\(\int \frac {x (a+b x)^n}{(c+d x)^2} \, dx\) [944]

Optimal result

Integrand size = 16, antiderivative size = 99 \[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=-\frac {c (a+b x)^{1+n}}{d (b c-a d) (c+d x)}-\frac {(a d-b c (1+n)) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d (b c-a d)^2 (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 70} \[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=-\frac {(a+b x)^{n+1} (a d-b c (n+1)) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)^2}-\frac {c (a+b x)^{n+1}}{d (c+d x) (b c-a d)} \]

[In]

Int[(x*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

-((c*(a + b*x)^(1 + n))/(d*(b*c - a*d)*(c + d*x))) - ((a*d - b*c*(1 + n))*(a + b*x)^(1 + n)*Hypergeometric2F1[
1, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(d*(b*c - a*d)^2*(1 + n))

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 79

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+n} \left (\frac {c (-b c+a d)}{c+d x}+\frac {(-a d+b c (1+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{d (b c-a d)^2} \]

[In]

Integrate[(x*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

((a + b*x)^(1 + n)*((c*(-(b*c) + a*d))/(c + d*x) + ((-(a*d) + b*c*(1 + n))*Hypergeometric2F1[1, 1 + n, 2 + n,
(d*(a + b*x))/(-(b*c) + a*d)])/(1 + n)))/(d*(b*c - a*d)^2)

Maple [F]

\[\int \frac {x \left (b x +a \right )^{n}}{\left (d x +c \right )^{2}}d x\]

[In]

int(x*(b*x+a)^n/(d*x+c)^2,x)

[Out]

int(x*(b*x+a)^n/(d*x+c)^2,x)

Fricas [F]

\[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(x*(b*x+a)^n/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral((b*x + a)^n*x/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [F]

\[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\int \frac {x \left (a + b x\right )^{n}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(x*(b*x+a)**n/(d*x+c)**2,x)

[Out]

Integral(x*(a + b*x)**n/(c + d*x)**2, x)

Maxima [F]

\[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(x*(b*x+a)^n/(d*x+c)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x/(d*x + c)^2, x)

Giac [F]

\[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(x*(b*x+a)^n/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x/(d*x + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\int \frac {x\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int((x*(a + b*x)^n)/(c + d*x)^2,x)

[Out]

int((x*(a + b*x)^n)/(c + d*x)^2, x)