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

Optimal. Leaf size=268 \[ \frac{(A b-a B) (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (m+1)}+\frac{B (a+b x)^{m+2} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+2;-n,m+n;m+3;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (m+2)} \]

[Out]

((A*b - a*B)*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-m - n)*((b*(e + f*x))/(b*
e - a*f))^(m + n)*AppellF1[1 + m, -n, m + n, 2 + m, -((d*(a + b*x))/(b*c - a*d))
, -((f*(a + b*x))/(b*e - a*f))])/(b^2*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n) + (
B*(a + b*x)^(2 + m)*(c + d*x)^n*(e + f*x)^(-m - n)*((b*(e + f*x))/(b*e - a*f))^(
m + n)*AppellF1[2 + m, -n, m + n, 3 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a +
b*x))/(b*e - a*f))])/(b^2*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n)

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Rubi [A]  time = 0.718338, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{(A b-a B) (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (m+1)}+\frac{B (a+b x)^{m+2} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+2;-n,m+n;m+3;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-m - n)*((b*(e + f*x))/(b*
e - a*f))^(m + n)*AppellF1[1 + m, -n, m + n, 2 + m, -((d*(a + b*x))/(b*c - a*d))
, -((f*(a + b*x))/(b*e - a*f))])/(b^2*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n) + (
B*(a + b*x)^(2 + m)*(c + d*x)^n*(e + f*x)^(-m - n)*((b*(e + f*x))/(b*e - a*f))^(
m + n)*AppellF1[2 + m, -n, m + n, 3 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a +
b*x))/(b*e - a*f))])/(b^2*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n)

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Rubi in Sympy [A]  time = 164.387, size = 211, normalized size = 0.79 \[ \frac{B \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{m + n} \left (a + b x\right )^{m + 2} \left (c + d x\right )^{n} \left (e + f x\right )^{- m - n} \operatorname{appellf_{1}}{\left (m + 2,- n,m + n,m + 3,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{b^{2} \left (m + 2\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{m + n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (e + f x\right )^{- m - n} \left (A b - B a\right ) \operatorname{appellf_{1}}{\left (m + 1,- n,m + n,m + 2,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{b^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(b*(-c - d*x)/(a*d - b*c))**(-n)*(b*(-e - f*x)/(a*f - b*e))**(m + n)*(a + b*x)
**(m + 2)*(c + d*x)**n*(e + f*x)**(-m - n)*appellf1(m + 2, -n, m + n, m + 3, d*(
a + b*x)/(a*d - b*c), f*(a + b*x)/(a*f - b*e))/(b**2*(m + 2)) + (b*(-c - d*x)/(a
*d - b*c))**(-n)*(b*(-e - f*x)/(a*f - b*e))**(m + n)*(a + b*x)**(m + 1)*(c + d*x
)**n*(e + f*x)**(-m - n)*(A*b - B*a)*appellf1(m + 1, -n, m + n, m + 2, d*(a + b*
x)/(a*d - b*c), f*(a + b*x)/(a*f - b*e))/(b**2*(m + 1))

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Mathematica [A]  time = 4.65953, size = 0, normalized size = 0. \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.22, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( Bx+A \right ) \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-m-n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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