\(\int \frac {(c+d \sqrt {a+b x^2})^{3/2}}{x^2} \, dx\) [263]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx=-\frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x}-\frac {3 \sqrt {a} d \sqrt {-\frac {b x^2}{a}} \sqrt {c+d \sqrt {a+b x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{x \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.30 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.97 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx=\frac {-\sqrt {-c-\sqrt {a} d} \left (c^2+d^2 \left (a-2 b x^2\right )+2 c d \sqrt {a+b x^2}\right )-3 i \left (c+\sqrt {a} d\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right )|\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )+3 i \left (c+\sqrt {a} d\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )}{\sqrt {-c-\sqrt {a} d} x \sqrt {c+d \sqrt {a+b x^2}}} \] Input:

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

Output:

(-(Sqrt[-c - Sqrt[a]*d]*(c^2 + d^2*(a - 2*b*x^2) + 2*c*d*Sqrt[a + b*x^2])) 
 - (3*I)*(c + Sqrt[a]*d)*Sqrt[(d*(-Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt 
[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2]) 
]*(c + d*Sqrt[a + b*x^2])^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - Sqrt[a]*d]/S 
qrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqrt[a]*d)] + (3*I)*(c + 
 Sqrt[a]*d)*Sqrt[(d*(-Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])] 
*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt 
[a + b*x^2])^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c - Sqrt[a]*d]/Sqrt[c + d*Sqr 
t[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqrt[a]*d)])/(Sqrt[-c - Sqrt[a]*d]*x* 
Sqrt[c + d*Sqrt[a + b*x^2]])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx=\frac {-4 \sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}\, a \,d^{2}+2 \sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}\, c^{2}+\left (\int \frac {\sqrt {\sqrt {b \,x^{2}+a}\, d +c}}{b^{2} d^{2} x^{6}+2 a b \,d^{2} x^{4}-b \,c^{2} x^{4}+a^{2} d^{2} x^{2}-a \,c^{2} x^{2}}d x \right ) a^{3} c \,d^{3} x -\left (\int \frac {\sqrt {\sqrt {b \,x^{2}+a}\, d +c}}{b^{2} d^{2} x^{6}+2 a b \,d^{2} x^{4}-b \,c^{2} x^{4}+a^{2} d^{2} x^{2}-a \,c^{2} x^{2}}d x \right ) a^{2} c^{3} d x -\left (\int \frac {\sqrt {\sqrt {b \,x^{2}+a}\, d +c}\, x^{2}}{b^{2} d^{2} x^{4}+2 a b \,d^{2} x^{2}-b \,c^{2} x^{2}+a^{2} d^{2}-a \,c^{2}}d x \right ) a \,b^{2} c \,d^{3} x +\left (\int \frac {\sqrt {\sqrt {b \,x^{2}+a}\, d +c}\, x^{2}}{b^{2} d^{2} x^{4}+2 a b \,d^{2} x^{2}-b \,c^{2} x^{2}+a^{2} d^{2}-a \,c^{2}}d x \right ) b^{2} c^{3} d x +3 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}\, x^{2}}{b^{2} d^{2} x^{4}+2 a b \,d^{2} x^{2}-b \,c^{2} x^{2}+a^{2} d^{2}-a \,c^{2}}d x \right ) a \,b^{2} d^{4} x -\left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}\, x^{2}}{b^{2} d^{2} x^{4}+2 a b \,d^{2} x^{2}-b \,c^{2} x^{2}+a^{2} d^{2}-a \,c^{2}}d x \right ) b^{2} c^{2} d^{2} x -3 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}}{b^{2} d^{2} x^{6}+2 a b \,d^{2} x^{4}-b \,c^{2} x^{4}+a^{2} d^{2} x^{2}-a \,c^{2} x^{2}}d x \right ) a^{3} d^{4} x +5 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}}{b^{2} d^{2} x^{6}+2 a b \,d^{2} x^{4}-b \,c^{2} x^{4}+a^{2} d^{2} x^{2}-a \,c^{2} x^{2}}d x \right ) a^{2} c^{2} d^{2} x -2 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {\sqrt {b \,x^{2}+a}\, d +c}}{b^{2} d^{2} x^{6}+2 a b \,d^{2} x^{4}-b \,c^{2} x^{4}+a^{2} d^{2} x^{2}-a \,c^{2} x^{2}}d x \right ) a \,c^{4} x}{a d x} \] Input:

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

Output:

( - 4*sqrt(a + b*x**2)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**2 + 2*sqrt(a + b* 
x**2)*sqrt(sqrt(a + b*x**2)*d + c)*c**2 + int(sqrt(sqrt(a + b*x**2)*d + c) 
/(a**2*d**2*x**2 + 2*a*b*d**2*x**4 - a*c**2*x**2 + b**2*d**2*x**6 - b*c**2 
*x**4),x)*a**3*c*d**3*x - int(sqrt(sqrt(a + b*x**2)*d + c)/(a**2*d**2*x**2 
 + 2*a*b*d**2*x**4 - a*c**2*x**2 + b**2*d**2*x**6 - b*c**2*x**4),x)*a**2*c 
**3*d*x - int((sqrt(sqrt(a + b*x**2)*d + c)*x**2)/(a**2*d**2 + 2*a*b*d**2* 
x**2 - a*c**2 + b**2*d**2*x**4 - b*c**2*x**2),x)*a*b**2*c*d**3*x + int((sq 
rt(sqrt(a + b*x**2)*d + c)*x**2)/(a**2*d**2 + 2*a*b*d**2*x**2 - a*c**2 + b 
**2*d**2*x**4 - b*c**2*x**2),x)*b**2*c**3*d*x + 3*int((sqrt(a + b*x**2)*sq 
rt(sqrt(a + b*x**2)*d + c)*x**2)/(a**2*d**2 + 2*a*b*d**2*x**2 - a*c**2 + b 
**2*d**2*x**4 - b*c**2*x**2),x)*a*b**2*d**4*x - int((sqrt(a + b*x**2)*sqrt 
(sqrt(a + b*x**2)*d + c)*x**2)/(a**2*d**2 + 2*a*b*d**2*x**2 - a*c**2 + b** 
2*d**2*x**4 - b*c**2*x**2),x)*b**2*c**2*d**2*x - 3*int((sqrt(a + b*x**2)*s 
qrt(sqrt(a + b*x**2)*d + c))/(a**2*d**2*x**2 + 2*a*b*d**2*x**4 - a*c**2*x* 
*2 + b**2*d**2*x**6 - b*c**2*x**4),x)*a**3*d**4*x + 5*int((sqrt(a + b*x**2 
)*sqrt(sqrt(a + b*x**2)*d + c))/(a**2*d**2*x**2 + 2*a*b*d**2*x**4 - a*c**2 
*x**2 + b**2*d**2*x**6 - b*c**2*x**4),x)*a**2*c**2*d**2*x - 2*int((sqrt(a 
+ b*x**2)*sqrt(sqrt(a + b*x**2)*d + c))/(a**2*d**2*x**2 + 2*a*b*d**2*x**4 
- a*c**2*x**2 + b**2*d**2*x**6 - b*c**2*x**4),x)*a*c**4*x)/(a*d*x)