3.3101 \(\int \frac{(a+b x)^m (c+d x)^{1-m}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=108 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m F_1\left (m+1;m-1,2;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]

[Out]

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

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Rubi [A]  time = 0.216053, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m F_1\left (m+1;m-1,2;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.1598, size = 83, normalized size = 0.77 \[ - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right ) \operatorname{appellf_{1}}{\left (m + 1,2,m - 1,m + 2,\frac{f \left (a + b x\right )}{a f - b e},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{\left (m + 1\right ) \left (a f - b e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(1-m)/(f*x+e)**2,x)

[Out]

-(b*(-c - d*x)/(a*d - b*c))**m*(a + b*x)**(m + 1)*(c + d*x)**(-m)*(a*d - b*c)*ap
pellf1(m + 1, 2, m - 1, m + 2, f*(a + b*x)/(a*f - b*e), d*(a + b*x)/(a*d - b*c))
/((m + 1)*(a*f - b*e)**2)

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Mathematica [B]  time = 0.540565, size = 461, normalized size = 4.27 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (-\frac{d (m+2) (b c-a d) (b e-a f)^3 F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b f (a f-b e) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}+c \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^m \, _2F_1\left (m,m+1;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )-\frac{d e \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^m \, _2F_1\left (m,m+1;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{f}\right )}{(m+1) (e+f x) (b e-a f)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^(1 + m)*(-((d*(b*c - a*d)*(b*e - a*f)^3*(2 + m)*AppellF1[1 + m, m, 1,
2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*f*(-(b*e
) + a*f)*((b*c - a*d)*(b*e - a*f)*(2 + m)*AppellF1[1 + m, m, 1, 2 + m, (d*(a + b
*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + (a + b*x)*((-(b*c*f) + a*d*
f)*AppellF1[2 + m, m, 2, 3 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b
*e) + a*f)] + d*(-(b*e) + a*f)*m*AppellF1[2 + m, 1 + m, 1, 3 + m, (d*(a + b*x))/
(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])))) + c*(((b*e - a*f)*(c + d*x))/(
(b*c - a*d)*(e + f*x)))^m*Hypergeometric2F1[m, 1 + m, 2 + m, ((-(d*e) + c*f)*(a
+ b*x))/((b*c - a*d)*(e + f*x))] - (d*e*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e
+ f*x)))^m*Hypergeometric2F1[m, 1 + m, 2 + m, ((-(d*e) + c*f)*(a + b*x))/((b*c
- a*d)*(e + f*x))])/f))/((b*e - a*f)*(1 + m)*(c + d*x)^m*(e + f*x))

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m}}{ \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^2,x, algorithm="giac")

[Out]

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