\(\int \frac {(a x+b x^n)^{3/2}}{(c x)^{5/2}} \, dx\) [355]

Optimal result

Integrand size = 21, antiderivative size = 122 \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {2 a^{3/2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}} \] Output:

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

Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\frac {x \left (8 a^2 x^2+2 b^2 x^{2 n}+10 a b x^{1+n}-6 a^{3/2} \sqrt {b} x^{\frac {3+n}{2}} \sqrt {1+\frac {a x^{1-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {1}{2}-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{3 (-1+n) (c x)^{5/2} \sqrt {a x+b x^n}} \] Input:

Integrate[(a*x + b*x^n)^(3/2)/(c*x)^(5/2),x]
 

Output:

(x*(8*a^2*x^2 + 2*b^2*x^(2*n) + 10*a*b*x^(1 + n) - 6*a^(3/2)*Sqrt[b]*x^((3 
 + n)/2)*Sqrt[1 + (a*x^(1 - n))/b]*ArcSinh[(Sqrt[a]*x^(1/2 - n/2))/Sqrt[b] 
]))/(3*(-1 + n)*(c*x)^(5/2)*Sqrt[a*x + b*x^n])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, 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.

\(\displaystyle \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx\)

\(\Big \downarrow \) 1934

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

\(\Big \downarrow \) 1934

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

\(\Big \downarrow \) 1937

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

\(\Big \downarrow \) 1935

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

\(\Big \downarrow \) 219

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

Input:

Int[(a*x + b*x^n)^(3/2)/(c*x)^(5/2),x]
 

Output:

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

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]

\[\int \frac {\left (a x +b \,x^{n}\right )^{\frac {3}{2}}}{\left (c x \right )^{\frac {5}{2}}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*x+b*x^n)^(3/2)/(c*x)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a*x+b*x**n)**(3/2)/(c*x)**(5/2),x)
 

Output:

Integral((a*x + b*x**n)**(3/2)/(c*x)**(5/2), x)
 

Maxima [F]

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

integrate((a*x+b*x^n)^(3/2)/(c*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*x + b*x^n)^(3/2)/(c*x)^(5/2), x)
 

Giac [F]

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

integrate((a*x+b*x^n)^(3/2)/(c*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((a*x + b*x^n)^(3/2)/(c*x)^(5/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\frac {\sqrt {c}\, \left (2 x^{n +\frac {1}{2}} \sqrt {x^{n} b +a x}\, b +8 \sqrt {x}\, \sqrt {x^{n} b +a x}\, a x +3 \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a x}}{x^{n} b x +a \,x^{2}}d x \right ) a^{2} n \,x^{2}-3 \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a x}}{x^{n} b x +a \,x^{2}}d x \right ) a^{2} x^{2}\right )}{3 c^{3} x^{2} \left (n -1\right )} \] Input:

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

Output:

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