\(\int \frac {(c \sqrt {a+b x^2})^{3/2}}{x^2} \, dx\) [47]

Optimal result

Integrand size = 21, antiderivative size = 115 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x}+\frac {3 b x \left (c \sqrt {a+b x^2}\right )^{3/2}}{a+b x^2}-\frac {3 \sqrt {b} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \left (1+\frac {b x^2}{a}\right )^{3/4}} \] Output:

-(c*(b*x^2+a)^(1/2))^(3/2)/x+3*b*x*(c*(b*x^2+a)^(1/2))^(3/2)/(b*x^2+a)-3*b 
^(1/2)*(c*(b*x^2+a)^(1/2))^(3/2)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2 
))),2^(1/2))/a^(1/2)/(1+b*x^2/a)^(3/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 7.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.48 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{2},-\frac {b x^2}{a}\right )}{x \left (1+\frac {b x^2}{a}\right )^{3/4}} \] Input:

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

Output:

-(((c*Sqrt[a + b*x^2])^(3/2)*Hypergeometric2F1[-3/4, -1/2, 1/2, -((b*x^2)/ 
a)])/(x*(1 + (b*x^2)/a)^(3/4)))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2045, 247, 225, 212}

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.

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

\(\Big \downarrow \) 2045

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}dx}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 247

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {3 b \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 225

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {3 b \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {3 b \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\)

Input:

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

Output:

((c*Sqrt[a + b*x^2])^(3/2)*(-((1 + (b*x^2)/a)^(3/4)/x) + (3*b*((2*x)/(1 + 
(b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2]) 
/Sqrt[b]))/(2*a)))/(1 + (b*x^2)/a)^(3/4)
 

Defintions of rubi rules used

rule 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 225

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) 
, x] - Simp[a   Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ 
a, 0] && PosQ[b/a]
 

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 2045

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si 
mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)]   Int[u*(1 + b*(x^n/a))^(p*q) 
, x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {c}\, c \left (2 \left (b \,x^{2}+a \right )^{\frac {3}{4}}+3 \left (\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}}}{b \,x^{4}+a \,x^{2}}d x \right ) a x \right )}{x} \] Input:

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

Output:

(sqrt(c)*c*(2*(a + b*x**2)**(3/4) + 3*int((a + b*x**2)**(3/4)/(a*x**2 + b* 
x**4),x)*a*x))/x