Integrand size = 24, antiderivative size = 147 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {b (5 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}} \]
1/2*b*(-4*a*d+5*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^ (7/2)/(-a*d+b*c)^(1/2)-5/6*(d*x^2+c)^(1/2)/a^2/x^3+1/6*(-2*a*d+15*b*c)*(d* x^2+c)^(1/2)/a^3/c/x+1/2*(d*x^2+c)^(1/2)/a/x^3/(b*x^2+a)
Time = 0.87 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (15 b^2 c x^4-2 a b x^2 \left (-5 c+d x^2\right )-2 a^2 \left (c+d x^2\right )\right )}{6 a^3 c x^3 \left (a+b x^2\right )}-\frac {b (5 b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2} \sqrt {b c-a d}} \]
(Sqrt[c + d*x^2]*(15*b^2*c*x^4 - 2*a*b*x^2*(-5*c + d*x^2) - 2*a^2*(c + d*x ^2)))/(6*a^3*c*x^3*(a + b*x^2)) - (b*(5*b*c - 4*a*d)*ArcTan[(a*Sqrt[d] + b *x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(7/2)*S qrt[b*c - a*d])
Time = 0.33 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {371, 25, 445, 27, 445, 27, 291, 218}
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.
| \(\displaystyle \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 371 |
| \(\displaystyle \frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}-\frac {\int -\frac {4 d x^2+5 c}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {4 d x^2+5 c}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int \frac {c \left (10 b d x^2+15 b c-2 a d\right )}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{3 a c}-\frac {5 \sqrt {c+d x^2}}{3 a x^3}}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {10 b d x^2+15 b c-2 a d}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{3 a}-\frac {5 \sqrt {c+d x^2}}{3 a x^3}}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {3 b c (5 b c-4 a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {c+d x^2} (15 b c-2 a d)}{a c x}}{3 a}-\frac {5 \sqrt {c+d x^2}}{3 a x^3}}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 b (5 b c-4 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {c+d x^2} (15 b c-2 a d)}{a c x}}{3 a}-\frac {5 \sqrt {c+d x^2}}{3 a x^3}}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {-\frac {-\frac {3 b (5 b c-4 a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{a}-\frac {\sqrt {c+d x^2} (15 b c-2 a d)}{a c x}}{3 a}-\frac {5 \sqrt {c+d x^2}}{3 a x^3}}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {-\frac {3 b (5 b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^2} (15 b c-2 a d)}{a c x}}{3 a}-\frac {5 \sqrt {c+d x^2}}{3 a x^3}}{2 a}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}\) |
Sqrt[c + d*x^2]/(2*a*x^3*(a + b*x^2)) + ((-5*Sqrt[c + d*x^2])/(3*a*x^3) - (-(((15*b*c - 2*a*d)*Sqrt[c + d*x^2])/(a*c*x)) - (3*b*(5*b*c - 4*a*d)*ArcT an[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*Sqrt[b*c - a*d ]))/(3*a))/(2*a)
3.8.39.3.1 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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-(e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a *e*2*(p + 1))), x] + Simp[1/(a*2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1) *(c + d*x^2)^(q - 1)*Simp[c*(m + 2*(p + 1) + 1) + d*(m + 2*(p + q + 1) + 1) *x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && Lt Q[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Time = 3.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 x^{3}}+\frac {b c \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d -5 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2}}{a^{3} c}\) | \(116\) |
| risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 c \,a^{3} x^{3}}-\frac {b \left (-\frac {\left (a d -b c \right ) \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a 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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b}-\frac {\left (a d -b c \right ) \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a 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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b}+\frac {\left (3 a d -5 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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (3 a d -5 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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{3}}\) | \(888\) |
| default | \(\text {Expression too large to display}\) | \(2052\) |
1/a^3*(-1/3*(d*x^2+c)^(1/2)*(a*d*x^2-6*b*c*x^2+a*c)/x^3+1/2*b*c*(b*(d*x^2+ c)^(1/2)*x/(b*x^2+a)-(4*a*d-5*b*c)/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^( 1/2)/x*a/((a*d-b*c)*a)^(1/2))))/c
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (123) = 246\).
Time = 0.35 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.10 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, a^{3} b c^{2} - 2 \, a^{4} c d - {\left (15 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left ({\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x^{5} + {\left (a^{5} b c^{2} - a^{6} c d\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} b c^{2} - 2 \, a^{4} c d - {\left (15 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x^{5} + {\left (a^{5} b c^{2} - a^{6} c d\right )} x^{3}\right )}}\right ] \]
[1/24*(3*((5*b^3*c^2 - 4*a*b^2*c*d)*x^5 + (5*a*b^2*c^2 - 4*a^2*b*c*d)*x^3) *sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b *c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(2*a^3*b*c^2 - 2*a^4*c*d - (15*a*b^3*c^2 - 17*a^2*b^2*c*d + 2*a^3*b*d^2)*x^4 - 2*(5*a^ 2*b^2*c^2 - 6*a^3*b*c*d + a^4*d^2)*x^2)*sqrt(d*x^2 + c))/((a^4*b^2*c^2 - a ^5*b*c*d)*x^5 + (a^5*b*c^2 - a^6*c*d)*x^3), 1/12*(3*((5*b^3*c^2 - 4*a*b^2* c*d)*x^5 + (5*a*b^2*c^2 - 4*a^2*b*c*d)*x^3)*sqrt(a*b*c - a^2*d)*arctan(1/2 *sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*(2*a^3*b*c^2 - 2*a^4*c*d - (15 *a*b^3*c^2 - 17*a^2*b^2*c*d + 2*a^3*b*d^2)*x^4 - 2*(5*a^2*b^2*c^2 - 6*a^3* b*c*d + a^4*d^2)*x^2)*sqrt(d*x^2 + c))/((a^4*b^2*c^2 - a^5*b*c*d)*x^5 + (a ^5*b*c^2 - a^6*c*d)*x^3)]
\[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x^{4} \left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x^{4}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (123) = 246\).
Time = 1.08 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.46 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (5 \, b^{2} c \sqrt {d} - 4 \, a b d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b d^{\frac {3}{2}} - b^{2} c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a^{3}} - \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} + 6 \, b c^{3} \sqrt {d} - a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3}} \]
-1/2*(5*b^2*c*sqrt(d) - 4*a*b*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2 )*a^3) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c*sqrt(d) - 2*(sqrt(d)*x - s qrt(d*x^2 + c))^2*a*b*d^(3/2) - b^2*c^2*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d *x^2 + c))^2*a*d + b*c^2)*a^3) - 2/3*(6*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b* c*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d^(3/2) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) + 6*b*c^3*sqrt(d) - a*c^2*d^(3/2))/(((sqr t(d)*x - sqrt(d*x^2 + c))^2 - c)^3*a^3)
Timed out. \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {d\,x^2+c}}{x^4\,{\left (b\,x^2+a\right )}^2} \,d x \]