3.3130 \(\int (a+b x)^m (c+d x)^n (e+f x)^{-4-m-n} \, dx\)

Optimal. Leaf size=402 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (c f (n+1)-d e (m+n+3))+b^2 \left (-\left (-c^2 f^2 \left (n^2+3 n+2\right )+2 c d e f (n+1) (m+n+3)-d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right )\right )\right ) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (m+n+3) (b e-a f)^3 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-3}}{(m+n+3) (b e-a f) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2} (a d f (m+2)+b (c f (n+2)-d e (m+n+4)))}{(m+n+2) (m+n+3) (b e-a f)^2 (d e-c f)^2} \]

[Out]

-((f*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x)^(-3 - m - n))/((b*e - a*f)*(d
*e - c*f)*(3 + m + n))) + (f*(a*d*f*(2 + m) + b*(c*f*(2 + n) - d*e*(4 + m + n)))
*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x)^(-2 - m - n))/((b*e - a*f)^2*(d*e
- c*f)^2*(2 + m + n)*(3 + m + n)) + ((a^2*d^2*f^2*(2 + 3*m + m^2) + 2*a*b*d*f*(
1 + m)*(c*f*(1 + n) - d*e*(3 + m + n)) - b^2*(2*c*d*e*f*(1 + n)*(3 + m + n) - c^
2*f^2*(2 + 3*n + n^2) - d^2*e^2*(6 + m^2 + 5*n + n^2 + m*(5 + 2*n))))*(a + b*x)^
(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeometric2F1[1 + m, -n, 2 + m, -
(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)^3*(d*e - c*f)^2
*(1 + m)*(2 + m + n)*(3 + m + n)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)
))^n)

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Rubi [A]  time = 1.47261, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (c f (n+1)-d e (m+n+3))+b^2 \left (-\left (-c^2 f^2 \left (n^2+3 n+2\right )+2 c d e f (n+1) (m+n+3)-d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right )\right )\right ) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (m+n+3) (b e-a f)^3 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-3}}{(m+n+3) (b e-a f) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2} (a d f (m+2)+b c f (n+2)-b d e (m+n+4))}{(m+n+2) (m+n+3) (b e-a f)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

-((f*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x)^(-3 - m - n))/((b*e - a*f)*(d
*e - c*f)*(3 + m + n))) + (f*(a*d*f*(2 + m) + b*c*f*(2 + n) - b*d*e*(4 + m + n))
*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x)^(-2 - m - n))/((b*e - a*f)^2*(d*e
- c*f)^2*(2 + m + n)*(3 + m + n)) + ((a^2*d^2*f^2*(2 + 3*m + m^2) + 2*a*b*d*f*(
1 + m)*(c*f*(1 + n) - d*e*(3 + m + n)) - b^2*(2*c*d*e*f*(1 + n)*(3 + m + n) - c^
2*f^2*(2 + 3*n + n^2) - d^2*e^2*(6 + m^2 + 5*n + n^2 + m*(5 + 2*n))))*(a + b*x)^
(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeometric2F1[1 + m, -n, 2 + m, -
(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)^3*(d*e - c*f)^2
*(1 + m)*(2 + m + n)*(3 + m + n)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)
))^n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 168.072, size = 13018, normalized size = 32.38 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-4-m-n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**n*(f*x+e)**(-4-m-n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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