Integrand size = 21, antiderivative size = 142 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d) x}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {b (b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{5/2}} \]
1/2*b*(-4*a*d+b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(3 /2)/(-a*d+b*c)^(5/2)+1/2*d*(2*a*d+b*c)*x/a/c/(-a*d+b*c)^2/(d*x^2+c)^(1/2)+ 1/2*b*x/a/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^(1/2)
Time = 0.54 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {x \left (2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )}{2 a c (b c-a d)^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {b (b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} (b c-a d)^{5/2}} \]
(x*(2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(c + d*x^2)))/(2*a*c*(b*c - a*d)^2*( a + b*x^2)*Sqrt[c + d*x^2]) - (b*(b*c - 4*a*d)*ArcTan[(a*Sqrt[d] + b*x*(Sq rt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(3/2)*(b*c - a*d)^(5/2))
Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {316, 25, 402, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}-\frac {\int -\frac {2 b d x^2+b c-2 a d}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx}{2 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 b d x^2+b c-2 a d}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx}{2 a (b c-a d)}+\frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\int \frac {b c (b c-4 a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{c (b c-a d)}+\frac {d x (2 a d+b c)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a (b c-a d)}+\frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b (b c-4 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b c-a d}+\frac {d x (2 a d+b c)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a (b c-a d)}+\frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {b (b c-4 a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b c-a d}+\frac {d x (2 a d+b c)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a (b c-a d)}+\frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {b (b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} (b c-a d)^{3/2}}+\frac {d x (2 a d+b c)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a (b c-a d)}+\frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}\) |
(b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*Sqrt[c + d*x^2]) + ((d*(b*c + 2*a*d)*x) /(c*(b*c - a*d)*Sqrt[c + d*x^2]) + (b*(b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d ]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*(b*c - a*d)^(3/2)))/(2*a*(b*c - a*d))
3.8.71.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 3.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77
| method | result | size |
| pseudoelliptic | \(\frac {\frac {b c \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 a}+\frac {d^{2} x}{\sqrt {d \,x^{2}+c}}}{\left (a d -b c \right )^{2} c}\) | \(109\) |
| default | \(\text {Expression too large to display}\) | \(1928\) |
1/(a*d-b*c)^2*(1/2*b*c/a*(b*(d*x^2+c)^(1/2)*x/(b*x^2+a)-(4*a*d-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)))+d^2/(d*x^2 +c)^(1/2)*x)/c
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (122) = 244\).
Time = 0.52 (sec) , antiderivative size = 854, normalized size of antiderivative = 6.01 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {{\left (a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{4} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{2}\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 ({\left (a b^{3} c^{2} d + a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{3} + {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d + 2 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{3} c^{5} - 3 \, a^{4} b^{2} c^{4} d + 3 \, a^{5} b c^{3} d^{2} - a^{6} c^{2} d^{3} + {\left (a^{2} b^{4} c^{4} d - 3 \, a^{3} b^{3} c^{3} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{3} - a^{5} b c d^{4}\right )} x^{4} + {\left (a^{2} b^{4} c^{5} - 2 \, a^{3} b^{3} c^{4} d + 2 \, a^{5} b c^{2} d^{3} - a^{6} c d^{4}\right )} x^{2}\right )}}, \frac {{\left (a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{4} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{2}\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 ({\left (a b^{3} c^{2} d + a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{3} + {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d + 2 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{3} c^{5} - 3 \, a^{4} b^{2} c^{4} d + 3 \, a^{5} b c^{3} d^{2} - a^{6} c^{2} d^{3} + {\left (a^{2} b^{4} c^{4} d - 3 \, a^{3} b^{3} c^{3} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{3} - a^{5} b c d^{4}\right )} x^{4} + {\left (a^{2} b^{4} c^{5} - 2 \, a^{3} b^{3} c^{4} d + 2 \, a^{5} b c^{2} d^{3} - a^{6} c d^{4}\right )} x^{2}\right )}}\right ] \]
[1/8*((a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^2*d - 4*a*b^2*c*d^2)*x^4 + (b^3* c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d^2)*x^2)*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*((a*b^3*c^2*d + a^2*b^2*c*d^2 - 2*a^3*b*d^3)* x^3 + (a*b^3*c^3 - a^2*b^2*c^2*d + 2*a^3*b*c*d^2 - 2*a^4*d^3)*x)*sqrt(d*x^ 2 + c))/(a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3 + ( a^2*b^4*c^4*d - 3*a^3*b^3*c^3*d^2 + 3*a^4*b^2*c^2*d^3 - a^5*b*c*d^4)*x^4 + (a^2*b^4*c^5 - 2*a^3*b^3*c^4*d + 2*a^5*b*c^2*d^3 - a^6*c*d^4)*x^2), 1/4*( (a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^2*d - 4*a*b^2*c*d^2)*x^4 + (b^3*c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d^2)*x^2)*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*((a*b^3*c^2*d + a^2*b^2*c*d^2 - 2*a^3* b*d^3)*x^3 + (a*b^3*c^3 - a^2*b^2*c^2*d + 2*a^3*b*c*d^2 - 2*a^4*d^3)*x)*sq rt(d*x^2 + c))/(a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2* d^3 + (a^2*b^4*c^4*d - 3*a^3*b^3*c^3*d^2 + 3*a^4*b^2*c^2*d^3 - a^5*b*c*d^4 )*x^4 + (a^2*b^4*c^5 - 2*a^3*b^3*c^4*d + 2*a^5*b*c^2*d^3 - a^6*c*d^4)*x^2) ]
\[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (122) = 244\).
Time = 0.86 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.24 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {d^{2} x}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {d x^{2} + c}} - \frac {{\left (b^{2} c \sqrt {d} - 4 \, a b d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b d^{\frac {3}{2}} - b^{2} c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} \]
d^2*x/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(d*x^2 + c)) - 1/2*(b^2*c*s qrt(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))/((a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*s qrt(a*b*c*d - a^2*d^2)) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(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*(s qrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*(a*b^2*c^2 - 2*a^2*b*c*d + a^3* d^2))
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]