3.3059 \(\int \frac{(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=158 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+1)-b (c f m+d e)) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{m (m+1) (b e-a f)^2 (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x) (b c-a d) (d e-c f)} \]

[Out]

(d*(a + b*x)^(1 + m))/((b*c - a*d)*(d*e - c*f)*m*(c + d*x)^m*(e + f*x)) + ((a*d*
f*(1 + m) - b*(d*e + c*f*m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric
2F1[2, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a
*f)^2*(d*e - c*f)*m*(1 + m))

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Rubi [A]  time = 0.271747, antiderivative size = 158, 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 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+1)-b (c f m+d e)) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{m (m+1) (b e-a f)^2 (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x) (b c-a d) (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(a + b*x)^(1 + m))/((b*c - a*d)*(d*e - c*f)*m*(c + d*x)^m*(e + f*x)) + ((a*d*
f*(1 + m) - b*(d*e + c*f*m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric
2F1[2, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a
*f)^2*(d*e - c*f)*m*(1 + m))

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Rubi in Sympy [A]  time = 28.3313, size = 124, normalized size = 0.78 \[ - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m}}{\left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- a d f \left (m + 1\right ) + b c f m + b d e\right ){{}_{2}F_{1}\left (\begin{matrix} m + 1, 1 \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{\left (m + 1\right ) \left (a f - b e\right )^{2} \left (c f - d e\right )} \]

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)**2,x)

[Out]

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

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Mathematica [C]  time = 1.9125, size = 286, normalized size = 1.81 \[ -\frac{(b e-a f)^3 (a+b x)^{m+1} (c+d x)^{-m} \left ((b (e-f m x)-a f (m+1)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (m+1) (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )}{(e+f x) (a f-b e)^3 \left (f (m+1) (a+b x)^2 (d e-c f) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )+f (m+1) (a+b x) (c+d x) (a f-b e) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+b (c+d x) (e+f x) (b e-a f)\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((b*e - a*f)^3*(a + b*x)^(1 + m)*((-(a*f*(1 + m)) + b*(e - f*m*x))*HurwitzLerc
hPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + f*(1 + m)*(a +
b*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m]
))/((-(b*e) + a*f)^3*(c + d*x)^m*(e + f*x)*(b*(b*e - a*f)*(c + d*x)*(e + f*x) +
f*(-(b*e) + a*f)*(1 + m)*(a + b*x)*(c + d*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b
*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + f*(d*e - c*f)*(1 + m)*(a + b*x)^2*Hurw
itzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m])))

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Maple [F]  time = 0.122, 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)