Integrand size = 22, antiderivative size = 77 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A}{a x \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-a B) x}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac {2 (4 A b-a B) x}{3 a^3 \sqrt {a+b x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 77, 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, 198, 197} \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 x (4 A b-a B)}{3 a^3 \sqrt {a+b x^2}}-\frac {x (4 A b-a B)}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{a x \left (a+b x^2\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{a x \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-a B) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{a} \\ & = -\frac {A}{a x \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-a B) x}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac {(2 (4 A b-a B)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a^2} \\ & = -\frac {A}{a x \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-a B) x}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac {2 (4 A b-a B) x}{3 a^3 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-3 a^2 A-12 a A b x^2+3 a^2 B x^2-8 A b^2 x^4+2 a b B x^4}{3 a^3 x \left (a+b x^2\right )^{3/2}} \]
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Time = 2.87 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {\left (-x^{2} B +A \right ) a^{2}+4 x^{2} b \left (-\frac {x^{2} B}{6}+A \right ) a +\frac {8 A \,b^{2} x^{4}}{3}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} x \,a^{3}}\) | \(55\) |
gosper | \(-\frac {8 A \,b^{2} x^{4}-2 B a b \,x^{4}+12 a A b \,x^{2}-3 a^{2} B \,x^{2}+3 a^{2} A}{3 x \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}\) | \(59\) |
trager | \(-\frac {8 A \,b^{2} x^{4}-2 B a b \,x^{4}+12 a A b \,x^{2}-3 a^{2} B \,x^{2}+3 a^{2} A}{3 x \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}\) | \(59\) |
risch | \(-\frac {A \sqrt {b \,x^{2}+a}}{a^{3} x}-\frac {\sqrt {b \,x^{2}+a}\, x \left (5 A \,b^{2} x^{2}-2 B a b \,x^{2}+6 a b A -3 a^{2} B \right )}{3 a^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(84\) |
default | \(B \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+A \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )\) | \(92\) |
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (2 \, {\left (B a b - 4 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} + 3 \, {\left (B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (71) = 142\).
Time = 6.68 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.44 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {3 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {12 a b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {8 b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) + B \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {2 \, B x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {B x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {8 \, A b x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {4 \, A b x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a x} \]
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Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.31 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {{\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{2}}{a^{5} b} + \frac {3 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )}}{a^{5} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{2}} \]
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Time = 5.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {A\,a^2-8\,A\,{\left (b\,x^2+a\right )}^2+B\,a^2\,x^2+4\,A\,a\,\left (b\,x^2+a\right )+2\,B\,a\,x^2\,\left (b\,x^2+a\right )}{3\,a^3\,x\,{\left (b\,x^2+a\right )}^{3/2}} \]
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