Integrand size = 22, antiderivative size = 84 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac {(4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b (4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {464, 277, 270} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {2 b \left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{315 a^3 x^5}+\frac {\left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{63 a^2 x^7}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9} \]
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Rule 270
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}-\frac {(4 A b-9 a B) \int \frac {\left (a+b x^2\right )^{3/2}}{x^8} \, dx}{9 a} \\ & = -\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac {(4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}+\frac {(2 b (4 A b-9 a B)) \int \frac {\left (a+b x^2\right )^{3/2}}{x^6} \, dx}{63 a^2} \\ & = -\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac {(4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b (4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {\left (a+b x^2\right )^{5/2} \left (-35 a^2 A+20 a A b x^2-45 a^2 B x^2-8 A b^2 x^4+18 a b B x^4\right )}{315 a^3 x^9} \]
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Time = 2.85 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {\left (\left (\frac {9 x^{2} B}{7}+A \right ) a^{2}-\frac {4 x^{2} b \left (\frac {9 x^{2} B}{10}+A \right ) a}{7}+\frac {8 A \,b^{2} x^{4}}{35}\right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 x^{9} a^{3}}\) | \(55\) |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (8 A \,b^{2} x^{4}-18 B a b \,x^{4}-20 a A b \,x^{2}+45 a^{2} B \,x^{2}+35 a^{2} A \right )}{315 x^{9} a^{3}}\) | \(59\) |
default | \(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )\) | \(102\) |
trager | \(-\frac {\left (8 A \,b^{4} x^{8}-18 B a \,b^{3} x^{8}-4 A a \,b^{3} x^{6}+9 B \,a^{2} b^{2} x^{6}+3 A \,a^{2} b^{2} x^{4}+72 B \,a^{3} b \,x^{4}+50 A \,a^{3} b \,x^{2}+45 B \,a^{4} x^{2}+35 A \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{315 x^{9} a^{3}}\) | \(107\) |
risch | \(-\frac {\left (8 A \,b^{4} x^{8}-18 B a \,b^{3} x^{8}-4 A a \,b^{3} x^{6}+9 B \,a^{2} b^{2} x^{6}+3 A \,a^{2} b^{2} x^{4}+72 B \,a^{3} b \,x^{4}+50 A \,a^{3} b \,x^{2}+45 B \,a^{4} x^{2}+35 A \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{315 x^{9} a^{3}}\) | \(107\) |
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none
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {{\left (2 \, {\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} - {\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 35 \, A a^{4} - 3 \, {\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} x^{4} - 5 \, {\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, a^{3} x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1408 vs. \(2 (78) = 156\).
Time = 3.17 (sec) , antiderivative size = 1408, normalized size of antiderivative = 16.76 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=\text {Too large to display} \]
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Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{315 \, a^{3} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{7 \, a x^{7}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{9 \, a x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (72) = 144\).
Time = 0.32 (sec) , antiderivative size = 400, normalized size of antiderivative = 4.76 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {4 \, {\left (315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} B b^{\frac {7}{2}} - 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {7}{2}} + 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {9}{2}} + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {7}{2}} + 1260 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {9}{2}} - 819 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {7}{2}} + 1764 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {9}{2}} + 441 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {7}{2}} + 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {9}{2}} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {7}{2}} + 144 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {9}{2}} + 81 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {7}{2}} - 36 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {9}{2}} - 9 \, B a^{7} b^{\frac {7}{2}} + 4 \, A a^{6} b^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]
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Time = 6.47 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.02 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {4\,A\,b^3\,\sqrt {b\,x^2+a}}{315\,a^2\,x^3}-\frac {10\,A\,b\,\sqrt {b\,x^2+a}}{63\,x^7}-\frac {B\,a\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {8\,B\,b\,\sqrt {b\,x^2+a}}{35\,x^5}-\frac {A\,b^2\,\sqrt {b\,x^2+a}}{105\,a\,x^5}-\frac {A\,a\,\sqrt {b\,x^2+a}}{9\,x^9}-\frac {8\,A\,b^4\,\sqrt {b\,x^2+a}}{315\,a^3\,x}-\frac {B\,b^2\,\sqrt {b\,x^2+a}}{35\,a\,x^3}+\frac {2\,B\,b^3\,\sqrt {b\,x^2+a}}{35\,a^2\,x} \]
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