Integrand size = 24, antiderivative size = 205 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^2-4 a b c d+4 a^2 d^2\right ) \sqrt {c+d x^2}}{2 a^2 c^2 (b c-a d)^2 x}-\frac {3 b^2 (b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{5/2}} \]
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Time = 0.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {483, 593, 597, 12, 385, 211} \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {3 b^2 (b c-2 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{5/2}}-\frac {\sqrt {c+d x^2} \left (4 a^2 d^2-4 a b c d+3 b^2 c^2\right )}{2 a^2 c^2 x (b c-a d)^2}+\frac {b}{2 a x \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {d (2 a d+b c)}{2 a c x \sqrt {c+d x^2} (b c-a d)^2} \]
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Rule 12
Rule 211
Rule 385
Rule 483
Rule 593
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\int \frac {-3 b c+2 a d-4 b d x^2}{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)}{2 a c (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\int \frac {-3 b^2 c^2+4 a b c d-4 a^2 d^2-2 b d (b c+2 a d) x^2}{x^2 \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)}{2 a c (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^2-4 a b c d+4 a^2 d^2\right ) \sqrt {c+d x^2}}{2 a^2 c^2 (b c-a d)^2 x}+\frac {\int -\frac {3 b^2 c^2 (b c-2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 c^2 (b c-a d)^2} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^2-4 a b c d+4 a^2 d^2\right ) \sqrt {c+d x^2}}{2 a^2 c^2 (b c-a d)^2 x}-\frac {\left (3 b^2 (b c-2 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 (b c-a d)^2} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^2-4 a b c d+4 a^2 d^2\right ) \sqrt {c+d x^2}}{2 a^2 c^2 (b c-a d)^2 x}-\frac {\left (3 b^2 (b c-2 a d)\right ) \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^2 (b c-a d)^2} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^2-4 a b c d+4 a^2 d^2\right ) \sqrt {c+d x^2}}{2 a^2 c^2 (b c-a d)^2 x}-\frac {3 b^2 (b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-3 b^3 c^2 x^2 \left (c+d x^2\right )-2 a^3 d^2 \left (c+2 d x^2\right )+2 a^2 b d \left (2 c^2+c d x^2-2 d^2 x^4\right )+2 a b^2 c \left (-c^2+c d x^2+2 d^2 x^4\right )}{2 a^2 c^2 (b c-a d)^2 x \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {3 b^2 (b c-2 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2} (b c-a d)^{5/2}} \]
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Time = 3.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}}{a^{2} x}-\frac {b^{2} c^{2} \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {3 \left (2 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 a^{2} \left (a d -b c \right )^{2}}-\frac {d^{3} x}{\left (a d -b c \right )^{2} \sqrt {d \,x^{2}+c}}}{c^{2}}\) | \(141\) |
risch | \(\text {Expression too large to display}\) | \(1246\) |
default | \(\text {Expression too large to display}\) | \(1962\) |
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Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (181) = 362\).
Time = 0.58 (sec) , antiderivative size = 1018, normalized size of antiderivative = 4.97 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2}\right )} x^{5} + {\left (b^{4} c^{4} - a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2}\right )} x^{3} + {\left (a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d\right )} x\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 (2 \, a^{2} b^{3} c^{4} - 6 \, a^{3} b^{2} c^{3} d + 6 \, a^{4} b c^{2} d^{2} - 2 \, a^{5} c d^{3} + {\left (3 \, a b^{4} c^{3} d - 7 \, a^{2} b^{3} c^{2} d^{2} + 8 \, a^{3} b^{2} c d^{3} - 4 \, a^{4} b d^{4}\right )} x^{4} + {\left (3 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 6 \, a^{4} b c d^{3} - 4 \, a^{5} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left ({\left (a^{3} b^{4} c^{5} d - 3 \, a^{4} b^{3} c^{4} d^{2} + 3 \, a^{5} b^{2} c^{3} d^{3} - a^{6} b c^{2} d^{4}\right )} x^{5} + {\left (a^{3} b^{4} c^{6} - 2 \, a^{4} b^{3} c^{5} d + 2 \, a^{6} b c^{3} d^{3} - a^{7} c^{2} d^{4}\right )} x^{3} + {\left (a^{4} b^{3} c^{6} - 3 \, a^{5} b^{2} c^{5} d + 3 \, a^{6} b c^{4} d^{2} - a^{7} c^{3} d^{3}\right )} x\right )}}, -\frac {3 \, {\left ({\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2}\right )} x^{5} + {\left (b^{4} c^{4} - a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2}\right )} x^{3} + {\left (a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d\right )} x\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 (2 \, a^{2} b^{3} c^{4} - 6 \, a^{3} b^{2} c^{3} d + 6 \, a^{4} b c^{2} d^{2} - 2 \, a^{5} c d^{3} + {\left (3 \, a b^{4} c^{3} d - 7 \, a^{2} b^{3} c^{2} d^{2} + 8 \, a^{3} b^{2} c d^{3} - 4 \, a^{4} b d^{4}\right )} x^{4} + {\left (3 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 6 \, a^{4} b c d^{3} - 4 \, a^{5} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{3} b^{4} c^{5} d - 3 \, a^{4} b^{3} c^{4} d^{2} + 3 \, a^{5} b^{2} c^{3} d^{3} - a^{6} b c^{2} d^{4}\right )} x^{5} + {\left (a^{3} b^{4} c^{6} - 2 \, a^{4} b^{3} c^{5} d + 2 \, a^{6} b c^{3} d^{3} - a^{7} c^{2} d^{4}\right )} x^{3} + {\left (a^{4} b^{3} c^{6} - 3 \, a^{5} b^{2} c^{5} d + 3 \, a^{6} b c^{4} d^{2} - a^{7} c^{3} d^{3}\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (181) = 362\).
Time = 0.93 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.70 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d^{3} x}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt {d x^{2} + c}} + \frac {3 \, {\left (b^{3} c \sqrt {d} - 2 \, a b^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{3} c^{2} \sqrt {d} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{2} c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b d^{\frac {5}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{3} \sqrt {d} + 18 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} d^{\frac {3}{2}} - 20 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {5}{2}} + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {7}{2}} + 3 \, b^{3} c^{4} \sqrt {d} - 4 \, a b^{2} c^{3} d^{\frac {3}{2}} + 2 \, a^{2} b c^{2} d^{\frac {5}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )} {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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