Integrand size = 22, antiderivative size = 53 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9}+\frac {(2 A b-9 a B) \left (a+b x^2\right )^{7/2}}{63 a^2 x^7} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {464, 270} \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {\left (a+b x^2\right )^{7/2} (2 A b-9 a B)}{63 a^2 x^7}-\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9}-\frac {(2 A b-9 a B) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{9 a} \\ & = -\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9}+\frac {(2 A b-9 a B) \left (a+b x^2\right )^{7/2}}{63 a^2 x^7} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (-7 a A+2 A b x^2-9 a B x^2\right )}{63 a^2 x^9} \]
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Time = 2.91 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (\left (\frac {9 x^{2} B}{7}+A \right ) a -\frac {2 A b \,x^{2}}{7}\right )}{9 x^{9} a^{2}}\) | \(36\) |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-2 A b \,x^{2}+9 B a \,x^{2}+7 A a \right )}{63 x^{9} a^{2}}\) | \(37\) |
default | \(-\frac {B \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )\) | \(58\) |
trager | \(-\frac {\left (-2 A \,b^{4} x^{8}+9 B a \,b^{3} x^{8}+A a \,b^{3} x^{6}+27 B \,a^{2} b^{2} x^{6}+15 A \,a^{2} b^{2} x^{4}+27 B \,a^{3} b \,x^{4}+19 A \,a^{3} b \,x^{2}+9 B \,a^{4} x^{2}+7 A \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{63 x^{9} a^{2}}\) | \(106\) |
risch | \(-\frac {\left (-2 A \,b^{4} x^{8}+9 B a \,b^{3} x^{8}+A a \,b^{3} x^{6}+27 B \,a^{2} b^{2} x^{6}+15 A \,a^{2} b^{2} x^{4}+27 B \,a^{3} b \,x^{4}+19 A \,a^{3} b \,x^{2}+9 B \,a^{4} x^{2}+7 A \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{63 x^{9} a^{2}}\) | \(106\) |
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (45) = 90\).
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {{\left ({\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} + {\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{6} + 7 \, A a^{4} + 3 \, {\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{4} + {\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{63 \, a^{2} x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1489 vs. \(2 (46) = 92\).
Time = 4.10 (sec) , antiderivative size = 1489, normalized size of antiderivative = 28.09 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=\text {Too large to display} \]
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none
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{7 \, a x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{9 \, a x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (45) = 90\).
Time = 0.32 (sec) , antiderivative size = 456, normalized size of antiderivative = 8.60 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {2 \, {\left (63 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} B b^{\frac {7}{2}} - 126 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} B a b^{\frac {7}{2}} + 126 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} A b^{\frac {9}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a^{2} b^{\frac {7}{2}} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A a b^{\frac {9}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{3} b^{\frac {7}{2}} + 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a^{2} b^{\frac {9}{2}} + 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{4} b^{\frac {7}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{3} b^{\frac {9}{2}} - 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{5} b^{\frac {7}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{4} b^{\frac {9}{2}} + 198 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{6} b^{\frac {7}{2}} + 54 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{5} b^{\frac {9}{2}} - 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{7} b^{\frac {7}{2}} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{6} b^{\frac {9}{2}} + 9 \, B a^{8} b^{\frac {7}{2}} - 2 \, A a^{7} b^{\frac {9}{2}}\right )}}{63 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]
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Time = 7.06 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {2\,A\,b^4\,\sqrt {b\,x^2+a}}{63\,a^2\,x}-\frac {5\,A\,b^2\,\sqrt {b\,x^2+a}}{21\,x^5}-\frac {B\,a^2\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {3\,B\,b^2\,\sqrt {b\,x^2+a}}{7\,x^3}-\frac {A\,b^3\,\sqrt {b\,x^2+a}}{63\,a\,x^3}-\frac {A\,a^2\,\sqrt {b\,x^2+a}}{9\,x^9}-\frac {B\,b^3\,\sqrt {b\,x^2+a}}{7\,a\,x}-\frac {19\,A\,a\,b\,\sqrt {b\,x^2+a}}{63\,x^7}-\frac {3\,B\,a\,b\,\sqrt {b\,x^2+a}}{7\,x^5} \]
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