\(\int \frac {\sqrt {c+d x^2}}{x^3 (a+b x^2)} \, dx\) [923]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {-\frac {a \sqrt {c+d x^2}}{x^2}-2 \sqrt {b} \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {354, 110, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-(Sqrt[c + d*x^2]/(a*x^2)) - ((-2*(2*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/S 
qrt[c]])/(a*Sqrt[c]) + (4*Sqrt[b]*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c 
+ d*x^2])/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 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 = 0.74 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {x^{2} d +c}\, a}{2 x^{2}}-\frac {\left (a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\left (a d -b c \right ) b \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a^{2}}\)	\(95\)
risch	\(-\frac {\sqrt {x^{2} d +c}}{2 a \,x^{2}}-\frac {\frac {\left (a d -2 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{a \sqrt {c}}-\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}}{2 a}\)	\(377\)
default	\(\frac {-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+\frac {d \left (\sqrt {x^{2} d +c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )\right )}{2 c}}{a}-\frac {b \left (\sqrt {x^{2} d +c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )\right )}{a^{2}}+\frac {b \left (\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}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x -\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -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 )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 a^{2}}+\frac {b \left (\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}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x +\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -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 )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 a^{2}}\)	\(759\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 714, normalized size of antiderivative = 6.32 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\left [\frac {\sqrt {b^{2} c - a b d} c x^{2} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b c - a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c x^{2}}, -\frac {2 \, {\left (2 \, b c - a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} \sqrt {-c}}{c}\right ) - \sqrt {b^{2} c - a b d} c x^{2} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c x^{2}}, -\frac {2 \, \sqrt {-b^{2} c + a b d} c x^{2} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c - a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c x^{2}}, -\frac {\sqrt {-b^{2} c + a b d} c x^{2} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c - a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} \sqrt {-c}}{c}\right ) + \sqrt {d x^{2} + c} a c}{2 \, a^{2} c x^{2}}\right ] \] Input:

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

Output:

[1/4*(sqrt(b^2*c - a*b*d)*c*x^2*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*(b*d*x^2 + 2*b*c - a*d)*sqrt( 
b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - (2*b*c - a* 
d)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*sqr 
t(d*x^2 + c)*a*c)/(a^2*c*x^2), -1/4*(2*(2*b*c - a*d)*sqrt(-c)*x^2*arctan(s 
qrt(d*x^2 + c)*sqrt(-c)/c) - sqrt(b^2*c - a*b*d)*c*x^2*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*(b*d*x 
^2 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^ 
2 + a^2)) + 2*sqrt(d*x^2 + c)*a*c)/(a^2*c*x^2), -1/4*(2*sqrt(-b^2*c + a*b* 
d)*c*x^2*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)*sqrt(d*x 
^2 + c)/(b^2*c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x^2)) + (2*b*c - a*d)*sqr 
t(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*sqrt(d*x^ 
2 + c)*a*c)/(a^2*c*x^2), -1/2*(sqrt(-b^2*c + a*b*d)*c*x^2*arctan(-1/2*(b*d 
*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c)/(b^2*c^2 - a*b*c* 
d + (b^2*c*d - a*b*d^2)*x^2)) + (2*b*c - a*d)*sqrt(-c)*x^2*arctan(sqrt(d*x 
^2 + c)*sqrt(-c)/c) + sqrt(d*x^2 + c)*a*c)/(a^2*c*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c}} - \frac {\sqrt {d x^{2} + c}}{2 \, a x^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {b^3\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{2\,\left (\frac {a\,b^3\,d^5}{2}-\frac {b^4\,c\,d^4}{2}\right )}\right )\,\sqrt {b^2\,c-a\,b\,d}}{a^2}-\frac {\sqrt {d\,x^2+c}}{2\,a\,x^2}-\frac {\mathrm {atanh}\left (\frac {b^4\,\sqrt {c}\,d^4\,\sqrt {d\,x^2+c}}{2\,\left (\frac {b^4\,c\,d^4}{2}-\frac {3\,a\,b^3\,d^5}{4}+\frac {a^2\,b^2\,d^6}{4\,c}\right )}-\frac {3\,b^3\,d^5\,\sqrt {d\,x^2+c}}{4\,\sqrt {c}\,\left (\frac {a\,b^2\,d^6}{4\,c}-\frac {3\,b^3\,d^5}{4}+\frac {b^4\,c\,d^4}{2\,a}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^2+c}}{4\,c^{3/2}\,\left (\frac {b^2\,d^6}{4\,c}-\frac {3\,b^3\,d^5}{4\,a}+\frac {b^4\,c\,d^4}{2\,a^2}\right )}\right )\,\left (a\,d-2\,b\,c\right )}{2\,a^2\,\sqrt {c}} \] Input:

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

Output:

(atanh((b^3*d^4*(c + d*x^2)^(1/2)*(b^2*c - a*b*d)^(1/2))/(2*((a*b^3*d^5)/2 
 - (b^4*c*d^4)/2)))*(b^2*c - a*b*d)^(1/2))/a^2 - (c + d*x^2)^(1/2)/(2*a*x^ 
2) - (atanh((b^4*c^(1/2)*d^4*(c + d*x^2)^(1/2))/(2*((b^4*c*d^4)/2 - (3*a*b 
^3*d^5)/4 + (a^2*b^2*d^6)/(4*c))) - (3*b^3*d^5*(c + d*x^2)^(1/2))/(4*c^(1/ 
2)*((a*b^2*d^6)/(4*c) - (3*b^3*d^5)/4 + (b^4*c*d^4)/(2*a))) + (b^2*d^6*(c 
+ d*x^2)^(1/2))/(4*c^(3/2)*((b^2*d^6)/(4*c) - (3*b^3*d^5)/(4*a) + (b^4*c*d 
^4)/(2*a^2))))*(a*d - 2*b*c))/(2*a^2*c^(1/2))
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 898, normalized size of antiderivative = 7.95 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*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)))*x**2 + 2*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*d*x**2 - 2*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)))*b*c*x**2 - 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)* 
log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqr 
t(c + d*x**2) + sqrt(d)*sqrt(b)*x)*x**2 + sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*s 
qrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c)*sqrt(a*d - b*c)*log(sqrt(2*s 
qrt(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)*x**2 - sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
- 2*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*d*x**2 + sqrt(b)*sqrt( 
2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c)*log( - sqrt(2*sqrt(d)*sqr 
t(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*s 
qrt(b)*x)*b*c*x**2 + sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a* 
d + b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sq...