Integrand size = 22, antiderivative size = 77 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^3}{a x}+\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^3}{3 b}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {472, 211} \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3}{a^{3/2} b^{5/2}}+\frac {d^2 x (3 b c-a d)}{b^2}-\frac {c^3}{a x}+\frac {d^3 x^3}{3 b} \]
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Rule 211
Rule 472
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 (3 b c-a d)}{b^2}+\frac {c^3}{a x^2}+\frac {d^3 x^2}{b}+\frac {(-b c+a d)^3}{a b^2 \left (a+b x^2\right )}\right ) \, dx \\ & = -\frac {c^3}{a x}+\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^3}{3 b}-\frac {(b c-a d)^3 \int \frac {1}{a+b x^2} \, dx}{a b^2} \\ & = -\frac {c^3}{a x}+\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^3}{3 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^3}{a x}+\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^3}{3 b}+\frac {(-b c+a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}} \]
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Time = 2.70 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23
method | result | size |
default | \(-\frac {d^{2} \left (-\frac {1}{3} b d \,x^{3}+a d x -3 b c x \right )}{b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \,b^{2} \sqrt {a b}}-\frac {c^{3}}{a x}\) | \(95\) |
risch | \(\frac {d^{3} x^{3}}{3 b}-\frac {d^{3} a x}{b^{2}}+\frac {3 d^{2} c x}{b}-\frac {c^{3}}{a x}-\frac {a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) d^{3}}{2 b^{2} \sqrt {-a b}}+\frac {3 a \ln \left (-\sqrt {-a b}\, x +a \right ) c \,d^{2}}{2 b \sqrt {-a b}}-\frac {3 \ln \left (-\sqrt {-a b}\, x +a \right ) c^{2} d}{2 \sqrt {-a b}}+\frac {b \ln \left (-\sqrt {-a b}\, x +a \right ) c^{3}}{2 \sqrt {-a b}\, a}+\frac {a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) d^{3}}{2 b^{2} \sqrt {-a b}}-\frac {3 a \ln \left (-\sqrt {-a b}\, x -a \right ) c \,d^{2}}{2 b \sqrt {-a b}}+\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) c^{2} d}{2 \sqrt {-a b}}-\frac {b \ln \left (-\sqrt {-a b}\, x -a \right ) c^{3}}{2 \sqrt {-a b}\, a}\) | \(268\) |
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Time = 0.25 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.29 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=\left [\frac {2 \, a^{2} b^{2} d^{3} x^{4} - 6 \, a b^{3} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} x \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{6 \, a^{2} b^{3} x}, \frac {a^{2} b^{2} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{3 \, a^{2} b^{3} x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (65) = 130\).
Time = 0.44 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.87 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=x \left (- \frac {a d^{3}}{b^{2}} + \frac {3 c d^{2}}{b}\right ) - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log {\left (\frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{3}}{3 b} - \frac {c^{3}}{a x} \]
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Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^{3}}{a x} + \frac {b d^{3} x^{3} + 3 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x}{3 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^{3}}{a x} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{2}} + \frac {b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x - 3 \, a b d^{3} x}{3 \, b^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.53 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=\frac {d^3\,x^3}{3\,b}-\frac {c^3}{a\,x}-x\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{3/2}\,b^{5/2}} \]
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