Integrand size = 21, antiderivative size = 163 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}} \]
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Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12, 385, 214} \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
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Rule 12
Rule 214
Rule 385
Rule 425
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-3 a d-2 b d x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)} \\ & = -\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 {\int \frac {8 b^2 c^2-8 a b c d+3 a^2 d^2}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2} \\ & = -\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 ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2} \\ & = -\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 {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)^2} \\ & = -\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 ) \tanh ^{-1}\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}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\frac {d x \sqrt {a+b x^2} \left (-2 b c \left (4 c+3 d x^2\right )+a d \left (5 c+3 d x^2\right )\right )}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{8 c^{5/2} (-b c+a d)^{5/2}} \]
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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 | \(\text {Expression too large to display}\) | \(1843\) |
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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 ] \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{3}} \,d x } \]
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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 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=-\frac {1}{8} \, b^{\frac {5}{2}} {\left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{2} c^{2} d - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b c d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} d^{3} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{3} c^{3} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{2} c^{2} d + 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} d^{3} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{2} c^{2} d - 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} d^{3} + 6 \, a^{4} b c d^{2} - 3 \, a^{5} d^{3}\right )}}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2}}\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3} \,d x \]
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