Integrand size = 21, antiderivative size = 142 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d) x}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {b (b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 142, 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, 211} \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {b (b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{5/2}}+\frac {d x (2 a d+b c)}{2 a c \sqrt {c+d x^2} (b c-a d)^2}+\frac {b x}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 425
Rule 541
Rubi steps \begin{align*} \text {integral}& = \frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\int \frac {-b c+2 a d-2 b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{2 a (b c-a d)} \\ & = \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 {\int \frac {b c (b c-4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a c (b c-a d)^2} \\ & = \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)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a (b c-a d)^2} \\ & = \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)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a (b c-a d)^2} \\ & = \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) \tan ^{-1}\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}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {x \left (2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )}{2 a c (b c-a d)^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {b (b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} (b c-a d)^{5/2}} \]
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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 | \(\text {Expression too large to display}\) | \(1928\) |
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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 ] \]
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\[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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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 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {d^{2} x}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {d x^{2} + c}} - \frac {{\left (b^{2} c \sqrt {d} - 4 \, a b d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b d^{\frac {3}{2}} - b^{2} c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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