3.3049 \(\int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=83 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2} \]

[Out]

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

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Rubi [A]  time = 0.0840944, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4965, size = 65, normalized size = 0.78 \[ \frac{\left (a + b x\right )^{m - 1} \left (c + d x\right )^{- m + 1} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} - m + 1, 2 \\ - m + 2 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{\left (- m + 1\right ) \left (c f - d e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.625097, size = 113, normalized size = 1.36 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^m \, _2F_1\left (m,m+1;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (e+f x) (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + m)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^m*Hypergeom
etric2F1[m, 1 + m, 2 + m, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))])/(
(b*e - a*f)*(1 + m)*(c + d*x)^m*(e + f*x))

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^m/((f^2*x^2 + 2*e*f*x + e^2)*(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)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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