\(\int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx\) [3185]

Optimal result

Integrand size = 20, antiderivative size = 85 \[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\frac {B (d+e x)^{1+m}}{b e (1+m)}-\frac {(A b-a B) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e) (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {81, 70} \[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\frac {B (d+e x)^{m+1}}{b e (m+1)}-\frac {(A b-a B) (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)} \]

[In]

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

[Out]

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

Rule 70

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

Rule 81

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\frac {(d+e x)^{1+m} \left (B (b d-a e)+(-A b e+a B e) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )\right )}{b e (b d-a e) (1+m)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{b x +a}d x\]

[In]

int((B*x+A)*(e*x+d)^m/(b*x+a),x)

[Out]

int((B*x+A)*(e*x+d)^m/(b*x+a),x)

Fricas [F]

\[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b x + a} \,d x } \]

[In]

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

[Out]

integral((B*x + A)*(e*x + d)^m/(b*x + a), x)

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{m}}{a + b x}\, dx \]

[In]

integrate((B*x+A)*(e*x+d)**m/(b*x+a),x)

[Out]

Integral((A + B*x)*(d + e*x)**m/(a + b*x), x)

Maxima [F]

\[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b x + a} \,d x } \]

[In]

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

[Out]

integrate((B*x + A)*(e*x + d)^m/(b*x + a), x)

Giac [F]

\[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b x + a} \,d x } \]

[In]

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

[Out]

integrate((B*x + A)*(e*x + d)^m/(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^m}{a+b\,x} \,d x \]

[In]

int(((A + B*x)*(d + e*x)^m)/(a + b*x),x)

[Out]

int(((A + B*x)*(d + e*x)^m)/(a + b*x), x)