\(\int \frac {\sqrt {x} (c+d x)^n}{(a+b x^2)^2} \, dx\) [251]

Optimal result

Integrand size = 22, antiderivative size = 392 \[ \int \frac {\sqrt {x} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\frac {d \sqrt {x} (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right )}+\frac {b x^{3/2} (c-d x) (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {\left (b c^2+a d^2 (1-2 n)+2 \sqrt {-a} \sqrt {b} c d n\right ) \sqrt {x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 (-a)^{3/2} \sqrt {b} \left (b c^2+a d^2\right )}+\frac {\left (b c^2+a d^2 (1-2 n)-2 \sqrt {-a} \sqrt {b} c d n\right ) \sqrt {x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 (-a)^{3/2} \sqrt {b} \left (b c^2+a d^2\right )}-\frac {c d (1+2 n) \sqrt {x} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {d x}{c}\right )}{2 a \left (b c^2+a d^2\right )} \] Output:

1/2*d*x^(1/2)*(d*x+c)^(1+n)/a/(a*d^2+b*c^2)+1/2*b*x^(3/2)*(-d*x+c)*(d*x+c) 
^(1+n)/a/(a*d^2+b*c^2)/(b*x^2+a)-1/4*(b*c^2+a*d^2*(1-2*n)+2*(-a)^(1/2)*b^( 
1/2)*c*d*n)*x^(1/2)*(d*x+c)^n*AppellF1(1/2,1,-n,3/2,-b^(1/2)*x/(-a)^(1/2), 
-d*x/c)/(-a)^(3/2)/b^(1/2)/(a*d^2+b*c^2)/((1+d*x/c)^n)+1/4*(b*c^2+a*d^2*(1 
-2*n)-2*(-a)^(1/2)*b^(1/2)*c*d*n)*x^(1/2)*(d*x+c)^n*AppellF1(1/2,1,-n,3/2, 
b^(1/2)*x/(-a)^(1/2),-d*x/c)/(-a)^(3/2)/b^(1/2)/(a*d^2+b*c^2)/((1+d*x/c)^n 
)-1/2*c*d*(1+2*n)*x^(1/2)*(d*x+c)^n*hypergeom([1/2, -n],[3/2],-d*x/c)/a/(a 
*d^2+b*c^2)/(((d*x+c)/c)^n)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.65, 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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b \sqrt {x} (c+d x)^n}{2 a \left (-a b-b^2 x^2\right )}-\frac {b \sqrt {x} (c+d x)^n}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sqrt {x} (c+d x)^n}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{3/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},-\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{6 a^2}+\frac {x^{3/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{6 a^2}+\frac {x^{3/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {3}{2},2,-n,\frac {5}{2},-\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{6 a^2}+\frac {x^{3/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {3}{2},2,-n,\frac {5}{2},\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{6 a^2}\)

Input:

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

Output:

(x^(3/2)*(c + d*x)^n*AppellF1[3/2, 1, -n, 5/2, -((Sqrt[b]*x)/Sqrt[-a]), -( 
(d*x)/c)])/(6*a^2*(1 + (d*x)/c)^n) + (x^(3/2)*(c + d*x)^n*AppellF1[3/2, 1, 
 -n, 5/2, (Sqrt[b]*x)/Sqrt[-a], -((d*x)/c)])/(6*a^2*(1 + (d*x)/c)^n) + (x^ 
(3/2)*(c + d*x)^n*AppellF1[3/2, 2, -n, 5/2, -((Sqrt[b]*x)/Sqrt[-a]), -((d* 
x)/c)])/(6*a^2*(1 + (d*x)/c)^n) + (x^(3/2)*(c + d*x)^n*AppellF1[3/2, 2, -n 
, 5/2, (Sqrt[b]*x)/Sqrt[-a], -((d*x)/c)])/(6*a^2*(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]

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

Input:

int(x^(1/2)*(d*x+c)^n/(b*x^2+a)^2,x)
 

Output:

int(x^(1/2)*(d*x+c)^n/(b*x^2+a)^2,x)
 

Fricas [F]

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

integrate(x^(1/2)*(d*x+c)^n/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

integral((d*x + c)^n*sqrt(x)/(b^2*x^4 + 2*a*b*x^2 + a^2), x)
 

Sympy [F(-1)]

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

integrate(x**(1/2)*(d*x+c)**n/(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^(1/2)*(d*x+c)^n/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^(1/2)*(d*x+c)^n/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((x^(1/2)*(c + d*x)^n)/(a + b*x^2)^2,x)
 

Output:

int((x^(1/2)*(c + d*x)^n)/(a + b*x^2)^2, x)
 

Reduce [F]

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

int(x^(1/2)*(d*x+c)^n/(b*x^2+a)^2,x)
 

Output:

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