Integrand size = 24, antiderivative size = 129 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b \left (a+b x^2\right )}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {b c-a d} (2 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 b^{3/2}} \]
-c^(3/2)*arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^2+1/2*(a*d+2*b*c)*arctanh(b^(1 /2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^(1/2)/a^2/b^(3/2)+1/2*(-a *d+b*c)*(d*x^2+c)^(1/2)/a/b/(b*x^2+a)
Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {a (b c-a d) \sqrt {c+d x^2}}{b \left (a+b x^2\right )}+\frac {\sqrt {-b c+a d} (2 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}-2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2} \]
((a*(b*c - a*d)*Sqrt[c + d*x^2])/(b*(a + b*x^2)) + (Sqrt[-(b*c) + a*d]*(2* b*c + a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/b^(3/2) - 2*c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2)
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {354, 109, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (d x^2+c\right )^{3/2}}{x^2 \left (b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 b c^2+d (b c+a d) x^2}{2 x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{a b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 b c^2+d (b c+a d) x^2}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{a b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 b c^2 \int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {(b c-a d) (a d+2 b c) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{a b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {4 b c^2 \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{a d}-\frac {2 (b c-a d) (a d+2 b c) \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{a d}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{a b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \sqrt {b c-a d} (a d+2 b c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {4 b c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{a b \left (a+b x^2\right )}\right )\) |
(((b*c - a*d)*Sqrt[c + d*x^2])/(a*b*(a + b*x^2)) + ((-4*b*c^(3/2)*ArcTanh[ Sqrt[c + d*x^2]/Sqrt[c]])/a + (2*Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sq rt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(2*a*b))/2
3.8.45.3.1 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_))^(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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
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[(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]
Time = 3.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(-\frac {-\left (b \,x^{2}+a \right ) \left (a d +2 b c \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (2 b \,c^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+a \sqrt {d \,x^{2}+c}\, \left (a d -b c \right )\right )}{2 \sqrt {\left (a d -b c \right ) b}\, a^{2} b \left (b \,x^{2}+a \right )}\) | \(140\) |
| default | \(\text {Expression too large to display}\) | \(3437\) |
-1/2/((a*d-b*c)*b)^(1/2)*(-(b*x^2+a)*(a*d+2*b*c)*(a*d-b*c)*arctan(b*(d*x^2 +c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*(2*b*c^(3/2)*(b*x^2+a)* arctanh((d*x^2+c)^(1/2)/c^(1/2))+a*(d*x^2+c)^(1/2)*(a*d-b*c)))/a^2/b/(b*x^ 2+a)
Time = 0.44 (sec) , antiderivative size = 883, normalized size of antiderivative = 6.84 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=\left [\frac {{\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \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^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 4 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {8 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \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^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {{\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {{\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 4 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}\right ] \]
[1/8*((2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x^2)*sqrt((b*c - a*d)/b)*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^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/ (b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(b^2*c*x^2 + a*b*c)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 4*(a*b*c - a^2*d)*sqrt(d*x^2 + c) )/(a^2*b^2*x^2 + a^3*b), 1/8*(8*(b^2*c*x^2 + a*b*c)*sqrt(-c)*arctan(sqrt(- c)/sqrt(d*x^2 + c)) + (2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x^2)*sqrt((b*c - a*d)/b)*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^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt( (b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(a*b*c - a^2*d)*sqrt(d*x^ 2 + c))/(a^2*b^2*x^2 + a^3*b), 1/4*((2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x ^2)*sqrt(-(b*c - a*d)/b)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/b)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + 2*(b^2*c* x^2 + a*b*c)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(a*b*c - a^2*d)*sqrt(d*x^2 + c))/(a^2*b^2*x^2 + a^3*b), 1/4*((2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x^2)*sqrt(-(b*c - a*d)/b)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/b)/(b*c^2 - a*c*d + (b*c* d - a*d^2)*x^2)) + 4*(b^2*c*x^2 + a*b*c)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x ^2 + c)) + 2*(a*b*c - a^2*d)*sqrt(d*x^2 + c))/(a^2*b^2*x^2 + a^3*b)]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{x \left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{2} x} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=\frac {c^{2} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{2} b} + \frac {\sqrt {d x^{2} + c} b c d - \sqrt {d x^{2} + c} a d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} a b} \]
c^2*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^2*sqrt(-c)) - 1/2*(2*b^2*c^2 - a*b *c*d - a^2*d^2)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2* c + a*b*d)*a^2*b) + 1/2*(sqrt(d*x^2 + c)*b*c*d - sqrt(d*x^2 + c)*a*d^2)/(( (d*x^2 + c)*b - b*c + a*d)*a*b)
Time = 7.19 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.78 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )^2} \, dx=-\frac {\mathrm {atanh}\left (\frac {d^6\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{2\,\left (\frac {c^2\,d^6}{2}+\frac {b\,c^3\,d^5}{a}-\frac {3\,b^2\,c^4\,d^4}{2\,a^2}\right )}+\frac {c\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{c^3\,d^5+\frac {a\,c^2\,d^6}{2\,b}-\frac {3\,b\,c^4\,d^4}{2\,a}}-\frac {3\,b\,c^2\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{2\,\left (a\,c^3\,d^5-\frac {3\,b\,c^4\,d^4}{2}+\frac {a^2\,c^2\,d^6}{2\,b}\right )}\right )\,\sqrt {c^3}}{a^2}-\frac {\mathrm {atanh}\left (\frac {5\,c^2\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {b^4\,c-a\,b^3\,d}}{4\,\left (\frac {a^2\,c\,d^7}{4}+\frac {b^2\,c^3\,d^5}{4}-\frac {3\,b^3\,c^4\,d^4}{2\,a}+a\,b\,c^2\,d^6\right )}+\frac {3\,c^3\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {b^4\,c-a\,b^3\,d}}{2\,\left (a^2\,c^2\,d^6-\frac {3\,b^2\,c^4\,d^4}{2}+\frac {a^3\,c\,d^7}{4\,b}+\frac {a\,b\,c^3\,d^5}{4}\right )}+\frac {c\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {b^4\,c-a\,b^3\,d}}{4\,\left (b^2\,c^2\,d^6+\frac {a\,b\,c\,d^7}{4}+\frac {b^3\,c^3\,d^5}{4\,a}-\frac {3\,b^4\,c^4\,d^4}{2\,a^2}\right )}\right )\,\left (a\,d+2\,b\,c\right )\,\sqrt {-b^3\,\left (a\,d-b\,c\right )}}{2\,a^2\,b^3}-\frac {d\,\sqrt {d\,x^2+c}\,\left (a\,d-b\,c\right )}{2\,a\,b\,\left (b\,\left (d\,x^2+c\right )+a\,d-b\,c\right )} \]
- (atanh((d^6*(c + d*x^2)^(1/2)*(c^3)^(1/2))/(2*((c^2*d^6)/2 + (b*c^3*d^5) /a - (3*b^2*c^4*d^4)/(2*a^2))) + (c*d^5*(c + d*x^2)^(1/2)*(c^3)^(1/2))/(c^ 3*d^5 + (a*c^2*d^6)/(2*b) - (3*b*c^4*d^4)/(2*a)) - (3*b*c^2*d^4*(c + d*x^2 )^(1/2)*(c^3)^(1/2))/(2*(a*c^3*d^5 - (3*b*c^4*d^4)/2 + (a^2*c^2*d^6)/(2*b) )))*(c^3)^(1/2))/a^2 - (atanh((5*c^2*d^5*(c + d*x^2)^(1/2)*(b^4*c - a*b^3* d)^(1/2))/(4*((a^2*c*d^7)/4 + (b^2*c^3*d^5)/4 - (3*b^3*c^4*d^4)/(2*a) + a* b*c^2*d^6)) + (3*c^3*d^4*(c + d*x^2)^(1/2)*(b^4*c - a*b^3*d)^(1/2))/(2*(a^ 2*c^2*d^6 - (3*b^2*c^4*d^4)/2 + (a^3*c*d^7)/(4*b) + (a*b*c^3*d^5)/4)) + (c *d^6*(c + d*x^2)^(1/2)*(b^4*c - a*b^3*d)^(1/2))/(4*(b^2*c^2*d^6 + (a*b*c*d ^7)/4 + (b^3*c^3*d^5)/(4*a) - (3*b^4*c^4*d^4)/(2*a^2))))*(a*d + 2*b*c)*(-b ^3*(a*d - b*c))^(1/2))/(2*a^2*b^3) - (d*(c + d*x^2)^(1/2)*(a*d - b*c))/(2* a*b*(b*(c + d*x^2) + a*d - b*c))