\(\int \frac {\sqrt {c+d x^2}}{x^3 (a+b x^2)} \, dx\) [683]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 101, 162, 65, 214} \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 \sqrt {c}}-\frac {\sqrt {b} \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2}-\frac {\sqrt {c+d x^2}}{2 a x^2} \]

[In]

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

[Out]

-1/2*Sqrt[c + d*x^2]/(a*x^2) + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*Sqrt[c]) - (Sqrt[b]*Sqr
t[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/a^2

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 101

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

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(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 214

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 457

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x^2 (a+b x)} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {c+d x^2}}{2 a x^2}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (-2 b c+a d)-\frac {b d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a} \\ & = -\frac {\sqrt {c+d x^2}}{2 a x^2}+\frac {(b (b c-a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2}-\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2} \\ & = -\frac {\sqrt {c+d x^2}}{2 a x^2}+\frac {(b (b c-a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 d}-\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d} \\ & = -\frac {\sqrt {c+d x^2}}{2 a x^2}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 \sqrt {c}}-\frac {\sqrt {b} \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(-((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)

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {x^{2} b \left (c^{\frac {3}{2}} b -a d \sqrt {c}\right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\left (x^{2} \left (a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+\sqrt {d \,x^{2}+c}\, a \sqrt {c}\right ) \sqrt {\left (a d -b c \right ) b}}{2}}{\sqrt {\left (a d -b c \right ) b}\, \sqrt {c}\, a^{2} x^{2}}\)	\(121\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}}{2 a \,x^{2}}-\frac {-\frac {\left (-a d +2 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a \sqrt {c}}-\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}}{2 a}\)	\(379\)
default	\(\frac {-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+\frac {d \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )}{2 c}}{a}-\frac {b \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )}{a^{2}}+\frac {b \left (\sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\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 {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\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 {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\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 {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\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 {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\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 {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\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}}\)	\(759\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 708, normalized size of antiderivative = 6.27 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\left [\frac {\sqrt {b^{2} c - a b d} c x^{2} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b c - a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c x^{2}}, -\frac {2 \, {\left (2 \, b c - a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - \sqrt {b^{2} c - a b d} c x^{2} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c x^{2}}, -\frac {2 \, \sqrt {-b^{2} c + a b d} c x^{2} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c - a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c x^{2}}, -\frac {\sqrt {-b^{2} c + a b d} c x^{2} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c - a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + \sqrt {d x^{2} + c} a c}{2 \, a^{2} c x^{2}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c}} - \frac {\sqrt {d x^{2} + c}}{2 \, a x^{2}} \]

[In]

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

[Out]

(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 = 5.60 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {b^3\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{2\,\left (\frac {a\,b^3\,d^5}{2}-\frac {b^4\,c\,d^4}{2}\right )}\right )\,\sqrt {b^2\,c-a\,b\,d}}{a^2}-\frac {\sqrt {d\,x^2+c}}{2\,a\,x^2}-\frac {\mathrm {atanh}\left (\frac {b^4\,\sqrt {c}\,d^4\,\sqrt {d\,x^2+c}}{2\,\left (\frac {b^4\,c\,d^4}{2}-\frac {3\,a\,b^3\,d^5}{4}+\frac {a^2\,b^2\,d^6}{4\,c}\right )}-\frac {3\,b^3\,d^5\,\sqrt {d\,x^2+c}}{4\,\sqrt {c}\,\left (\frac {a\,b^2\,d^6}{4\,c}-\frac {3\,b^3\,d^5}{4}+\frac {b^4\,c\,d^4}{2\,a}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^2+c}}{4\,c^{3/2}\,\left (\frac {b^2\,d^6}{4\,c}-\frac {3\,b^3\,d^5}{4\,a}+\frac {b^4\,c\,d^4}{2\,a^2}\right )}\right )\,\left (a\,d-2\,b\,c\right )}{2\,a^2\,\sqrt {c}} \]

[In]

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

[Out]

(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))