2*f - (7*a^2*b*c - 5*a^3*
d)*e*f^2)*n)*x)*(f*x + e)^(n - 4)/((6*b^3*e^3 - 18*a*b^2*e^2*f + 18*a^2*b*e*f^2
- 6*a^3*f^3 - (b^3*e^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 - a^3*f^3)*n^3 + 6*(b^3*e
^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 - a^3*f^3)*n^2 - 11*(b^3*e^3 - 3*a*b^2*e^2*f
+ 3*a^2*b*e*f^2 - a^3*f^3)*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)**(-4+n)/((b*x+a)**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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