\(\int \frac {(c+d x)^n}{x^{3/2} (a+b x^2)^2} \, dx\) [253]

Optimal result

Integrand size = 22, antiderivative size = 424 \[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {\left (5 b c^2+4 a d^2\right ) (c+d x)^{1+n}}{2 a^2 c \left (b c^2+a d^2\right ) \sqrt {x}}+\frac {b (c-d x) (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right ) \sqrt {x} \left (a+b x^2\right )}-\frac {\sqrt {b} \left (5 b c^2+a d^2 (5-2 n)+2 \sqrt {-a} \sqrt {b} c d n\right ) \sqrt {x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 (-a)^{5/2} \left (b c^2+a d^2\right )}+\frac {\sqrt {b} \left (5 b c^2+a d^2 (5-2 n)-2 \sqrt {-a} \sqrt {b} c d n\right ) \sqrt {x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 (-a)^{5/2} \left (b c^2+a d^2\right )}+\frac {d \left (5 b c^2+4 a d^2\right ) (1+2 n) \sqrt {x} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {d x}{c}\right )}{2 a^2 c \left (b c^2+a d^2\right )} \] Output:

-1/2*(4*a*d^2+5*b*c^2)*(d*x+c)^(1+n)/a^2/c/(a*d^2+b*c^2)/x^(1/2)+1/2*b*(-d 
*x+c)*(d*x+c)^(1+n)/a/(a*d^2+b*c^2)/x^(1/2)/(b*x^2+a)-1/4*b^(1/2)*(5*b*c^2 
+a*d^2*(5-2*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)^(5/2)/(a*d^2+b*c^2)/((1+d*x/c) 
^n)+1/4*b^(1/2)*(5*b*c^2+a*d^2*(5-2*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)^(5/2)/( 
a*d^2+b*c^2)/((1+d*x/c)^n)+1/2*d*(4*a*d^2+5*b*c^2)*(1+2*n)*x^(1/2)*(d*x+c) 
^n*hypergeom([1/2, -n],[3/2],-d*x/c)/a^2/c/(a*d^2+b*c^2)/(((d*x+c)/c)^n)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.60, 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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b (c+d x)^n}{2 a x^{3/2} \left (-a b-b^2 x^2\right )}-\frac {b (c+d x)^n}{4 a x^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b (c+d x)^n}{4 a x^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x + c)^n*sqrt(x)/(b^2*x^6 + 2*a*b*x^4 + a^2*x^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\int \frac {\left (d x +c \right )^{n}}{\sqrt {x}\, a^{2} x +2 \sqrt {x}\, a b \,x^{3}+\sqrt {x}\, b^{2} x^{5}}d x \] Input:

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

Output:

int((c + d*x)**n/(sqrt(x)*a**2*x + 2*sqrt(x)*a*b*x**3 + sqrt(x)*b**2*x**5) 
,x)