Integrand size = 24, antiderivative size = 178 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}} \]
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Time = 0.17 (sec) , antiderivative size = 178, 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} \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {b^3 \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}-\frac {\sqrt {c+d x^2} (b c-4 a d) (3 b c-2 a d)}{3 a c^3 x (b c-a d)^2}-\frac {d (7 b c-4 a d)}{3 c^2 x \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c x \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 483
Rule 593
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {3 b c-4 a d-4 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c (b c-a d)} \\ & = -\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {\int \frac {(b c-4 a d) (3 b c-2 a d)-2 b d (7 b c-4 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c^2 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {\int \frac {3 b^3 c^3}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a c^3 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {-3 b^2 c^2 \left (c+d x^2\right )^2-a^2 d^2 \left (3 c^2+12 c d x^2+8 d^2 x^4\right )+a b c d \left (6 c^2+21 c d x^2+14 d^2 x^4\right )}{3 a c^3 (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac {b^3 \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{3/2} (b c-a d)^{5/2}} \]
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Time = 3.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}}{a x}-\frac {b^{3} c^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{a \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) a}}+\frac {d^{3} x^{3}}{3 \left (a d -b c \right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {d^{2} \left (2 a d -3 b c \right ) x}{\left (a d -b c \right )^{2} \sqrt {d \,x^{2}+c}}}{c^{3}}\) | \(144\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}}{c^{3} a x}-\frac {\frac {b \,d^{2} a \left (\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{3 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{3 c \left (x +\frac {\sqrt {-c d}}{d}\right )}\right )}{4 \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right )}+\frac {b \,d^{2} a \left (-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{3 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{3 c \left (x -\frac {\sqrt {-c d}}{d}\right )}\right )}{4 \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right )}+\frac {b^{5} d^{2} c^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \left (b \sqrt {-c d}+d \sqrt {-a b}\right )^{2} \left (d \sqrt {-a b}-b \sqrt {-c d}\right )^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {b^{5} d^{2} c^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \left (b \sqrt {-c d}+d \sqrt {-a b}\right )^{2} \left (d \sqrt {-a b}-b \sqrt {-c d}\right )^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {b^{2} d^{3} a \left (3 a d -5 b c \right ) \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{4 c \left (b \sqrt {-c d}+d \sqrt {-a b}\right )^{2} \left (d \sqrt {-a b}-b \sqrt {-c d}\right )^{2} \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {b^{2} d^{3} a \left (3 a d -5 b c \right ) \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{4 c \left (b \sqrt {-c d}+d \sqrt {-a b}\right )^{2} \left (d \sqrt {-a b}-b \sqrt {-c d}\right )^{2} \left (x -\frac {\sqrt {-c d}}{d}\right )}}{a \,c^{2}}\) | \(998\) |
default | \(\text {Expression too large to display}\) | \(1463\) |
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Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (156) = 312\).
Time = 0.51 (sec) , antiderivative size = 934, normalized size of antiderivative = 5.25 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{3} c^{3} d^{2} x^{5} + 2 \, b^{3} c^{4} d x^{3} + b^{3} c^{5} 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 (3 \, a b^{3} c^{5} - 9 \, a^{2} b^{2} c^{4} d + 9 \, a^{3} b c^{3} d^{2} - 3 \, a^{4} c^{2} d^{3} + {\left (3 \, a b^{3} c^{3} d^{2} - 17 \, a^{2} b^{2} c^{2} d^{3} + 22 \, a^{3} b c d^{4} - 8 \, a^{4} d^{5}\right )} x^{4} + 3 \, {\left (2 \, a b^{3} c^{4} d - 9 \, a^{2} b^{2} c^{3} d^{2} + 11 \, a^{3} b c^{2} d^{3} - 4 \, a^{4} c d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{2} b^{3} c^{6} d^{2} - 3 \, a^{3} b^{2} c^{5} d^{3} + 3 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}\right )} x^{5} + 2 \, {\left (a^{2} b^{3} c^{7} d - 3 \, a^{3} b^{2} c^{6} d^{2} + 3 \, a^{4} b c^{5} d^{3} - a^{5} c^{4} d^{4}\right )} x^{3} + {\left (a^{2} b^{3} c^{8} - 3 \, a^{3} b^{2} c^{7} d + 3 \, a^{4} b c^{6} d^{2} - a^{5} c^{5} d^{3}\right )} x\right )}}, -\frac {3 \, {\left (b^{3} c^{3} d^{2} x^{5} + 2 \, b^{3} c^{4} d x^{3} + b^{3} c^{5} 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 (3 \, a b^{3} c^{5} - 9 \, a^{2} b^{2} c^{4} d + 9 \, a^{3} b c^{3} d^{2} - 3 \, a^{4} c^{2} d^{3} + {\left (3 \, a b^{3} c^{3} d^{2} - 17 \, a^{2} b^{2} c^{2} d^{3} + 22 \, a^{3} b c d^{4} - 8 \, a^{4} d^{5}\right )} x^{4} + 3 \, {\left (2 \, a b^{3} c^{4} d - 9 \, a^{2} b^{2} c^{3} d^{2} + 11 \, a^{3} b c^{2} d^{3} - 4 \, a^{4} c d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{2} b^{3} c^{6} d^{2} - 3 \, a^{3} b^{2} c^{5} d^{3} + 3 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}\right )} x^{5} + 2 \, {\left (a^{2} b^{3} c^{7} d - 3 \, a^{3} b^{2} c^{6} d^{2} + 3 \, a^{4} b c^{5} d^{3} - a^{5} c^{4} d^{4}\right )} x^{3} + {\left (a^{2} b^{3} c^{8} - 3 \, a^{3} b^{2} c^{7} d + 3 \, a^{4} b c^{6} d^{2} - a^{5} c^{5} d^{3}\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (156) = 312\).
Time = 0.79 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {b^{3} \sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\frac {{\left (8 \, b^{3} c^{5} d^{4} - 21 \, a b^{2} c^{4} d^{5} + 18 \, a^{2} b c^{3} d^{6} - 5 \, a^{3} c^{2} d^{7}\right )} x^{2}}{b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}} + \frac {3 \, {\left (3 \, b^{3} c^{6} d^{3} - 8 \, a b^{2} c^{5} d^{4} + 7 \, a^{2} b c^{4} d^{5} - 2 \, a^{3} c^{3} d^{6}\right )}}{b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {2 \, \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} a c^{2}} \]
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Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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