3.144 \(\int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-1-m-n} \, dx\)

Optimal. Leaf size=283 \[ \frac{B (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b f (m+1)}-\frac{(a+b x)^{m+1} (B e-A f) (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{f (m+1) (b e-a f)} \]

[Out]

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

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Rubi [A]  time = 0.807217, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{B (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b f (m+1)}-\frac{(a+b x)^{m+1} (B e-A f) (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{f (m+1) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(A + B*x)*(c + d*x)^n*(e + f*x)^(-1 - m - n),x]

[Out]

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

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Rubi in Sympy [A]  time = 171.238, size = 218, normalized size = 0.77 \[ \frac{B \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{m + n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (e + f x\right )^{- m - n} \operatorname{appellf_{1}}{\left (m + 1,- n,m + n,m + 2,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{b f \left (m + 1\right )} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{m + n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (e + f x\right )^{- m - n} \left (A f - B e\right ) \operatorname{appellf_{1}}{\left (m + 1,- n,m + n + 1,m + 2,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{f \left (m + 1\right ) \left (a f - b e\right )} \]

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

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Mathematica [B]  time = 2.85626, size = 576, normalized size = 2.04 \[ \frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{\left (A-\frac{B e}{f}\right ) F_1\left (m+1;-n,m+n+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{(e+f x) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;-n,m+n+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (d n (a f-b e) F_1\left (m+2;1-n,m+n+1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+f (m+n+1) (b c-a d) F_1\left (m+2;-n,m+n+2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}+\frac{B F_1\left (m+1;-n,m+n;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{f \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;-n,m+n;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (d n (a f-b e) F_1\left (m+2;1-n,m+n;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+f (m+n) (b c-a d) F_1\left (m+2;-n,m+n+1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}\right )}{b (m+1)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(A + B*x)*(c + d*x)^n*(e + f*x)^(-1 - m - n),x]

[Out]

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

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Maple [F]  time = 0.208, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( Bx+A \right ) \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-1-m-n}\, dx \]

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)^(-1-m-n),x)

[Out]

int((b*x+a)^m*(B*x+A)*(d*x+c)^n*(f*x+e)^(-1-m-n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 1}\,{d x} \]

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 - 1),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 1}, x\right ) \]

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 - 1),x, algorithm="fricas")

[Out]

integral((B*x + A)*(b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 1), 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)**(-1-m-n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 1}\,{d x} \]

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 - 1),x, algorithm="giac")

[Out]

integrate((B*x + A)*(b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 1), x)