3.3067 \(\int \frac{(a+b x)^m (c+d x)^{-2-m}}{e+f x} \, dx\)

Optimal. Leaf size=135 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.218258, antiderivative size = 135, 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 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.206, size = 104, normalized size = 0.77 \[ \frac{d \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1}}{\left (m + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} m + 1, 1 \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{\left (m + 1\right ) \left (a f - b e\right ) \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 6.67473, size = 578, normalized size = 4.28 \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (m^2 \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )-\frac{f m (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )}{a f-b e}+5 m \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )-\frac{3 f (a+b x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )}{a f-b e}+6 \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )-\frac{f (a+b x)^2 (c f-d e) \, _2F_1\left (2,m+3;m+4;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(c+d x) (b e-a f)^2}+\frac{(a+b x) (d e-c f) \, _2F_1\left (2,m+3;m+4;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(c+d x) (b e-a f)}\right )}{(m+3) (a f-b e) \left (-\frac{b (m+2) (e+f x) \Phi \left (\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)},1,m+2\right )}{b e-a f}+\frac{b (a+b x) (e+f x) (c f-d e) \, _2F_1\left (2,m+3;m+4;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+3) (c+d x) (b e-a f)^2}+\frac{-a d (m+1)+b c (m+2)+b d x}{b c-a d}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((a + b*x)^(1 + m)*(c + d*x)^(-2 - m)*(6*HurwitzLerchPhi[((d*e - c*f)*(a + b*x
))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + 5*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*
x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + m^2*HurwitzLerchPhi[((d*e - c*f)*(a + b
*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] - (3*f*(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) - (f*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) + ((d*e - c*f)*(a + b*x)*Hypergeometric2F1[2, 3 + m, 4 + m, ((d*
e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)*(c + d*x)) - (f*(-(d*
e) + c*f)*(a + b*x)^2*Hypergeometric2F1[2, 3 + m, 4 + m, ((d*e - c*f)*(a + b*x))
/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^2*(c + d*x))))/((-(b*e) + a*f)*(3 + m)*(
(-(a*d*(1 + m)) + b*c*(2 + m) + b*d*x)/(b*c - a*d) - (b*(2 + m)*(e + f*x)*Hurwit
zLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m])/(b*e - a*f
) + (b*(-(d*e) + c*f)*(a + b*x)*(e + f*x)*Hypergeometric2F1[2, 3 + m, 4 + m, ((d
*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^2*(3 + m)*(c + d*x))
)))

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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