Integrand size = 21, antiderivative size = 313 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}} \]
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Time = 0.31 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12, 385, 214} \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\frac {d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}+\frac {b x \left (-3 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{12 a^2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)^3}+\frac {d x \sqrt {a+b x^2} \left (9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac {d x}{4 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (b c-a d)}+\frac {b x (3 a d+4 b c)}{12 a c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]
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Rule 12
Rule 214
Rule 385
Rule 425
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-3 a d-6 b d x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)} \\ & = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}-\frac {\int \frac {-8 b^2 c^2+24 a b c d-9 a^2 d^2-4 b d (4 b c+3 a d) x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx}{12 a c (b c-a d)^2} \\ & = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {\int \frac {a d \left (8 b^2 c^2+36 a b c d-9 a^2 d^2\right )+2 b d \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{12 a^2 c (b c-a d)^3} \\ & = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\int \frac {3 a^2 d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{24 a^2 c^2 (b c-a d)^4} \\ & = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\left (d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^4} \\ & = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\left (d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)^4} \\ & = -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}} \\ \end{align*}
Time = 3.26 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\frac {\frac {\sqrt {c} x \left (16 b^5 c^3 x^2 \left (c+d x^2\right )^2+8 a b^4 c^2 \left (3 c-11 d x^2\right ) \left (c+d x^2\right )^2+3 a^5 d^4 \left (5 c+3 d x^2\right )+3 a^3 b^2 d^3 x^2 \left (-32 c^2-23 c d x^2+3 d^2 x^4\right )+6 a^4 b d^3 \left (-8 c^2-2 c d x^2+3 d^2 x^4\right )-6 a^2 b^3 c d \left (16 c^3+32 c^2 d x^2+24 c d^2 x^4+7 d^3 x^6\right )\right )}{a^2 (b c-a d)^4 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}-\frac {9 d^2 (-4 b c+a d)^2 \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(-b c+a d)^{9/2}}+\frac {24 a b c d^3 \text {arctanh}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {b c-a d}}\right )}{(b c-a d)^{9/2}}}{24 c^{5/2}} \]
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Time = 2.57 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{2} \left (a^{2} d^{2}-\frac {16}{3} a b c d +16 b^{2} c^{2}\right ) d^{2} a^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )-\frac {5 x \sqrt {\left (a d -b c \right ) c}\, \left (\frac {3 a^{3} x^{2} \left (b \,x^{2}+a \right )^{2} d^{5}}{5}+\left (b \,x^{2}+a \right )^{2} \left (-\frac {14 b \,x^{2}}{5}+a \right ) c \,a^{2} d^{4}-\frac {16 \left (\frac {11}{6} b^{3} x^{6}+3 a \,b^{2} x^{4}+2 a^{2} b \,x^{2}+a^{3}\right ) b \,c^{2} a \,d^{3}}{5}-\frac {64 x^{2} b^{3} \left (-\frac {1}{12} b^{2} x^{4}+\frac {19}{24} a b \,x^{2}+a^{2}\right ) c^{3} d^{2}}{5}-\frac {32 b^{3} \left (-\frac {1}{3} b^{2} x^{4}+\frac {5}{12} a b \,x^{2}+a^{2}\right ) c^{4} d}{5}+\frac {8 b^{4} \left (\frac {2 b \,x^{2}}{3}+a \right ) c^{5}}{5}\right )}{3}\right )}{8 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) c}\, \left (d \,x^{2}+c \right )^{2} c^{2} \left (a d -b c \right )^{4} a^{2}}\) | \(302\) |
default | \(\text {Expression too large to display}\) | \(7121\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (285) = 570\).
Time = 2.76 (sec) , antiderivative size = 2250, normalized size of antiderivative = 7.19 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1010 vs. \(2 (285) = 570\).
Time = 1.72 (sec) , antiderivative size = 1010, normalized size of antiderivative = 3.23 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\frac {{\left (\frac {{\left (2 \, b^{10} c^{5} - 19 \, a b^{9} c^{4} d + 56 \, a^{2} b^{8} c^{3} d^{2} - 74 \, a^{3} b^{7} c^{2} d^{3} + 46 \, a^{4} b^{6} c d^{4} - 11 \, a^{5} b^{5} d^{5}\right )} x^{2}}{a^{2} b^{9} c^{8} - 8 \, a^{3} b^{8} c^{7} d + 28 \, a^{4} b^{7} c^{6} d^{2} - 56 \, a^{5} b^{6} c^{5} d^{3} + 70 \, a^{6} b^{5} c^{4} d^{4} - 56 \, a^{7} b^{4} c^{3} d^{5} + 28 \, a^{8} b^{3} c^{2} d^{6} - 8 \, a^{9} b^{2} c d^{7} + a^{10} b d^{8}} + \frac {3 \, {\left (a b^{9} c^{5} - 8 \, a^{2} b^{8} c^{4} d + 22 \, a^{3} b^{7} c^{3} d^{2} - 28 \, a^{4} b^{6} c^{2} d^{3} + 17 \, a^{5} b^{5} c d^{4} - 4 \, a^{6} b^{4} d^{5}\right )}}{a^{2} b^{9} c^{8} - 8 \, a^{3} b^{8} c^{7} d + 28 \, a^{4} b^{7} c^{6} d^{2} - 56 \, a^{5} b^{6} c^{5} d^{3} + 70 \, a^{6} b^{5} c^{4} d^{4} - 56 \, a^{7} b^{4} c^{3} d^{5} + 28 \, a^{8} b^{3} c^{2} d^{6} - 8 \, a^{9} b^{2} c d^{7} + a^{10} b d^{8}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{\frac {5}{2}} c^{2} d^{2} - 16 \, a b^{\frac {3}{2}} c d^{3} + 3 \, a^{2} \sqrt {b} d^{4}\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 )}{8 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c d^{4} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{5} + 112 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} d^{2} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d^{3} + 66 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{4} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{5} + 88 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d^{3} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{4} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{5} + 14 \, a^{4} b^{\frac {3}{2}} c d^{4} - 3 \, a^{5} \sqrt {b} d^{5}}{4 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\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 )}^{2}} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^3} \,d x \]
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