\(\int \frac {\sqrt {c+d x^2}}{x^3 (a+b x^2)^2} \, dx\) [994]

Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b c-a d}} \] Output:

-b*(d*x^2+c)^(1/2)/a^2/(b*x^2+a)-1/2*(d*x^2+c)^(1/2)/a/x^2/(b*x^2+a)+1/2*( 
-a*d+4*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^3/c^(1/2)-1/2*b^(1/2)*(-3*a 
*d+4*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+b*c) 
^(1/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a \left (a+2 b x^2\right ) \sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )}+\frac {\sqrt {b} (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a^3} \] Input:

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

Output:

(-((a*(a + 2*b*x^2)*Sqrt[c + d*x^2])/(x^2*(a + b*x^2))) + (Sqrt[b]*(4*b*c 
- 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/Sqrt[-(b*c) 
 + a*d] + ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c])/(2*a^3 
)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {354, 110, 27, 168, 27, 174, 73, 221}

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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 \sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a \sqrt {c}}}{a}+\frac {4 b \sqrt {c+d x^2}}{a \left (a+b x^2\right )}}{2 a}-\frac {\sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {-4 \left (b \,x^{2}+a \right ) \left (b c -\frac {3 a d}{4}\right ) \sqrt {c}\, b \,x^{2} \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (x^{2} \left (b \,x^{2}+a \right ) \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{\sqrt {c}}\right )+a \left (2 b \,x^{2}+a \right ) \sqrt {c}\, \sqrt {x^{2} d +c}\right )}{2 \sqrt {\left (a d -b c \right ) b}\, \sqrt {c}\, a^{3} x^{2} \left (b \,x^{2}+a \right )}\)	\(151\)
risch	\(-\frac {\sqrt {x^{2} d +c}}{2 a^{2} x^{2}}-\frac {\frac {\left (a d -4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{a \sqrt {c}}-\frac {\left (a d -2 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 )}{a \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -2 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 )}{a \sqrt {-\frac {a d -b c}{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 )}{2 \sqrt {-a 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 )}{2 \sqrt {-a b}}}{2 a^{2}}\)	\(903\)
default	\(\text {Expression too large to display}\)	\(2071\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1049, normalized size of antiderivative = 6.60 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/8*(((4*b^2*c^2 - 3*a*b*c*d)*x^4 + (4*a*b*c^2 - 3*a^2*c*d)*x^2)*sqrt(b/ 
(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2 
*c*d - 3*a*b*d^2)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a* 
b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^ 
2)) + 2*((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt(c)*log(-(d*x^ 
2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 4*(2*a*b*c*x^2 + a^2*c)*sqrt(d 
*x^2 + c))/(a^3*b*c*x^4 + a^4*c*x^2), -1/8*(4*((4*b^2*c - a*b*d)*x^4 + (4* 
a*b*c - a^2*d)*x^2)*sqrt(-c)*arctan(sqrt(d*x^2 + c)*sqrt(-c)/c) + ((4*b^2* 
c^2 - 3*a*b*c*d)*x^4 + (4*a*b*c^2 - 3*a^2*c*d)*x^2)*sqrt(b/(b*c - a*d))*lo 
g((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^ 
2)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqr 
t(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(2*a*b* 
c*x^2 + a^2*c)*sqrt(d*x^2 + c))/(a^3*b*c*x^4 + a^4*c*x^2), 1/4*(((4*b^2*c^ 
2 - 3*a*b*c*d)*x^4 + (4*a*b*c^2 - 3*a^2*c*d)*x^2)*sqrt(-b/(b*c - a*d))*arc 
tan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d* 
x^2 + b*c)) - ((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt(c)*log( 
-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(2*a*b*c*x^2 + a^2*c)* 
sqrt(d*x^2 + c))/(a^3*b*c*x^4 + a^4*c*x^2), 1/4*(((4*b^2*c^2 - 3*a*b*c*d)* 
x^4 + (4*a*b*c^2 - 3*a^2*c*d)*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^ 
2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x^{2} + c} b c d + \sqrt {d x^{2} + c} a d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )} a^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 2.11 (sec) , antiderivative size = 1193, normalized size of antiderivative = 7.50 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(atan((((-b*(a*d - b*c))^(1/2)*(((c + d*x^2)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5 
*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 - (((2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3)/a^6 
- ((8*a^7*b^2*d^3 - 16*a^6*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^( 
1/2)*(3*a*d - 4*b*c))/(4*a^4*(a^4*d - a^3*b*c)))*(-b*(a*d - b*c))^(1/2)*(3 
*a*d - 4*b*c))/(4*(a^4*d - a^3*b*c)))*(3*a*d - 4*b*c)*1i)/(4*(a^4*d - a^3* 
b*c)) + ((-b*(a*d - b*c))^(1/2)*(((c + d*x^2)^(1/2)*(5*a^2*b^3*d^4 + 16*b^ 
5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + (((2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3)/a^6 
 + ((8*a^7*b^2*d^3 - 16*a^6*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^ 
(1/2)*(3*a*d - 4*b*c))/(4*a^4*(a^4*d - a^3*b*c)))*(-b*(a*d - b*c))^(1/2)*( 
3*a*d - 4*b*c))/(4*(a^4*d - a^3*b*c)))*(3*a*d - 4*b*c)*1i)/(4*(a^4*d - a^3 
*b*c)))/(((3*a^2*b^3*d^5)/2 + 8*b^5*c^2*d^3 - 8*a*b^4*c*d^4)/a^6 - ((-b*(a 
*d - b*c))^(1/2)*(((c + d*x^2)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16* 
a*b^4*c*d^3))/a^4 - (((2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3)/a^6 - ((8*a^7*b^2* 
d^3 - 16*a^6*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 
4*b*c))/(4*a^4*(a^4*d - a^3*b*c)))*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)) 
/(4*(a^4*d - a^3*b*c)))*(3*a*d - 4*b*c))/(4*(a^4*d - a^3*b*c)) + ((-b*(a*d 
 - b*c))^(1/2)*(((c + d*x^2)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a* 
b^4*c*d^3))/a^4 + (((2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3)/a^6 + ((8*a^7*b^2*d^ 
3 - 16*a^6*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4* 
b*c))/(4*a^4*(a^4*d - a^3*b*c)))*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)...
 

Reduce [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 3176, normalized size of antiderivative = 19.97 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a 
*d - b*c)*sqrt(a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b) 
*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a**2*d*x**2 + 8*s 
qrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b* 
c)*sqrt(a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2 
*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*c*x**2 - 6*sqrt(d)*s 
qrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*sqrt( 
a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d) 
*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*d*x**4 + 8*sqrt(d)*sqrt(b)*s 
qrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*sqrt(a*d - b* 
c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a) 
*sqrt(a*d - b*c) + 2*a*d - b*c)))*b**2*c*x**4 + 6*sqrt(b)*sqrt(2*sqrt(d)*s 
qrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b 
*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a**3* 
d**2*x**2 - 14*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b* 
c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a) 
*sqrt(a*d - b*c) + 2*a*d - b*c)))*a**2*b*c*d*x**2 + 6*sqrt(b)*sqrt(2*sqrt( 
d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt( 
d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a 
**2*b*d**2*x**4 + 8*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*...