3.3048 \(\int \frac{(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx\)

Optimal. Leaf size=128 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f m}-\frac{(a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \]

[Out]

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

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Rubi [A]  time = 0.228846, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f m}-\frac{(a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.0179, size = 95, normalized size = 0.74 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, m \\ m + 1 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{f m} - \frac{\left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, 1 \\ m + 1 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{f m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.868285, size = 292, normalized size = 2.28 \[ -\frac{(m+2) (b c-a d) (b e-a f)^2 (a+b x)^{m+1} (c+d x)^{-m} F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) (e+f x) (a f-b e) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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