3.3102 \(\int \frac{(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx\)

Optimal. Leaf size=85 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^3} \]

[Out]

((b*c - a*d)^2*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[3, 1 + m,
2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^3*(1 + m))

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Rubi [A]  time = 0.101297, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^3,x]

[Out]

((b*c - a*d)^2*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[3, 1 + m,
2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^3*(1 + m))

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Rubi in Sympy [A]  time = 11.751, size = 68, normalized size = 0.8 \[ - \frac{\left (a + b x\right )^{m - 2} \left (c + d x\right )^{- m + 2} \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - m + 2, 3 \\ - m + 3 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{\left (- m + 2\right ) \left (c f - d e\right )^{3}} \]

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)**3,x)

[Out]

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

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Mathematica [C]  time = 2.84042, size = 933, normalized size = 10.98 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (d (2 b e-2 a f) (e+f x) \left ((b e-a f) (c+d x) (a f (m+1)+b (f m x-e)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m\right )-(a+b x) \left ((a f (m+1) (d (e-f x)-2 c f)+b (c f (e (m+2)-f m x)+d e (f (2 m+1) x-e))) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (c f-d e) (m+1) (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )\right ) \, _2F_1\left (m,m+1;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right ) \left (\frac{(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^m-d e (b e-a f)^2 (m+1) (c+d x) \left ((-2 b e+a f (m+1)+b f (m-1) x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m\right )-2 (a f (m+1)+b (f m x-e)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (m+1) (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )+c f (b e-a f)^2 (m+1) (c+d x) \left ((-2 b e+a f (m+1)+b f (m-1) x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m\right )-2 (a f (m+1)+b (f m x-e)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (m+1) (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )\right )}{f (2 b e-2 a f) (b e-a f) (m+1) (e+f x)^2 \left ((b e-a f) (c+d x) (a f (m+1)+b (f m x-e)) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m\right )-(a+b x) \left ((a f (m+1) (d (e-f x)-2 c f)+b (c f (e (m+2)-f m x)+d e (f (2 m+1) x-e))) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+1\right )+f (c f-d e) (m+1) (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^3,x]

[Out]

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

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m}}{ \left ( fx+e \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^3,x, algorithm="giac")

[Out]

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