3.3100 \(\int \frac{(a+b x)^m (c+d x)^{1-m}}{e+f x} \, dx\)

Optimal. Leaf size=220 \[ \frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}-\frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f^2 m}+\frac{d (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f (m+1)} \]

[Out]

((d*e - c*f)*(a + b*x)^m*Hypergeometric2F1[1, m, 1 + m, ((d*e - c*f)*(a + b*x))/
((b*e - a*f)*(c + d*x))])/(f^2*m*(c + d*x)^m) - ((d*e - c*f)*(a + b*x)^m*((b*(c
+ d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, m, 1 + m, -((d*(a + b*x))/(b*c - a*d
))])/(f^2*m*(c + d*x)^m) + (d*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hy
pergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*f*(1 + m)*(c
+ d*x)^m)

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Rubi [A]  time = 0.39908, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}-\frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f^2 m}+\frac{d (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d*e - c*f)*(a + b*x)^m*Hypergeometric2F1[1, m, 1 + m, ((d*e - c*f)*(a + b*x))/
((b*e - a*f)*(c + d*x))])/(f^2*m*(c + d*x)^m) - ((d*e - c*f)*(a + b*x)^m*((b*(c
+ d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, m, 1 + m, -((d*(a + b*x))/(b*c - a*d
))])/(f^2*m*(c + d*x)^m) + (d*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hy
pergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*f*(1 + m)*(c
+ d*x)^m)

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Rubi in Sympy [A]  time = 53.7334, size = 178, normalized size = 0.81 \[ - \frac{\left (a + b x\right )^{m - 1} \left (c + d x\right )^{- m + 1} \left (a f - b e\right ){{}_{2}F_{1}\left (\begin{matrix} - m + 1, 1 \\ - m + 2 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{f^{2} \left (- m + 1\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m} \left (c f - d e\right ){{}_{2}F_{1}\left (\begin{matrix} m, m \\ m + 1 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{f^{2} m} + \frac{d \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b f \left (m + 1\right )} \]

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),x)

[Out]

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

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Mathematica [C]  time = 1.35865, size = 622, normalized size = 2.83 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (b (m+1) (c+d x) (e+f x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right ) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )-d e (m-1) (m+2) (a+b x) (a d-b c) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-c f (m-1) (m+2) (a+b x) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )}{b f (1-m) (m+1) (e+f x) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(-(d*(-(b*c) + a*d)*e*(b*e - a*f)*(-1 + m)*(2 + m)*(a + b*x)*Appell
F1[1 + m, m, 1, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f
)]) - c*(b*c - a*d)*f*(b*e - a*f)*(-1 + m)*(2 + m)*(a + b*x)*AppellF1[1 + m, m,
1, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + (b*(1 +
m)*(c + d*x)*(e + f*x)*((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)]))*Hypergeometric2F1[
1 - m, -m, 2 - m, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*c) + a*d))^m))
/(b*f*(1 - m)*(1 + m)*(c + d*x)^m*(e + f*x)*((b*c - a*d)*(b*e - a*f)*(2 + m)*App
ellF1[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)]
)))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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