Integrand size = 21, antiderivative size = 79 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {390, 385, 214} \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\frac {b x}{a \sqrt {a+b x^2} (b c-a d)}-\frac {d \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}} \]
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Rule 214
Rule 385
Rule 390
Rubi steps \begin{align*} \text {integral}& = \frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{b c-a d} \\ & = \frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b c-a d} \\ & = \frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\frac {b x}{\left (a b c-a^2 d\right ) \sqrt {a+b x^2}}-\frac {d \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} (-b c+a d)^{3/2}} \]
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Time = 2.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16
method | result | size |
pseudoelliptic | \(-\frac {d \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right ) a \sqrt {b \,x^{2}+a}+b x \sqrt {\left (a d -b c \right ) c}}{a \left (a d -b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {\left (a d -b c \right ) c}}\) | \(92\) |
default | \(-\frac {\frac {d}{\left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}+\frac {2 b \sqrt {-c d}\, \left (2 b \left (x +\frac {\sqrt {-c d}}{d}\right )-\frac {2 b \sqrt {-c d}}{d}\right )}{\left (a d -b c \right ) \left (\frac {4 b \left (a d -b c \right )}{d}+\frac {4 b^{2} c}{d}\right ) \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}-\frac {d \ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{\left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}}{2 \sqrt {-c d}}+\frac {\frac {d}{\left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}-\frac {2 b \sqrt {-c d}\, \left (2 b \left (x -\frac {\sqrt {-c d}}{d}\right )+\frac {2 b \sqrt {-c d}}{d}\right )}{\left (a d -b c \right ) \left (\frac {4 b \left (a d -b c \right )}{d}+\frac {4 b^{2} c}{d}\right ) \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}-\frac {d \ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{\left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}}{2 \sqrt {-c d}}\) | \(723\) |
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (67) = 134\).
Time = 0.32 (sec) , antiderivative size = 441, normalized size of antiderivative = 5.58 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\left [\frac {4 \, {\left (b^{2} c^{2} - a b c d\right )} \sqrt {b x^{2} + a} x - {\left (a b d x^{2} + a^{2} d\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{2} c^{2} - a b c d\right )} \sqrt {b x^{2} + a} x + {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{2 \, {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=-\frac {\sqrt {b} d \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (b c - a d\right )}} + \frac {b x}{{\left (a b c - a^{2} d\right )} \sqrt {b x^{2} + a}} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \]
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