\(\int \frac {x^{5/2} (c+d x)^n}{(a+b x^2)^2} \, dx\) [249]

Optimal result

Integrand size = 22, antiderivative size = 464 \[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=-\frac {d \sqrt {x} (c+d x)^{1+n}}{2 b \left (b c^2+a d^2\right )}-\frac {c x^{3/2} (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right )}+\frac {d x^{5/2} (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right )}+\frac {b x^{7/2} (c-d x) (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )}+\frac {\left (3 b c^2-2 \sqrt {-a} \sqrt {b} c d n+a d^2 (3+2 n)\right ) \sqrt {x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 \sqrt {-a} b^{3/2} \left (b c^2+a d^2\right )}-\frac {\left (3 b c^2+2 \sqrt {-a} \sqrt {b} c d n+a d^2 (3+2 n)\right ) \sqrt {x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 \sqrt {-a} b^{3/2} \left (b c^2+a d^2\right )}+\frac {c d (1+2 n) \sqrt {x} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {d x}{c}\right )}{2 b \left (b c^2+a d^2\right )} \] Output:

-1/2*d*x^(1/2)*(d*x+c)^(1+n)/b/(a*d^2+b*c^2)-1/2*c*x^(3/2)*(d*x+c)^(1+n)/a 
/(a*d^2+b*c^2)+1/2*d*x^(5/2)*(d*x+c)^(1+n)/a/(a*d^2+b*c^2)+1/2*b*x^(7/2)*( 
-d*x+c)*(d*x+c)^(1+n)/a/(a*d^2+b*c^2)/(b*x^2+a)+1/4*(3*b*c^2-2*(-a)^(1/2)* 
b^(1/2)*c*d*n+a*d^2*(3+2*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)^(1/2)/b^(3/2)/(a*d^2+b*c^2)/((1+d*x/c)^n)-1/ 
4*(3*b*c^2+2*(-a)^(1/2)*b^(1/2)*c*d*n+a*d^2*(3+2*n))*x^(1/2)*(d*x+c)^n*App 
ellF1(1/2,1,-n,3/2,b^(1/2)*x/(-a)^(1/2),-d*x/c)/(-a)^(1/2)/b^(3/2)/(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)/b/(a*d^2+b*c^2)/(((d*x+c)/c)^n)
 

Mathematica [F]

\[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx \] Input:

Integrate[(x^(5/2)*(c + d*x)^n)/(a + b*x^2)^2,x]
 

Output:

Integrate[(x^(5/2)*(c + d*x)^n)/(a + b*x^2)^2, x]
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.55, 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.

\(\displaystyle \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b x^{5/2} (c+d x)^n}{2 a \left (-a b-b^2 x^2\right )}-\frac {b x^{5/2} (c+d x)^n}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b x^{5/2} (c+d x)^n}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {7}{2},1,-n,\frac {9}{2},-\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{14 a^2}+\frac {x^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {7}{2},1,-n,\frac {9}{2},\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{14 a^2}+\frac {x^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {7}{2},2,-n,\frac {9}{2},-\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{14 a^2}+\frac {x^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {7}{2},2,-n,\frac {9}{2},\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{14 a^2}\)

Input:

Int[(x^(5/2)*(c + d*x)^n)/(a + b*x^2)^2,x]
 

Output:

(x^(7/2)*(c + d*x)^n*AppellF1[7/2, 1, -n, 9/2, -((Sqrt[b]*x)/Sqrt[-a]), -( 
(d*x)/c)])/(14*a^2*(1 + (d*x)/c)^n) + (x^(7/2)*(c + d*x)^n*AppellF1[7/2, 1 
, -n, 9/2, (Sqrt[b]*x)/Sqrt[-a], -((d*x)/c)])/(14*a^2*(1 + (d*x)/c)^n) + ( 
x^(7/2)*(c + d*x)^n*AppellF1[7/2, 2, -n, 9/2, -((Sqrt[b]*x)/Sqrt[-a]), -(( 
d*x)/c)])/(14*a^2*(1 + (d*x)/c)^n) + (x^(7/2)*(c + d*x)^n*AppellF1[7/2, 2, 
 -n, 9/2, (Sqrt[b]*x)/Sqrt[-a], -((d*x)/c)])/(14*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]

\[\int \frac {x^{\frac {5}{2}} \left (d x +c \right )^{n}}{\left (b \,x^{2}+a \right )^{2}}d x\]

Input:

int(x^(5/2)*(d*x+c)^n/(b*x^2+a)^2,x)
 

Output:

int(x^(5/2)*(d*x+c)^n/(b*x^2+a)^2,x)
 

Fricas [F]

\[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(x^(5/2)*(d*x+c)^n/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

integral((d*x + c)^n*x^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**(5/2)*(d*x+c)**n/(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(x^(5/2)*(d*x+c)^n/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^n*x^(5/2)/(b*x^2 + a)^2, x)
 

Giac [F]

\[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(x^(5/2)*(d*x+c)^n/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)^n*x^(5/2)/(b*x^2 + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{5/2}\,{\left (c+d\,x\right )}^n}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:

int((x^(5/2)*(c + d*x)^n)/(a + b*x^2)^2,x)
 

Output:

int((x^(5/2)*(c + d*x)^n)/(a + b*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {x^{5/2} (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\text {too large to display} \] Input:

int(x^(5/2)*(d*x+c)^n/(b*x^2+a)^2,x)
 

Output:

(4*sqrt(x)*(c + d*x)**n*a*d*n + 6*sqrt(x)*(c + d*x)**n*a*d + 6*sqrt(x)*(c 
+ d*x)**n*b*c*x - 4*int((c + d*x)**n/(2*sqrt(x)*a**2*c*n - sqrt(x)*a**2*c 
+ 2*sqrt(x)*a**2*d*n*x - sqrt(x)*a**2*d*x + 4*sqrt(x)*a*b*c*n*x**2 - 2*sqr 
t(x)*a*b*c*x**2 + 4*sqrt(x)*a*b*d*n*x**3 - 2*sqrt(x)*a*b*d*x**3 + 2*sqrt(x 
)*b**2*c*n*x**4 - sqrt(x)*b**2*c*x**4 + 2*sqrt(x)*b**2*d*n*x**5 - sqrt(x)* 
b**2*d*x**5),x)*a**3*c*d*n**2 - 4*int((c + d*x)**n/(2*sqrt(x)*a**2*c*n - s 
qrt(x)*a**2*c + 2*sqrt(x)*a**2*d*n*x - sqrt(x)*a**2*d*x + 4*sqrt(x)*a*b*c* 
n*x**2 - 2*sqrt(x)*a*b*c*x**2 + 4*sqrt(x)*a*b*d*n*x**3 - 2*sqrt(x)*a*b*d*x 
**3 + 2*sqrt(x)*b**2*c*n*x**4 - sqrt(x)*b**2*c*x**4 + 2*sqrt(x)*b**2*d*n*x 
**5 - sqrt(x)*b**2*d*x**5),x)*a**3*c*d*n + 3*int((c + d*x)**n/(2*sqrt(x)*a 
**2*c*n - sqrt(x)*a**2*c + 2*sqrt(x)*a**2*d*n*x - sqrt(x)*a**2*d*x + 4*sqr 
t(x)*a*b*c*n*x**2 - 2*sqrt(x)*a*b*c*x**2 + 4*sqrt(x)*a*b*d*n*x**3 - 2*sqrt 
(x)*a*b*d*x**3 + 2*sqrt(x)*b**2*c*n*x**4 - sqrt(x)*b**2*c*x**4 + 2*sqrt(x) 
*b**2*d*n*x**5 - sqrt(x)*b**2*d*x**5),x)*a**3*c*d - 4*int((c + d*x)**n/(2* 
sqrt(x)*a**2*c*n - sqrt(x)*a**2*c + 2*sqrt(x)*a**2*d*n*x - sqrt(x)*a**2*d* 
x + 4*sqrt(x)*a*b*c*n*x**2 - 2*sqrt(x)*a*b*c*x**2 + 4*sqrt(x)*a*b*d*n*x**3 
 - 2*sqrt(x)*a*b*d*x**3 + 2*sqrt(x)*b**2*c*n*x**4 - sqrt(x)*b**2*c*x**4 + 
2*sqrt(x)*b**2*d*n*x**5 - sqrt(x)*b**2*d*x**5),x)*a**2*b*c*d*n**2*x**2 - 4 
*int((c + d*x)**n/(2*sqrt(x)*a**2*c*n - sqrt(x)*a**2*c + 2*sqrt(x)*a**2*d* 
n*x - sqrt(x)*a**2*d*x + 4*sqrt(x)*a*b*c*n*x**2 - 2*sqrt(x)*a*b*c*x**2 ...