\(\int \frac {x^{3/2} (c+d x)^n}{a+b x^2} \, dx\) [244]

Optimal result

Integrand size = 22, antiderivative size = 128 \[ \int \frac {x^{3/2} (c+d x)^n}{a+b x^2} \, dx=\frac {x^{5/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},-n,1,\frac {7}{2},-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{5 a}+\frac {x^{5/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},-n,1,\frac {7}{2},-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{5 a} \] Output:

1/5*x^(5/2)*(d*x+c)^n*AppellF1(5/2,1,-n,7/2,-b^(1/2)*x/(-a)^(1/2),-d*x/c)/ 
a/((1+d*x/c)^n)+1/5*x^(5/2)*(d*x+c)^n*AppellF1(5/2,1,-n,7/2,b^(1/2)*x/(-a) 
^(1/2),-d*x/c)/a/((1+d*x/c)^n)
 

Mathematica [F]

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

Integrate[(x^(3/2)*(c + d*x)^n)/(a + b*x^2),x]
 

Output:

Integrate[(x^(3/2)*(c + d*x)^n)/(a + b*x^2), x]
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

\(\displaystyle \int \frac {x^{3/2} (c+d x)^n}{a+b x^2} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {\sqrt {-a} x^{3/2} (c+d x)^n}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} x^{3/2} (c+d x)^n}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{5/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},-\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{5 a}+\frac {x^{5/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{5 a}\)

Input:

Int[(x^(3/2)*(c + d*x)^n)/(a + b*x^2),x]
 

Output:

(x^(5/2)*(c + d*x)^n*AppellF1[5/2, 1, -n, 7/2, -((Sqrt[b]*x)/Sqrt[-a]), -( 
(d*x)/c)])/(5*a*(1 + (d*x)/c)^n) + (x^(5/2)*(c + d*x)^n*AppellF1[5/2, 1, - 
n, 7/2, (Sqrt[b]*x)/Sqrt[-a], -((d*x)/c)])/(5*a*(1 + (d*x)/c)^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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x^(3/2)*(d*x+c)^n/(b*x^2+a),x, algorithm="fricas")
 

Output:

integral((d*x + c)^n*x^(3/2)/(b*x^2 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{3/2} (c+d x)^n}{a+b x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate(x^(3/2)*(d*x+c)^n/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^n*x^(3/2)/(b*x^2 + a), x)
 

Giac [F]

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

integrate(x^(3/2)*(d*x+c)^n/(b*x^2+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^n*x^(3/2)/(b*x^2 + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^{3/2} (c+d x)^n}{a+b x^2} \, dx=\frac {2 \sqrt {x}\, \left (d x +c \right )^{n}-2 \left (\int \frac {\sqrt {x}\, \left (d x +c \right )^{n} x^{2}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) b d n -\left (\int \frac {\sqrt {x}\, \left (d x +c \right )^{n}}{b d \,x^{4}+b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) a c -2 \left (\int \frac {\sqrt {x}\, \left (d x +c \right )^{n}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a d n -\left (\int \frac {\sqrt {x}\, \left (d x +c \right )^{n}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a d}{b} \] Input:

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

Output:

(2*sqrt(x)*(c + d*x)**n - 2*int((sqrt(x)*(c + d*x)**n*x**2)/(a*c + a*d*x + 
 b*c*x**2 + b*d*x**3),x)*b*d*n - int((sqrt(x)*(c + d*x)**n)/(a*c*x + a*d*x 
**2 + b*c*x**3 + b*d*x**4),x)*a*c - 2*int((sqrt(x)*(c + d*x)**n)/(a*c + a* 
d*x + b*c*x**2 + b*d*x**3),x)*a*d*n - int((sqrt(x)*(c + d*x)**n)/(a*c + a* 
d*x + b*c*x**2 + b*d*x**3),x)*a*d)/b