\(\int \frac {\sqrt {c+d x^2}}{x^2 (a+b x^2)^2} \, dx\) [993]

Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {(3 b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} \sqrt {b c-a d}} \] Output:

-3/2*(d*x^2+c)^(1/2)/a^2/x+1/2*(d*x^2+c)^(1/2)/a/x/(b*x^2+a)-1/2*(-2*a*d+3 
*b*c)*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2))/a^(5/2)/(-a*d+b*c 
)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1650\) vs. \(2(113)=226\).

Time = 8.02 (sec) , antiderivative size = 1650, normalized size of antiderivative = 14.60 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(-((Sqrt[a]*(2*a + 3*b*x^2)*(4*c^2 + 5*c*d*x^2 + d^2*x^4 - 4*c^(3/2)*Sqrt[ 
c + d*x^2] - 3*Sqrt[c]*d*x^2*Sqrt[c + d*x^2]))/(x*(a + b*x^2)*(-4*c^(3/2) 
- 3*Sqrt[c]*d*x^2 + 4*c*Sqrt[c + d*x^2] + d*x^2*Sqrt[c + d*x^2]))) + (5*a* 
b^(3/2)*c^(3/2)*d*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - 
a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2 
*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*b^2*c^2*ArcTan[(Sqrt 
[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - S 
qrt[c + d*x^2]))])/((b*c - a*d)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[ 
b*c - a*d]]) + (2*a^2*d^2*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqr 
t[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)*Sqrt 
[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*b^(5/2)*c^(5/2)*Ar 
cTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(S 
qrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[ 
b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*a^2*Sqrt[b]*Sqrt[c]*d^2*ArcTan[(Sqrt[2*b 
*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[ 
c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqr 
t[b*c - a*d]]) + (3*b^2*c^2*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*S 
qrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)*Sq 
rt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*a^2*d^2*ArcTan[( 
Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt...
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {371, 25, 445, 27, 291, 218}

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.

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

\(\Big \downarrow \) 371

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 371

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-(e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a 
*e*2*(p + 1))), x] + Simp[1/(a*2*(p + 1))   Int[(e*x)^m*(a + b*x^2)^(p + 1) 
*(c + d*x^2)^(q - 1)*Simp[c*(m + 2*(p + 1) + 1) + d*(m + 2*(p + q + 1) + 1) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && Lt 
Q[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {x^{2} d +c}}{x}-\frac {\sqrt {x^{2} d +c}\, b x}{2 \left (b \,x^{2}+a \right )}+\frac {\left (2 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{2 \sqrt {\left (a d -b c \right ) a}}}{a^{2}}\)	\(92\)
risch	\(-\frac {\sqrt {x^{2} d +c}}{a^{2} x}-\frac {\frac {\left (a d -b c \right ) \left (\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b}+\frac {\left (a d -b c \right ) \left (\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b}+\frac {\left (a d -3 b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -3 b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}}{a^{2}}\)	\(865\)
default	\(\text {Expression too large to display}\)	\(2024\)

Input:

int((d*x^2+c)^(1/2)/x^2/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (93) = 186\).

Time = 0.15 (sec) , antiderivative size = 458, normalized size of antiderivative = 4.05 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\left [\frac {{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, a^{2} b c - 2 \, a^{3} d + 3 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left ({\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3} + {\left (a^{4} b c - a^{5} d\right )} x\right )}}, -\frac {{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} b c - 2 \, a^{3} d + 3 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3} + {\left (a^{4} b c - a^{5} d\right )} x\right )}}\right ] \] Input:

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

Output:

[1/8*(((3*b^2*c - 2*a*b*d)*x^3 + (3*a*b*c - 2*a^2*d)*x)*sqrt(-a*b*c + a^2* 
d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4 
*a^2*c*d)*x^2 - 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d* 
x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(2*a^2*b*c - 2*a^3*d + 3*(a*b^2 
*c - a^2*b*d)*x^2)*sqrt(d*x^2 + c))/((a^3*b^2*c - a^4*b*d)*x^3 + (a^4*b*c 
- a^5*d)*x), -1/4*(((3*b^2*c - 2*a*b*d)*x^3 + (3*a*b*c - 2*a^2*d)*x)*sqrt( 
a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sq 
rt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*(2*a^ 
2*b*c - 2*a^3*d + 3*(a*b^2*c - a^2*b*d)*x^2)*sqrt(d*x^2 + c))/((a^3*b^2*c 
- a^4*b*d)*x^3 + (a^4*b*c - a^5*d)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (93) = 186\).

Time = 0.36 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {{\left (3 \, b c \sqrt {d} - 2 \, a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{2}} + \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )} a^{2}} \] Input:

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

Output:

1/2*(3*b*c*sqrt(d) - 2*a*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c) 
)^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a^2 
) + (3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d 
*x^2 + c))^4*a*d^(3/2) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) + 
 10*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c*d^(3/2) + 3*b*c^3*sqrt(d))/(((sqrt 
(d)*x - sqrt(d*x^2 + c))^6*b - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c + 4*( 
sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c 
^2 - 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c*d - b*c^3)*a^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 1426, normalized size of antiderivative = 12.62 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 
 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**3*d**2*x 
- 18*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
 - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*b*c*d 
*x + 8*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b* 
c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*b*d 
**2*x**3 + 9*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a* 
d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a* 
b**2*c**2*x - 18*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqr 
t(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x 
)*a*b**2*c*d*x**3 + 9*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a 
)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt 
(b)*x)*b**3*c**2*x**3 + 8*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt( 
a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqr 
t(b)*x)*a**3*d**2*x - 18*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a 
)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt 
(b)*x)*a**2*b*c*d*x + 8*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a) 
*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt( 
b)*x)*a**2*b*d**2*x**3 + 9*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt 
(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)...