\(\int \frac {x (c+d x)^n}{a+b x^2} \, dx\) [217]

Optimal result

Integrand size = 18, antiderivative size = 145 \[ \int \frac {x (c+d x)^n}{a+b x^2} \, dx=-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 b \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) (1+n)}-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 b \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04 \[ \int \frac {x (c+d x)^n}{a+b x^2} \, dx=-\frac {(c+d x)^{1+n} \left (\left (\sqrt {b} c+\sqrt {-a} d\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )+\left (\sqrt {b} c-\sqrt {-a} d\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )\right )}{2 \sqrt {b} \left (b c^2+a d^2\right ) (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {(c+d x)^n}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {(c+d x)^n}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 \sqrt {b} (n+1) \left (\sqrt {b} c-\sqrt {-a} d\right )}-\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 \sqrt {b} (n+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (c+d x)^n}{a+b x^2} \, dx=\frac {\left (d x +c \right )^{n}-\left (\int \frac {\left (d x +c \right )^{n}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a d n +\left (\int \frac {\left (d x +c \right )^{n} x}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) b c n}{b n} \] Input:

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

Output:

((c + d*x)**n - int((c + d*x)**n/(a*c + a*d*x + b*c*x**2 + b*d*x**3),x)*a* 
d*n + int(((c + d*x)**n*x)/(a*c + a*d*x + b*c*x**2 + b*d*x**3),x)*b*c*n)/( 
b*n)