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]

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

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Rubi [A]  time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 1.00 \[ \frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (2,m+1;m+2;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

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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fricas [F]  time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m}}{{\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} {\left (d x + c\right )}^{m}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{2} {\left (d x + c\right )}^{m}}\,{d x} \]

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)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m}}{\left (f x +e \right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{2} {\left (d x + c\right )}^{m}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^m} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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