\(\int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx\) [3118]

Optimal result

Integrand size = 26, antiderivative size = 85 \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {133} \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \operatorname {Hypergeometric2F1}\left (3,m+1,m+2,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^3} \]

[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))

Rule 133

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)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (1+m)} \]

[In]

Integrate[((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))

Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{1-m}}{\left (f x +e \right )^{3}}d x\]

[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)

Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}} \,d x } \]

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m}}{{\left (e+f\,x\right )}^3} \,d x \]

[In]

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

[Out]

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