3.3122 \(\int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-3+n} \, dx\)

Optimal. Leaf size=237 \[ -\frac{(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} (a d f (m+1)-b (d e (2-n)-c f (-m-n+1))) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \, _2F_1\left (m+1,m+n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (2-n) (b e-a f)^2 (d e-c f)}-\frac{f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1}}{(2-n) (b e-a f) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.300035, antiderivative size = 235, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} (a d f (m+1)+b c f (-m-n+1)-b d e (2-n)) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \, _2F_1\left (m+1,m+n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (2-n) (b e-a f)^2 (d e-c f)}-\frac{f (a+b x)^{m+1} (e+f x)^{n-2} (c+d x)^{-m-n+1}}{(2-n) (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-3 + n),x]

[Out]

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

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Rubi in Sympy [A]  time = 54.6796, size = 192, normalized size = 0.81 \[ - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - n + 1} \left (e + f x\right )^{n - 2}}{\left (- n + 2\right ) \left (a f - b e\right ) \left (c f - d e\right )} - \frac{\left (\frac{\left (e + f x\right ) \left (- a d + b c\right )}{\left (a + b x\right ) \left (c f - d e\right )}\right )^{- n + 2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - n + 1} \left (e + f x\right )^{n - 2} \left (a d f \left (m + 1\right ) + b c f \left (- m - n + 1\right ) - b d e \left (- n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - m - n + 1, - n + 2 \\ - m - n + 2 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{\left (- n + 2\right ) \left (a d - b c\right ) \left (a f - b e\right ) \left (c f - d e\right ) \left (- m - n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(-m-n)*(f*x+e)**(-3+n),x)

[Out]

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

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Mathematica [B]  time = 73.0204, size = 5197, normalized size = 21.93 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-3 + n),x]

[Out]

Result too large to show

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Maple [F]  time = 0.237, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-m-n} \left ( fx+e \right ) ^{-3+n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-3+n),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-3+n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 3),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 3), 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)**(-m-n)*(f*x+e)**(-3+n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 3),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 3), x)