3.3065 \(\int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx\)

Optimal. Leaf size=128 \[ \frac {(a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac {d (a+b x)}{b c-a d}\right )}{f m}-\frac {(a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {105, 70, 69, 131} \[ \frac {(a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac {d (a+b x)}{b c-a d}\right )}{f m}-\frac {(a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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

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Mathematica [A]  time = 0.04, size = 106, normalized size = 0.83 \[ \frac {(a+b x)^m (c+d x)^{-m} \left (\left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;\frac {d (a+b x)}{a d-b c}\right )-\, _2F_1\left (1,m;m+1;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{f m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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