Integrand size = 21, antiderivative size = 126 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )-\frac {(b+a c)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{c^{3/2}} \]
a^(3/2)*arctanh(((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^(1/2))-(a*c+b)^(3/2)*a rctanh(c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))/c^(3/2)+b* ((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c
Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c}-\frac {(-b-a c)^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {-b-a c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{b+a c}\right )}{c^{3/2}}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right ) \]
(b*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/c - ((-b - a*c)^(3/2)*ArcTan[(Sq rt[c]*Sqrt[-b - a*c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(b + a*c)])/c^ (3/2) + a^(3/2)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]]
Time = 0.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2057, 2053, 2052, 25, 27, 381, 397, 219, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}{x}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (\frac {a d x^2+b+a c}{d x^2+c}\right )^{3/2}}{x^2}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int -\frac {x^8}{d \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b d \int \frac {x^8}{d \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b \int \frac {x^8}{\left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 381 |
| \(\displaystyle b \left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c}-\frac {\int \frac {a (b+a c)-(b+2 a c) x^4}{\left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{c}\right )\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle b \left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c}-\frac {\frac {(a c+b)^2 \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}-\frac {a^2 c \int \frac {1}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}}{c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle b \left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c}-\frac {\frac {(a c+b)^2 \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}-\frac {a^{3/2} c \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{b}}{c}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle b \left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c}-\frac {\frac {(a c+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{b \sqrt {c}}-\frac {a^{3/2} c \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{b}}{c}\right )\) |
b*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/c - (-((a^(3/2)*c*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/b) + ((b + a*c)^(3/2)*ArcTanh[(Sq rt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(b*Sqrt[c]))/ c)
3.4.34.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q }, x] && NeQ[b*c - a*d, 0] && GtQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2 , p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(651\) vs. \(2(108)=216\).
Time = 0.13 (sec) , antiderivative size = 652, normalized size of antiderivative = 5.17
| method | result | size |
| default | \(\frac {\left (\ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} c^{2} d^{2} x^{2}-\sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a c d \,x^{2}-\sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b d \,x^{2}+\ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} c^{3} d -\sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a \,c^{2}-\sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b c +2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a \,d^{2}}\, b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{2 \sqrt {a \,d^{2}}\, c^{2} \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\) | \(652\) |
1/2*(ln(1/2*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b* c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a^2*c^2*d^2*x^2-(a*d^2)^(1/2)*( a*c^2+b*c)^(1/2)*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*(a*c^2+b*c)^(1/2)*(a*d^ 2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a*c*d*x^2-(a*d^2)^( 1/2)*(a*c^2+b*c)^(1/2)*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*(a*c^2+b*c)^(1/2) *(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*b*d*x^2+ln(1/ 2*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*( a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a^2*c^3*d-(a*d^2)^(1/2)*(a*c^2+b*c)^(1/2) *ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^ 2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a*c^2-(a*d^2)^(1/2)*(a*c^2+b*c)^(1/ 2)*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d* x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*b*c+2*((a*d*x^2+a*c+b)*(d*x^2+c)) ^(1/2)*(a*d^2)^(1/2)*b*c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*d^2)^(1/2)/ c^2/((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)
Time = 0.40 (sec) , antiderivative size = 1073, normalized size of antiderivative = 8.52 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\left [\frac {a^{\frac {3}{2}} c \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + {\left (a c + b\right )} \sqrt {\frac {a c + b}{c}} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a c^{2} + b c\right )} d^{2} x^{4} + 2 \, a c^{4} + 2 \, b c^{3} + {\left (4 \, a c^{3} + 3 \, b c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {\frac {a c + b}{c}}}{x^{4}}\right ) + 4 \, b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, c}, -\frac {2 \, \sqrt {-a} a c \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - {\left (a c + b\right )} \sqrt {\frac {a c + b}{c}} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a c^{2} + b c\right )} d^{2} x^{4} + 2 \, a c^{4} + 2 \, b c^{3} + {\left (4 \, a c^{3} + 3 \, b c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {\frac {a c + b}{c}}}{x^{4}}\right ) - 4 \, b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, c}, \frac {a^{\frac {3}{2}} c \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 2 \, {\left (a c + b\right )} \sqrt {-\frac {a c + b}{c}} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {-\frac {a c + b}{c}}}{2 \, {\left (a^{2} c^{2} + {\left (a^{2} c + a b\right )} d x^{2} + 2 \, a b c + b^{2}\right )}}\right ) + 4 \, b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, c}, -\frac {\sqrt {-a} a c \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - {\left (a c + b\right )} \sqrt {-\frac {a c + b}{c}} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {-\frac {a c + b}{c}}}{2 \, {\left (a^{2} c^{2} + {\left (a^{2} c + a b\right )} d x^{2} + 2 \, a b c + b^{2}\right )}}\right ) - 2 \, b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, c}\right ] \]
[1/4*(a^(3/2)*c*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 + 8*a*b*c + b^2 + 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a )*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))) + (a*c + b)*sqrt((a*c + b)/c)*log (((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c^2 + b*c)*d^2*x^4 + 2*a*c^4 + 2*b*c^3 + (4*a*c^3 + 3*b*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d *x^2 + c))*sqrt((a*c + b)/c))/x^4) + 4*b*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/c, -1/4*(2*sqrt(-a)*a*c*arctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqrt(-a)* sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*d*x^2 + a^2*c + a*b)) - (a*c + b)*sqrt((a*c + b)/c)*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2 *a*c^2 + b*c)*d^2*x^4 + 2*a*c^4 + 2*b*c^3 + (4*a*c^3 + 3*b*c^2)*d*x^2)*sqr t((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt((a*c + b)/c))/x^4) - 4*b*sqrt((a*d *x^2 + a*c + b)/(d*x^2 + c)))/c, 1/4*(a^(3/2)*c*log(8*a^2*d^2*x^4 + 8*a^2* c^2 + 8*(2*a^2*c + a*b)*d*x^2 + 8*a*b*c + b^2 + 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))) + 2*(a*c + b)*sqrt(-(a*c + b)/c)*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(-(a*c + b)/c)/(a^2*c^2 + (a^2*c + a*b)*d*x^2 + 2*a*b*c + b^2)) + 4*b*sqrt((a*d*x^2 + a*c + b)/(d*x ^2 + c)))/c, -1/2*(sqrt(-a)*a*c*arctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqrt...
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x}\, dx \]
Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right ) + \frac {b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c} + \frac {{\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{2 \, \sqrt {{\left (a c + b\right )} c} c} \]
-1/2*a^(3/2)*log(-(sqrt(a) - sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(sqrt( a) + sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))) + b*sqrt((a*d*x^2 + a*c + b)/ (d*x^2 + c))/c + 1/2*(a^2*c^2 + 2*a*b*c + b^2)*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/(sqrt((a*c + b)*c)*c)
Exception generated. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x} \,d x \]