\(\int \sqrt {c x} (\frac {a}{x}+b x^n)^{3/2} \, dx\) [357]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \sqrt {c x} \left (\frac {a}{x}+b x^n\right )^{3/2} \, dx=\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{1+n}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (1+n)}-\frac {2 a^{3/2} c \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(1+n) \sqrt {c x}} \] Output:

2*a*(c*x)^(1/2)*(a/x+b*x^n)^(1/2)/(1+n)+2/3*(c*x)^(3/2)*(a/x+b*x^n)^(3/2)/ 
c/(1+n)-2*a^(3/2)*c*x^(1/2)*arctanh(a^(1/2)/x^(1/2)/(a/x+b*x^n)^(1/2))/(1+ 
n)/(c*x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83 \[ \int \sqrt {c x} \left (\frac {a}{x}+b x^n\right )^{3/2} \, dx=\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n} \left (\sqrt {a+b x^{1+n}} \left (4 a+b x^{1+n}\right )-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^{1+n}}}{\sqrt {a}}\right )\right )}{3 (1+n) \sqrt {a+b x^{1+n}}} \] Input:

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

Output:

(2*Sqrt[c*x]*Sqrt[a/x + b*x^n]*(Sqrt[a + b*x^(1 + n)]*(4*a + b*x^(1 + n)) 
- 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^(1 + n)]/Sqrt[a]]))/(3*(1 + n)*Sqrt[a + b 
*x^(1 + n)])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1934, 1934, 1937, 1935, 219}

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.

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

\(\Big \downarrow \) 1934

\(\displaystyle a c \int \frac {\sqrt {b x^n+\frac {a}{x}}}{\sqrt {c x}}dx+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)}\)

\(\Big \downarrow \) 1934

\(\displaystyle a c \left (a c \int \frac {1}{(c x)^{3/2} \sqrt {b x^n+\frac {a}{x}}}dx+\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (n+1)}\right )+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)}\)

\(\Big \downarrow \) 1937

\(\displaystyle a c \left (\frac {a \sqrt {x} \int \frac {1}{x^{3/2} \sqrt {b x^n+\frac {a}{x}}}dx}{\sqrt {c x}}+\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (n+1)}\right )+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)}\)

\(\Big \downarrow \) 1935

\(\displaystyle a c \left (\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (n+1)}-\frac {2 a \sqrt {x} \int \frac {1}{1-\frac {a}{x \left (b x^n+\frac {a}{x}\right )}}d\frac {1}{\sqrt {x} \sqrt {b x^n+\frac {a}{x}}}}{(n+1) \sqrt {c x}}\right )+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)}\)

\(\Big \downarrow \) 219

\(\displaystyle a c \left (\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (n+1)}-\frac {2 \sqrt {a} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(n+1) \sqrt {c x}}\right )+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)}\)

Input:

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

Output:

(2*(c*x)^(3/2)*(a/x + b*x^n)^(3/2))/(3*c*(1 + n)) + a*c*((2*Sqrt[c*x]*Sqrt 
[a/x + b*x^n])/(c*(1 + n)) - (2*Sqrt[a]*Sqrt[x]*ArcTanh[Sqrt[a]/(Sqrt[x]*S 
qrt[a/x + b*x^n])])/((1 + n)*Sqrt[c*x]))
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1934

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*p*(n - j))), x] + Simp[a/c^j 
  Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, j, m 
, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] 
&& (IntegerQ[j] || GtQ[c, 0])
 

rule 1935

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp 
[-2/(n - j)   Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], 
 x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
 

rule 1937

Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] 
:> Simp[c^IntPart[m]*((c*x)^FracPart[m]/x^FracPart[m])   Int[x^m*(a*x^j + b 
*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2] && 
NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]
 

Maple [F]

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

Input:

int((c*x)^(1/2)*(a/x+b*x^n)^(3/2),x)
 

Output:

int((c*x)^(1/2)*(a/x+b*x^n)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \sqrt {c x} \left (\frac {a}{x}+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c*x)^(1/2)*(a/x+b*x^n)^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

integrate((c*x)**(1/2)*(a/x+b*x**n)**(3/2),x)
 

Output:

Integral(sqrt(c*x)*(a/x + b*x**n)**(3/2), x)
 

Maxima [F]

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

integrate((c*x)^(1/2)*(a/x+b*x^n)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((c*x)^(1/2)*(a/x+b*x^n)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((c*x)^(1/2)*(b*x^n + a/x)^(3/2),x)
 

Output:

int((c*x)^(1/2)*(b*x^n + a/x)^(3/2), x)
 

Reduce [F]

\[ \int \sqrt {c x} \left (\frac {a}{x}+b x^n\right )^{3/2} \, dx=\frac {\sqrt {c}\, \left (2 x^{n} \sqrt {x^{n} b x +a}\, b x +8 \sqrt {x^{n} b x +a}\, a +3 \left (\int \frac {\sqrt {x^{n} b x +a}}{x^{n} b \,x^{2}+a x}d x \right ) a^{2} n +3 \left (\int \frac {\sqrt {x^{n} b x +a}}{x^{n} b \,x^{2}+a x}d x \right ) a^{2}\right )}{3 n +3} \] Input:

int((c*x)^(1/2)*(a/x+b*x^n)^(3/2),x)
 

Output:

(sqrt(c)*(2*x**n*sqrt(x**n*b*x + a)*b*x + 8*sqrt(x**n*b*x + a)*a + 3*int(s 
qrt(x**n*b*x + a)/(x**n*b*x**2 + a*x),x)*a**2*n + 3*int(sqrt(x**n*b*x + a) 
/(x**n*b*x**2 + a*x),x)*a**2))/(3*(n + 1))