Integrand size = 17, antiderivative size = 57 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=-\frac {b \left (a+\frac {b}{c+d x^2}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {b}{a \left (c+d x^2\right )}\right )}{2 a^2 d (1+p)} \] Output:
-1/2*b*(a+b/(d*x^2+c))^(p+1)*hypergeom([2, p+1],[2+p],1+b/a/(d*x^2+c))/a^2 /d/(p+1)
Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.39 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=-\frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^p \left (1+\frac {a \left (c+d x^2\right )}{b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {a \left (c+d x^2\right )}{b}\right )}{2 d (-1+p)} \] Input:
Integrate[x*(a + b/(c + d*x^2))^p,x]
Output:
-1/2*((c + d*x^2)*(a + b/(c + d*x^2))^p*Hypergeometric2F1[1 - p, -p, 2 - p , -((a*(c + d*x^2))/b)])/(d*(-1 + p)*(1 + (a*(c + d*x^2))/b)^p)
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2024, 773, 75}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx\) |
\(\Big \downarrow \) 2024 |
\(\displaystyle \frac {\int \left (a+\frac {b}{d x^2+c}\right )^pd\left (d x^2+c\right )}{2 d}\) |
\(\Big \downarrow \) 773 |
\(\displaystyle -\frac {\int \left (d x^2+c\right )^2 \left (a+\frac {b}{d x^2+c}\right )^pd\frac {1}{d x^2+c}}{2 d}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {b \left (a+\frac {b}{c+d x^2}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {b}{a \left (d x^2+c\right )}+1\right )}{2 a^2 d (p+1)}\) |
Input:
Int[x*(a + b/(c + d*x^2))^p,x]
Output:
-1/2*(b*(a + b/(c + d*x^2))^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + b/(a*(c + d*x^2))])/(a^2*d*(1 + p))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && !IntegerQ[p]
Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[P q, x], r = Expon[Qr, x]}, Simp[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q]) Subst[ Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D [Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] & & PolyQ[Qr, x]
\[\int x \left (a +\frac {b}{d \,x^{2}+c}\right )^{p}d x\]
Input:
int(x*(a+b/(d*x^2+c))^p,x)
Output:
int(x*(a+b/(d*x^2+c))^p,x)
\[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b/(d*x^2+c))^p,x, algorithm="fricas")
Output:
integral(x*((a*d*x^2 + a*c + b)/(d*x^2 + c))^p, x)
\[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=\int x \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{p}\, dx \] Input:
integrate(x*(a+b/(d*x**2+c))**p,x)
Output:
Integral(x*((a*c + a*d*x**2 + b)/(c + d*x**2))**p, x)
\[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b/(d*x^2+c))^p,x, algorithm="maxima")
Output:
integrate((a + b/(d*x^2 + c))^p*x, x)
\[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b/(d*x^2+c))^p,x, algorithm="giac")
Output:
integrate((a + b/(d*x^2 + c))^p*x, x)
Time = 9.50 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.37 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=-\frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^p\,\left (d\,x^2+c\right )\,{{}}_2{\mathrm {F}}_1\left (1-p,-p;\ 2-p;\ -\frac {a\,\left (d\,x^2+c\right )}{b}\right )}{2\,d\,{\left (\frac {a\,\left (d\,x^2+c\right )}{b}+1\right )}^p\,\left (p-1\right )} \] Input:
int(x*(a + b/(c + d*x^2))^p,x)
Output:
-((a + b/(c + d*x^2))^p*(c + d*x^2)*hypergeom([1 - p, -p], 2 - p, -(a*(c + d*x^2))/b))/(2*d*((a*(c + d*x^2))/b + 1)^p*(p - 1))
\[ \int x \left (a+\frac {b}{c+d x^2}\right )^p \, dx=\int \frac {\left (a d \,x^{2}+a c +b \right )^{p} x}{\left (d \,x^{2}+c \right )^{p}}d x \] Input:
int(x*(a+b/(d*x^2+c))^p,x)
Output:
int(((a*c + a*d*x**2 + b)**p*x)/(c + d*x**2)**p,x)