Integrand size = 22, antiderivative size = 108 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac {4 b (2 A b-a B) x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {8 b (2 A b-a B) x}{3 a^4 \sqrt {a+b x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {464, 277, 198, 197} \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=\frac {8 b x (2 A b-a B)}{3 a^4 \sqrt {a+b x^2}}+\frac {4 b x (2 A b-a B)}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}-\frac {(6 A b-3 a B) \int \frac {1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{3 a} \\ & = -\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac {(4 b (2 A b-a B)) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{a^2} \\ & = -\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac {4 b (2 A b-a B) x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {(8 b (2 A b-a B)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a^3} \\ & = -\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac {4 b (2 A b-a B) x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {8 b (2 A b-a B) x}{3 a^4 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=\frac {16 A b^3 x^6+6 a^2 b x^2 \left (A-2 B x^2\right )-8 a b^2 x^4 \left (-3 A+B x^2\right )-a^3 \left (A+3 B x^2\right )}{3 a^4 x^3 \left (a+b x^2\right )^{3/2}} \]
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Time = 2.92 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\frac {\left (3 x^{2} B +A \right ) a^{3}-6 b \,x^{2} \left (-2 x^{2} B +A \right ) a^{2}-24 x^{4} b^{2} \left (-\frac {x^{2} B}{3}+A \right ) a -16 x^{6} b^{3} A}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} x^{3} a^{4}}\) | \(74\) |
gosper | \(-\frac {-16 x^{6} b^{3} A +8 x^{6} a \,b^{2} B -24 A a \,b^{2} x^{4}+12 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+3 B \,a^{3} x^{2}+a^{3} A}{3 x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4}}\) | \(82\) |
trager | \(-\frac {-16 x^{6} b^{3} A +8 x^{6} a \,b^{2} B -24 A a \,b^{2} x^{4}+12 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+3 B \,a^{3} x^{2}+a^{3} A}{3 x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4}}\) | \(82\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-8 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{4} x^{3}}+\frac {\sqrt {b \,x^{2}+a}\, x \left (8 A \,b^{2} x^{2}-5 B a b \,x^{2}+9 a b A -6 a^{2} B \right ) b}{3 a^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(102\) |
default | \(A \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )}{a}\right )+B \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )\) | \(140\) |
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Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left (8 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 12 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{4} + A a^{3} + 3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (99) = 198\).
Time = 10.06 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.85 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {5 a^{3} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {30 a^{2} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {40 a b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {16 b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}}\right ) + B \left (- \frac {3 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {12 a b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {8 b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {8 \, B b x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {4 \, B b x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {16 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A b^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {B}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a x} + \frac {2 \, A b}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (92) = 184\).
Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {x {\left (\frac {{\left (5 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{2}}{a^{7} b} + \frac {3 \, {\left (2 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3}\right )}}{a^{7} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} + 3 \, B a^{3} \sqrt {b} - 8 \, A a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{3}} \]
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Time = 5.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {16\,A\,{\left (b\,x^2+a\right )}^3+A\,a^3+B\,a^3\,x^2-24\,A\,a\,{\left (b\,x^2+a\right )}^2+6\,A\,a^2\,\left (b\,x^2+a\right )-8\,B\,a\,x^2\,{\left (b\,x^2+a\right )}^2+4\,B\,a^2\,x^2\,\left (b\,x^2+a\right )}{{\left (b\,x^2+a\right )}^{3/2}\,\left (\frac {3\,a^5\,x}{b}-\frac {3\,a^4\,x\,\left (b\,x^2+a\right )}{b}\right )} \]
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