3.3066 \(\int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx\)

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 = 0.022019, 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, Rules used = {131} \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2} \]

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))

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx &=\frac{(b c-a d) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (2,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^2 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0310563, size = 84, normalized size = 1.01 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (2,m+1;m+2;-\frac{(c f-d e) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(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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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) ^{2}}}\, dx \end{align*}

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. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{2}{\left (d x + c\right )}^{m}}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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

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, algorithm="giac")

[Out]

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