Integrand size = 24, antiderivative size = 115 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2}}{2 a c x^2}+\frac {(2 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{3/2}}-\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 \sqrt {b c-a d}} \]
1/2*(a*d+2*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^2/c^(3/2)-b^(3/2)*arcta nh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/a^2/(-a*d+b*c)^(1/2)-1/2*(d*x ^2+c)^(1/2)/a/c/x^2
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {-\frac {a \sqrt {c+d x^2}}{c x^2}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(2 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{2 a^2} \]
(-((a*Sqrt[c + d*x^2])/(c*x^2)) + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^ 2])/Sqrt[-(b*c) + a*d]])/Sqrt[-(b*c) + a*d] + ((2*b*c + a*d)*ArcTanh[Sqrt[ c + d*x^2]/Sqrt[c]])/c^(3/2))/(2*a^2)
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {354, 114, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {b d x^2+2 b c+a d}{2 x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a c}-\frac {\sqrt {c+d x^2}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {b d x^2+2 b c+a d}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{2 a c}-\frac {\sqrt {c+d x^2}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {(a d+2 b c) \int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {2 b^2 c \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a}}{2 a c}-\frac {\sqrt {c+d x^2}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {2 (a d+2 b c) \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{a d}-\frac {4 b^2 c \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{a d}}{2 a c}-\frac {\sqrt {c+d x^2}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {4 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (a d+2 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a \sqrt {c}}}{2 a c}-\frac {\sqrt {c+d x^2}}{a c x^2}\right )\) |
(-(Sqrt[c + d*x^2]/(a*c*x^2)) - ((-2*(2*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2] /Sqrt[c]])/(a*Sqrt[c]) + (4*b^(3/2)*c*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sq rt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(2*a*c))/2
3.8.7.3.1 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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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 = 3.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{\frac {3}{2}} b^{2} x^{2}+\sqrt {\left (a d -b c \right ) b}\, \left (-x^{2} \left (a d +2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+\sqrt {d \,x^{2}+c}\, a \sqrt {c}\right )}{2 \sqrt {\left (a d -b c \right ) b}\, c^{\frac {3}{2}} a^{2} x^{2}}\) | \(115\) |
| risch | \(-\frac {\sqrt {d \,x^{2}+c}}{2 a c \,x^{2}}-\frac {-\frac {\left (a d +2 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a \sqrt {c}}+\frac {b c \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}+\frac {b c \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}}{2 a c}\) | \(370\) |
| default | \(\frac {-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}}{a}+\frac {b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a^{2} \sqrt {c}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}\) | \(384\) |
-1/2/((a*d-b*c)*b)^(1/2)*(-2*arctan(b*(d*x^2+c)^(1/2)/((a*d-b*c)*b)^(1/2)) *c^(3/2)*b^2*x^2+((a*d-b*c)*b)^(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)))/c^(3/2)/a^2/x^2
Time = 0.33 (sec) , antiderivative size = 734, normalized size of antiderivative = 6.38 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [\frac {b c^{2} x^{2} \sqrt {\frac {b}{b c - a d}} \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 (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{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^{2} x^{2}}, \frac {b c^{2} x^{2} \sqrt {\frac {b}{b c - a d}} \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 (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, {\left (2 \, b c + a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c^{2} x^{2}}, \frac {2 \, b c^{2} x^{2} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\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^{2} x^{2}}, \frac {b c^{2} x^{2} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\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^{2} x^{2}}\right ] \]
[1/4*(b*c^2*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*b*c + a*d)*sqrt(c)*x^2*log(-(d*x^2 + 2*sqr t(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*sqrt(d*x^2 + c)*a*c)/(a^2*c^2*x^2), 1 /4*(b*c^2*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*(2*b*c + a*d)*sqrt(-c)*x^2*arctan(sqrt(-c)/sq rt(d*x^2 + c)) - 2*sqrt(d*x^2 + c)*a*c)/(a^2*c^2*x^2), 1/4*(2*b*c^2*x^2*sq rt(-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)) + (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^2* x^2), 1/2*(b*c^2*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)) - (2*b*c + a*d)*s qrt(-c)*x^2*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - sqrt(d*x^2 + c)*a*c)/(a^2*c ^2*x^2)]
\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} x^{3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {b^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c} c} - \frac {\sqrt {d x^{2} + c}}{2 \, a c x^{2}} \]
b^2*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)*c) - 1/2*sqrt(d*x^2 + c)/(a*c*x^2)
Time = 5.80 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.44 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\ln \left (\sqrt {d\,x^2+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}+b^6\,c^2+a^2\,b^4\,d^2-2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{2\,a^3\,d-2\,a^2\,b\,c}-\frac {\ln \left (\sqrt {d\,x^2+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}-b^6\,c^2-a^2\,b^4\,d^2+2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{2\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {\sqrt {d\,x^2+c}}{2\,a\,c\,x^2}-\frac {\mathrm {atan}\left (\frac {b^4\,d^4\,\sqrt {d\,x^2+c}\,3{}\mathrm {i}}{2\,\sqrt {c^3}\,\left (\frac {3\,b^4\,d^4}{2\,c}+\frac {5\,a\,b^3\,d^5}{4\,c^2}+\frac {a^2\,b^2\,d^6}{4\,c^3}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{4\,\sqrt {c^3}\,\left (\frac {5\,b^3\,d^5}{4\,a}+\frac {b^2\,d^6}{4\,c}+\frac {3\,b^4\,c\,d^4}{2\,a^2}\right )}+\frac {b^3\,d^5\,\sqrt {d\,x^2+c}\,5{}\mathrm {i}}{4\,\sqrt {c^3}\,\left (\frac {3\,b^4\,d^4}{2\,a}+\frac {5\,b^3\,d^5}{4\,c}+\frac {a\,b^2\,d^6}{4\,c^2}\right )}\right )\,\left (a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{2\,a^2\,\sqrt {c^3}} \]
(log((c + d*x^2)^(1/2)*(b^4*c - a*b^3*d)^(3/2) + b^6*c^2 + a^2*b^4*d^2 - 2 *a*b^5*c*d)*(b^4*c - a*b^3*d)^(1/2))/(2*a^3*d - 2*a^2*b*c) - (log((c + d*x ^2)^(1/2)*(b^4*c - a*b^3*d)^(3/2) - b^6*c^2 - a^2*b^4*d^2 + 2*a*b^5*c*d)*( b^4*c - a*b^3*d)^(1/2))/(2*(a^3*d - a^2*b*c)) - (c + d*x^2)^(1/2)/(2*a*c*x ^2) - (atan((b^4*d^4*(c + d*x^2)^(1/2)*3i)/(2*(c^3)^(1/2)*((3*b^4*d^4)/(2* c) + (5*a*b^3*d^5)/(4*c^2) + (a^2*b^2*d^6)/(4*c^3))) + (b^2*d^6*(c + d*x^2 )^(1/2)*1i)/(4*(c^3)^(1/2)*((5*b^3*d^5)/(4*a) + (b^2*d^6)/(4*c) + (3*b^4*c *d^4)/(2*a^2))) + (b^3*d^5*(c + d*x^2)^(1/2)*5i)/(4*(c^3)^(1/2)*((3*b^4*d^ 4)/(2*a) + (5*b^3*d^5)/(4*c) + (a*b^2*d^6)/(4*c^2))))*(a*d + 2*b*c)*1i)/(2 *a^2*(c^3)^(1/2))