3.3050 \(\int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^3} \, dx\)

Optimal. Leaf size=174 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} (b (2 d e-c f (1-m))-a d f (m+1)) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (m+1) (b e-a f)^3 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.273, antiderivative size = 173, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+1)-b c f (1-m)+2 b d e) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (m+1) (b e-a f)^3 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m/((c + d*x)^m*(e + f*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 32.6892, size = 141, normalized size = 0.81 \[ - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 1}}{2 \left (e + f x\right )^{2} \left (a f - b e\right ) \left (c f - d e\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (a d - b c\right ) \left (- a d f \left (m + 1\right ) - b c f \left (- m + 1\right ) + 2 b d e\right ){{}_{2}F_{1}\left (\begin{matrix} m + 1, 2 \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{2 \left (m + 1\right ) \left (a f - b e\right )^{3} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m/((d*x+c)**m)/(f*x+e)**3,x)

[Out]

-f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 1)/(2*(e + f*x)**2*(a*f - b*e)*(c*f - d*e
)) - (a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(a*d - b*c)*(-a*d*f*(m + 1) - b*c*f*
(-m + 1) + 2*b*d*e)*hyper((m + 1, 2), (m + 2,), (-a - b*x)*(-c*f + d*e)/((c + d*
x)*(a*f - b*e)))/(2*(m + 1)*(a*f - b*e)**3*(c*f - d*e))

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Mathematica [C]  time = 3.01598, size = 432, normalized size = 2.48 \[ \frac{(b e-a f)^4 (a+b x)^{m+1} (c+d x)^{-m} \left ((a f (m+1)-2 b e+b f (m-1) x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m\right )-2 (a f (m+1)+b (f m x-e)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (m+1) (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )}{(e+f x)^2 (2 b e-2 a f) (a f-b e)^3 \left ((b e-a f) (b (e-f m x)-a f (m+1)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m\right )+\frac{(a+b x) \left ((a f (m+1) (d (e-f x)-2 c f)+b (c f (e (m+2)-f m x)+d e (f (2 m+1) x-e))) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (m+1) (a+b x) (c f-d e) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )}{c+d x}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m/((c + d*x)^m*(e + f*x)^3),x]

[Out]

((b*e - a*f)^4*(a + b*x)^(1 + m)*((-2*b*e + a*f*(1 + m) + b*f*(-1 + m)*x)*Hurwit
zLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 2*(a*f*(1 + m
) + b*(-e + f*m*x))*HurwitzLerchPhi[((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]))/((2*b*e - 2*a*f)*(-(b*e) + a*f)^3*(c + d*x)^m*
(e + f*x)^2*((b*e - a*f)*(-(a*f*(1 + m)) + b*(e - f*m*x))*HurwitzLerchPhi[((d*e
- c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + ((a + b*x)*((a*f*(1 + m)*(-2*
c*f + d*(e - f*x)) + b*(c*f*(e*(2 + m) - f*m*x) + d*e*(-e + f*(1 + 2*m)*x)))*Hur
witzLerchPhi[((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)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f
)*(c + d*x)), 1, 2 + m]))/(c + d*x)))

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/((d*x+c)^m)/(f*x+e)^3,x)

[Out]

int((b*x+a)^m/((d*x+c)^m)/(f*x+e)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m}}{{\left (f x + e\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m/((f*x + e)^3*(d*x + c)^m),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m)/(f*x + e)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}}{{\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}{\left (d x + c\right )}^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m/((f*x + e)^3*(d*x + c)^m),x, algorithm="fricas")

[Out]

integral((b*x + a)^m/((f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3)*(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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{3}{\left (d x + c\right )}^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m/((f*x + e)^3*(d*x + c)^m),x, algorithm="giac")

[Out]

integrate((b*x + a)^m/((f*x + e)^3*(d*x + c)^m), x)