Integrand size = 22, antiderivative size = 74 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=-\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {d^3 x}{b}+\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} b^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 74, 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} \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3}{a^{5/2} b^{3/2}}+\frac {c^2 (b c-3 a d)}{a^2 x}-\frac {c^3}{3 a x^3}+\frac {d^3 x}{b} \]
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Rule 211
Rule 472
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3}{b}+\frac {c^3}{a x^4}+\frac {c^2 (-b c+3 a d)}{a^2 x^2}-\frac {(-b c+a d)^3}{a^2 b \left (a+b x^2\right )}\right ) \, dx \\ & = -\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {d^3 x}{b}+\frac {(b c-a d)^3 \int \frac {1}{a+b x^2} \, dx}{a^2 b} \\ & = -\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {d^3 x}{b}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} b^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=-\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {d^3 x}{b}+\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} b^{3/2}} \]
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Time = 2.71 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32
method | result | size |
default | \(\frac {d^{3} x}{b}-\frac {c^{3}}{3 a \,x^{3}}-\frac {c^{2} \left (3 a d -b c \right )}{x \,a^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{2} b \sqrt {a b}}\) | \(98\) |
risch | \(\frac {d^{3} x}{b}+\frac {-\frac {b \,c^{2} \left (3 a d -b c \right ) x^{2}}{a^{2}}-\frac {c^{3} b}{3 a}}{b \,x^{3}}-\frac {a \ln \left (-\sqrt {-a b}\, x -a \right ) d^{3}}{2 b \sqrt {-a b}}+\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) c \,d^{2}}{2 \sqrt {-a b}}-\frac {3 b \ln \left (-\sqrt {-a b}\, x -a \right ) c^{2} d}{2 \sqrt {-a b}\, a}+\frac {b^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) c^{3}}{2 \sqrt {-a b}\, a^{2}}+\frac {a \ln \left (-\sqrt {-a b}\, x +a \right ) d^{3}}{2 b \sqrt {-a b}}-\frac {3 \ln \left (-\sqrt {-a b}\, x +a \right ) c \,d^{2}}{2 \sqrt {-a b}}+\frac {3 b \ln \left (-\sqrt {-a b}\, x +a \right ) c^{2} d}{2 \sqrt {-a b}\, a}-\frac {b^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) c^{3}}{2 \sqrt {-a b}\, a^{2}}\) | \(272\) |
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Time = 0.26 (sec) , antiderivative size = 256, normalized size of antiderivative = 3.46 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=\left [\frac {6 \, a^{3} b d^{3} x^{4} - 2 \, a^{2} b^{2} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} x^{3} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{6 \, a^{3} b^{2} x^{3}}, \frac {3 \, a^{3} b d^{3} x^{4} - a^{2} b^{2} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} x^{3} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{3 \, a^{3} b^{2} x^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (65) = 130\).
Time = 0.63 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.99 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log {\left (\frac {a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x}{b} + \frac {- a c^{3} + x^{2} \left (- 9 a c^{2} d + 3 b c^{3}\right )}{3 a^{2} x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=\frac {d^{3} x}{b} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} b} - \frac {a c^{3} - 3 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x^{2}}{3 \, a^{2} x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=\frac {d^{3} x}{b} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} b} + \frac {3 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{2} x^{3}} \]
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Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.65 \[ \int \frac {\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx=\frac {d^3\,x}{b}-\frac {\frac {b\,c^3}{3\,a}+\frac {b\,c^2\,x^2\,\left (3\,a\,d-b\,c\right )}{a^2}}{b\,x^3}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{5/2}\,b^{3/2}} \]
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