Integrand size = 21, antiderivative size = 163 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}} \]
1/8*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b* x^2+a)^(1/2))/c^(5/2)/(-a*d+b*c)^(5/2)-1/4*d*x*(b*x^2+a)^(1/2)/c/(-a*d+b*c )/(d*x^2+c)^2-3/8*d*(-a*d+2*b*c)*x*(b*x^2+a)^(1/2)/c^2/(-a*d+b*c)^2/(d*x^2 +c)
Time = 0.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\frac {d x \sqrt {a+b x^2} \left (-2 b c \left (4 c+3 d x^2\right )+a d \left (5 c+3 d x^2\right )\right )}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{8 c^{5/2} (-b c+a d)^{5/2}} \]
(d*x*Sqrt[a + b*x^2]*(-2*b*c*(4*c + 3*d*x^2) + a*d*(5*c + 3*d*x^2)))/(8*c^ 2*(b*c - a*d)^2*(c + d*x^2)^2) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcT an[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[-(b*c) + a *d])])/(8*c^(5/2)*(-(b*c) + a*d)^(5/2))
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {316, 402, 27, 291, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {-2 b d x^2+4 b c-3 a d}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\int \frac {8 b^2 c^2-8 a b d c+3 a^2 d^2}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 c (b c-a d)}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)}\) |
-1/4*(d*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)^2) + ((-3*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(2*c*(b*c - a*d)*(c + d*x^2)) + ((8*b^2*c^2 - 8*a *b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])] )/(2*c^(3/2)*(b*c - a*d)^(3/2)))/(4*c*(b*c - a*d))
3.1.80.3.1 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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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 = 2.56 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \left (d \,x^{2}+c \right )^{2} \left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {8}{3} b^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{8}+\frac {5 x \left (-\frac {8 b \,c^{2}}{5}+d \left (-\frac {6 b \,x^{2}}{5}+a \right ) c +\frac {3 a \,d^{2} x^{2}}{5}\right ) \sqrt {b \,x^{2}+a}\, d \sqrt {\left (a d -b c \right ) c}}{8}}{\sqrt {\left (a d -b c \right ) c}\, \left (a d -b c \right )^{2} c^{2} \left (d \,x^{2}+c \right )^{2}}\) | \(149\) |
| default | \(\text {Expression too large to display}\) | \(1843\) |
5/8/((a*d-b*c)*c)^(1/2)*(-3/5*(d*x^2+c)^2*(a^2*d^2-8/3*a*b*c*d+8/3*b^2*c^2 )*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+x*(-8/5*b*c^2+d*(-6/5*b* x^2+a)*c+3/5*a*d^2*x^2)*(b*x^2+a)^(1/2)*d*((a*d-b*c)*c)^(1/2))/(a*d-b*c)^2 /c^2/(d*x^2+c)^2
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (143) = 286\).
Time = 0.49 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.30 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\left [\frac {{\left (8 \, b^{2} c^{4} - 8 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 4 \, {\left (3 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d - 13 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} x^{2}\right )}}, -\frac {{\left (8 \, b^{2} c^{4} - 8 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (3 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d - 13 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} x^{2}\right )}}\right ] \]
[1/32*((8*b^2*c^4 - 8*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 - 8*a*b*c *d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d - 8*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2) *sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) - 4*(3*(2*b^2*c^3* d^2 - 3*a*b*c^2*d^3 + a^2*c*d^4)*x^3 + (8*b^2*c^4*d - 13*a*b*c^3*d^2 + 5*a ^2*c^2*d^3)*x)*sqrt(b*x^2 + a))/(b^3*c^8 - 3*a*b^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3 + (b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^ 3*d^5)*x^4 + 2*(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c^4*d^ 4)*x^2), -1/16*((8*b^2*c^4 - 8*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 - 8*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d - 8*a*b*c^2*d^2 + 3*a^2*c* d^3)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a *d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c *d)*x)) + 2*(3*(2*b^2*c^3*d^2 - 3*a*b*c^2*d^3 + a^2*c*d^4)*x^3 + (8*b^2*c^ 4*d - 13*a*b*c^3*d^2 + 5*a^2*c^2*d^3)*x)*sqrt(b*x^2 + a))/(b^3*c^8 - 3*a*b ^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3 + (b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^3*d^5)*x^4 + 2*(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3* a^2*b*c^5*d^3 - a^3*c^4*d^4)*x^2)]
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (143) = 286\).
Time = 1.74 (sec) , antiderivative size = 538, normalized size of antiderivative = 3.30 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=-\frac {1}{8} \, b^{\frac {5}{2}} {\left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{2} c^{2} d - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b c d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} d^{3} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{3} c^{3} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{2} c^{2} d + 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} d^{3} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{2} c^{2} d - 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} d^{3} + 6 \, a^{4} b c d^{2} - 3 \, a^{5} d^{3}\right )}}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2}}\right )} \]
-1/8*b^(5/2)*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/((b^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*d^2)*sqrt(-b^2*c^2 + a*b*c*d)) + 2*(8*(sqrt(b )*x - sqrt(b*x^2 + a))^6*b^2*c^2*d - 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b *c*d^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*d^3 + 48*(sqrt(b)*x - sqrt( b*x^2 + a))^4*b^3*c^3 - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*c^2*d + 4 2*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b*c*d^2 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*d^3 + 40*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^2*c^2*d - 40*(s qrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b*c*d^2 + 9*(sqrt(b)*x - sqrt(b*x^2 + a) )^2*a^4*d^3 + 6*a^4*b*c*d^2 - 3*a^5*d^3)/((b^4*c^4 - 2*a*b^3*c^3*d + a^2*b ^2*c^2*d^2)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^2))
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3} \,d x \]