Integrand size = 22, antiderivative size = 146 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {16 b^2 (8 A b-5 a B) x}{15 a^5 \sqrt {a+b x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {464, 277, 198, 197} \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt {a+b x^2}}-\frac {8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}-\frac {(8 A b-5 a B) \int \frac {1}{x^4 \left (a+b x^2\right )^{5/2}} \, dx}{5 a} \\ & = -\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 b (8 A b-5 a B)) \int \frac {1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{5 a^2} \\ & = -\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {\left (8 b^2 (8 A b-5 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{5 a^3} \\ & = -\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 b^2 (8 A b-5 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^4} \\ & = -\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {16 b^2 (8 A b-5 a B) x}{15 a^5 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-128 A b^4 x^8+16 a b^3 x^6 \left (-12 A+5 B x^2\right )+24 a^2 b^2 x^4 \left (-2 A+5 B x^2\right )-a^4 \left (3 A+5 B x^2\right )+a^3 \left (8 A b x^2+30 b B x^4\right )}{15 a^5 x^5 \left (a+b x^2\right )^{3/2}} \]
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Time = 2.90 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {\left (-5 x^{2} B -3 A \right ) a^{4}+8 x^{2} b \left (\frac {15 x^{2} B}{4}+A \right ) a^{3}-48 x^{4} \left (-\frac {5 x^{2} B}{2}+A \right ) b^{2} a^{2}-192 x^{6} \left (-\frac {5 x^{2} B}{12}+A \right ) b^{3} a -128 A \,b^{4} x^{8}}{15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} x^{5} a^{5}}\) | \(95\) |
gosper | \(-\frac {128 A \,b^{4} x^{8}-80 B a \,b^{3} x^{8}+192 A a \,b^{3} x^{6}-120 B \,a^{2} b^{2} x^{6}+48 A \,a^{2} b^{2} x^{4}-30 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+5 B \,a^{4} x^{2}+3 A \,a^{4}}{15 x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{5}}\) | \(107\) |
trager | \(-\frac {128 A \,b^{4} x^{8}-80 B a \,b^{3} x^{8}+192 A a \,b^{3} x^{6}-120 B \,a^{2} b^{2} x^{6}+48 A \,a^{2} b^{2} x^{4}-30 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+5 B \,a^{4} x^{2}+3 A \,a^{4}}{15 x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{5}}\) | \(107\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (73 A \,b^{2} x^{4}-40 B a b \,x^{4}-14 a A b \,x^{2}+5 a^{2} B \,x^{2}+3 a^{2} A \right )}{15 a^{5} x^{5}}-\frac {\sqrt {b \,x^{2}+a}\, x \left (11 A \,b^{2} x^{2}-8 B a b \,x^{2}+12 a b A -9 a^{2} B \right ) b^{2}}{3 a^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(127\) |
default | \(B \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )}{a}\right )+A \left (-\frac {1}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {8 b \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )}{a}\right )}{5 a}\right )\) | \(188\) |
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Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (16 \, {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 24 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} - 3 \, A a^{4} + 6 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (141) = 282\).
Time = 15.02 (sec) , antiderivative size = 944, normalized size of antiderivative = 6.47 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {3 a^{6} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} + \frac {2 a^{5} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {35 a^{4} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {280 a^{3} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {560 a^{2} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {448 a b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {128 b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}}\right ) + B \left (- \frac {a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {5 a^{3} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {30 a^{2} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {40 a b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {16 b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {16 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, B b^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {128 \, A b^{3} x}{15 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b^{3} x}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} + \frac {2 \, B b}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x} - \frac {16 \, A b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} x} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}} + \frac {8 \, A b}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x^{3}} - \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (126) = 252\).
Time = 0.32 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.30 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {{\left (8 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}\right )} x^{2}}{a^{9} b} + \frac {3 \, {\left (3 \, B a^{6} b^{3} - 4 \, A a^{5} b^{4}\right )}}{a^{9} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} - 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} + 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 250 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} - 490 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 170 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} + 320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 40 \, B a^{5} b^{\frac {3}{2}} - 73 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{4}} \]
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Time = 5.48 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.58 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {a\,\left (\frac {b^2\,\left (73\,A\,b-40\,B\,a\right )}{18\,a^4}+\frac {b^2\,\left (86\,A\,b-35\,B\,a\right )}{30\,a^4}+\frac {a\,\left (\frac {28\,A\,b^4-10\,B\,a\,b^3}{45\,a^5}-\frac {b^3\,\left (86\,A\,b-35\,B\,a\right )}{18\,a^5}\right )}{b}\right )}{b}-\frac {b\,\left (73\,A\,b-40\,B\,a\right )}{30\,a^3}}{x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x^2\,\left (\frac {28\,A\,b^3-10\,B\,a\,b^2}{15\,a^5}-\frac {2\,b^2\,\left (26\,A\,b-15\,B\,a\right )}{5\,a^5}\right )-\frac {b\,\left (26\,A\,b-15\,B\,a\right )}{5\,a^4}}{x\,\sqrt {b\,x^2+a}}-\frac {\sqrt {b\,x^2+a}\,\left (5\,B\,a^3-14\,A\,a^2\,b\right )}{15\,a^6\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{5\,a^3\,x^5} \]
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