\(\int \frac {x (a+b x)^n}{c+d x} \, dx\) [938]

Optimal result

Integrand size = 16, antiderivative size = 78 \[ \int \frac {x (a+b x)^n}{c+d x} \, dx=\frac {(a+b x)^{1+n}}{b d (1+n)}-\frac {c (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d (b c-a d) (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 78, 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 = {81, 70} \[ \int \frac {x (a+b x)^n}{c+d x} \, dx=\frac {(a+b x)^{n+1}}{b d (n+1)}-\frac {c (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)} \]

[In]

Int[(x*(a + b*x)^n)/(c + d*x),x]

[Out]

(a + b*x)^(1 + n)/(b*d*(1 + n)) - (c*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -((d*(a + b*x))/(b*c
- a*d))])/(d*(b*c - a*d)*(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 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 {(a+b x)^{1+n}}{b d (1+n)}-\frac {c \int \frac {(a+b x)^n}{c+d x} \, dx}{d} \\ & = \frac {(a+b x)^{1+n}}{b d (1+n)}-\frac {c (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) (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int \frac {x (a+b x)^n}{c+d x} \, dx=-\frac {(a+b x)^{1+n} \left (-b c+a d+b c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )\right )}{b d (b c-a d) (1+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*x + a)^n*x/(d*x + c), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*(a + b*x)**n/(c + d*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate((b*x + a)^n*x/(d*x + c), x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x + a)^n*x/(d*x + c), x)

Mupad [F(-1)]

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

[In]

int((x*(a + b*x)^n)/(c + d*x),x)

[Out]

int((x*(a + b*x)^n)/(c + d*x), x)