\(\int \frac {(c \sqrt {a+b x^2})^{3/2}}{x^4} \, dx\) [48]

Optimal result

Integrand size = 21, antiderivative size = 154 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{3 x^3}-\frac {b \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a x}+\frac {b^2 x \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a \left (a+b x^2\right )}-\frac {b^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4}} \] Output:

-1/3*(c*(b*x^2+a)^(1/2))^(3/2)/x^3-1/2*b*(c*(b*x^2+a)^(1/2))^(3/2)/a/x+1/2 
*b^2*x*(c*(b*x^2+a)^(1/2))^(3/2)/a/(b*x^2+a)-1/2*b^(3/2)*(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^(3/2)/(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 = 10.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.37 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {1}{2},-\frac {b x^2}{a}\right )}{3 x^3 \left (1+\frac {b x^2}{a}\right )^{3/4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2045, 247, 264, 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.

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

\(\Big \downarrow \) 2045

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 225

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

\(\Big \downarrow \) 212

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

Input:

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

Output:

((c*Sqrt[a + b*x^2])^(3/2)*(-1/3*(1 + (b*x^2)/a)^(3/4)/x^3 + (b*(-((1 + (b 
*x^2)/a)^(3/4)/x) + (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)))/(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 264

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

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

Input:

int((c*(b*x^2+a)^(1/2))^(3/2)/x^4,x)
 

Output:

int((c*(b*x^2+a)^(1/2))^(3/2)/x^4,x)
 

Fricas [F]

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

integrate((c*(b*x^2+a)^(1/2))^(3/2)/x^4,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((c*(b*x**2+a)**(1/2))**(3/2)/x**4,x)
 

Output:

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

Maxima [F]

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

integrate((c*(b*x^2+a)^(1/2))^(3/2)/x^4,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((c*(b*x^2+a)^(1/2))^(3/2)/x^4,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((c*(a + b*x^2)^(1/2))^(3/2)/x^4,x)
 

Output:

int((c*(a + b*x^2)^(1/2))^(3/2)/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {c}\, c \left (-2 \left (b \,x^{2}+a \right )^{\frac {3}{4}}-3 \left (\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}}}{b \,x^{6}+a \,x^{4}}d x \right ) a \,x^{3}\right )}{3 x^{3}} \] Input:

int((c*(b*x^2+a)^(1/2))^(3/2)/x^4,x)
 

Output:

(sqrt(c)*c*( - 2*(a + b*x**2)**(3/4) - 3*int((a + b*x**2)**(3/4)/(a*x**4 + 
 b*x**6),x)*a*x**3))/(3*x**3)