Integrand size = 22, antiderivative size = 117 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac {4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac {8 b^2 (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{315 a^4 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {464, 277, 270} \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac {4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac {\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9} \]
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Rule 270
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}-\frac {(6 A b-9 a B) \int \frac {\sqrt {a+b x^2}}{x^8} \, dx}{9 a} \\ & = -\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}+\frac {(4 b (2 A b-3 a B)) \int \frac {\sqrt {a+b x^2}}{x^6} \, dx}{21 a^2} \\ & = -\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac {4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}-\frac {\left (8 b^2 (2 A b-3 a B)\right ) \int \frac {\sqrt {a+b x^2}}{x^4} \, dx}{105 a^3} \\ & = -\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac {4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac {8 b^2 (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{315 a^4 x^3} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {\left (a+b x^2\right )^{3/2} \left (16 A b^3 x^6-24 a b^2 x^4 \left (A+B x^2\right )+6 a^2 b x^2 \left (5 A+6 B x^2\right )-5 a^3 \left (7 A+9 B x^2\right )\right )}{315 a^4 x^9} \]
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Time = 2.87 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(-\frac {\left (\left (\frac {9 x^{2} B}{7}+A \right ) a^{3}-\frac {6 x^{2} \left (\frac {6 x^{2} B}{5}+A \right ) b \,a^{2}}{7}+\frac {24 b^{2} x^{4} \left (x^{2} B +A \right ) a}{35}-\frac {16 x^{6} b^{3} A}{35}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 x^{9} a^{4}}\) | \(73\) |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-16 x^{6} b^{3} A +24 x^{6} a \,b^{2} B +24 A a \,b^{2} x^{4}-36 B \,a^{2} b \,x^{4}-30 A \,a^{2} b \,x^{2}+45 B \,a^{3} x^{2}+35 a^{3} A \right )}{315 x^{9} a^{4}}\) | \(83\) |
trager | \(-\frac {\left (-16 A \,b^{4} x^{8}+24 B a \,b^{3} x^{8}+8 A a \,b^{3} x^{6}-12 B \,a^{2} b^{2} x^{6}-6 A \,a^{2} b^{2} x^{4}+9 B \,a^{3} b \,x^{4}+5 A \,a^{3} b \,x^{2}+45 B \,a^{4} x^{2}+35 A \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{315 x^{9} a^{4}}\) | \(107\) |
risch | \(-\frac {\left (-16 A \,b^{4} x^{8}+24 B a \,b^{3} x^{8}+8 A a \,b^{3} x^{6}-12 B \,a^{2} b^{2} x^{6}-6 A \,a^{2} b^{2} x^{4}+9 B \,a^{3} b \,x^{4}+5 A \,a^{3} b \,x^{2}+45 B \,a^{4} x^{2}+35 A \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{315 x^{9} a^{4}}\) | \(107\) |
default | \(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{9}}-\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )\) | \(150\) |
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {{\left (8 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} - 4 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + 35 \, A a^{4} + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4} + 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, a^{4} x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (112) = 224\).
Time = 1.92 (sec) , antiderivative size = 957, normalized size of antiderivative = 8.18 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=- \frac {35 A a^{7} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {110 A a^{6} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {114 A a^{5} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {40 A a^{4} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {5 A a^{3} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {30 A a^{2} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {40 A a b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {16 A b^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {15 B a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {33 B a^{4} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {17 B a^{3} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {3 B a^{2} b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {12 B a b^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {8 B b^{\frac {19}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} \]
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Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{105 \, a^{3} x^{3}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{315 \, a^{4} x^{3}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{105 \, a^{3} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{7 \, a x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{21 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{9 \, a x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (101) = 202\).
Time = 0.30 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.94 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {16 \, {\left (210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B b^{\frac {7}{2}} - 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a b^{\frac {7}{2}} + 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A b^{\frac {9}{2}} + 63 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{2} b^{\frac {7}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a b^{\frac {9}{2}} - 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{3} b^{\frac {7}{2}} + 168 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{2} b^{\frac {9}{2}} + 108 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{4} b^{\frac {7}{2}} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{3} b^{\frac {9}{2}} - 27 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{5} b^{\frac {7}{2}} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{4} b^{\frac {9}{2}} + 3 \, B a^{6} b^{\frac {7}{2}} - 2 \, A a^{5} b^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]
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Time = 5.79 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx=\frac {2\,A\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^5}-\frac {B\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {A\,b\,\sqrt {b\,x^2+a}}{63\,a\,x^7}-\frac {B\,b\,\sqrt {b\,x^2+a}}{35\,a\,x^5}-\frac {A\,\sqrt {b\,x^2+a}}{9\,x^9}-\frac {8\,A\,b^3\,\sqrt {b\,x^2+a}}{315\,a^3\,x^3}+\frac {16\,A\,b^4\,\sqrt {b\,x^2+a}}{315\,a^4\,x}+\frac {4\,B\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^3}-\frac {8\,B\,b^3\,\sqrt {b\,x^2+a}}{105\,a^3\,x} \]
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