Integrand size = 21, antiderivative size = 143 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {d (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 143, 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} \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=-\frac {d (4 b c-a d) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}}+\frac {b x (a d+2 b c)}{2 a c \sqrt {a+b x^2} (b c-a d)^2}-\frac {d x}{2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)} \]
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Rule 12
Rule 214
Rule 385
Rule 425
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-2 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = \frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {\int \frac {a d (4 b c-a d)}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 a c (b c-a d)^2} \\ & = \frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {(d (4 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)^2} \\ & = \frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {(d (4 b c-a d)) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 c (b c-a d)^2} \\ & = \frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\frac {\frac {\sqrt {c} x \left (a^2 d^2+a b d^2 x^2+2 b^2 c \left (c+d x^2\right )\right )}{a (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}}{2 c^{3/2}} \]
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Time = 2.45 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(\frac {-a d \sqrt {b \,x^{2}+a}\, \left (d \,x^{2}+c \right ) \left (a d -4 b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+x \sqrt {\left (a d -b c \right ) c}\, \left (2 b^{2} c^{2}+2 x^{2} b^{2} c d +a \,d^{2} \left (b \,x^{2}+a \right )\right )}{2 \sqrt {b \,x^{2}+a}\, \sqrt {\left (a d -b c \right ) c}\, c \left (d \,x^{2}+c \right ) \left (a d -b c \right )^{2} a}\) | \(150\) |
default | \(\text {Expression too large to display}\) | \(1906\) |
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (123) = 246\).
Time = 0.49 (sec) , antiderivative size = 864, normalized size of antiderivative = 6.04 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\left [-\frac {{\left (4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} 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 ({\left (2 \, b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} + {\left (a b^{4} c^{5} d - 3 \, a^{2} b^{3} c^{4} d^{2} + 3 \, a^{3} b^{2} c^{3} d^{3} - a^{4} b c^{2} d^{4}\right )} x^{4} + {\left (a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 2 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4}\right )} x^{2}\right )}}, \frac {{\left (4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} 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 ({\left (2 \, b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} + {\left (a b^{4} c^{5} d - 3 \, a^{2} b^{3} c^{4} d^{2} + 3 \, a^{3} b^{2} c^{3} d^{3} - a^{4} b c^{2} d^{4}\right )} x^{4} + {\left (a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 2 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (123) = 246\).
Time = 0.87 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\frac {b^{2} x}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b x^{2} + a}} + \frac {{\left (4 \, b^{\frac {3}{2}} c d - a \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c d - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d^{2} + a^{2} \sqrt {b} d^{2}}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^2} \,d x \]
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