Integrand size = 19, antiderivative size = 98 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^3}{3 b^2}+\frac {d^3 x^5}{5 b}+\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {398, 211} \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3}{\sqrt {a} b^{7/2}}+\frac {d^2 x^3 (3 b c-a d)}{3 b^2}+\frac {d^3 x^5}{5 b} \]
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Rule 211
Rule 398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3}+\frac {d^2 (3 b c-a d) x^2}{b^2}+\frac {d^3 x^4}{b}+\frac {b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3}{b^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^3}{3 b^2}+\frac {d^3 x^5}{5 b}+\frac {(b c-a d)^3 \int \frac {1}{a+b x^2} \, dx}{b^3} \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^3}{3 b^2}+\frac {d^3 x^5}{5 b}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d x \left (15 a^2 d^2-5 a b d \left (9 c+d x^2\right )+3 b^2 \left (15 c^2+5 c d x^2+d^2 x^4\right )\right )}{15 b^3}+\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}} \]
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Time = 2.65 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {d \left (\frac {1}{5} b^{2} d^{2} x^{5}-\frac {1}{3} x^{3} a b \,d^{2}+x^{3} b^{2} c d +a^{2} d^{2} x -3 a b c d x +3 b^{2} c^{2} x \right )}{b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(116\) |
risch | \(\frac {d^{3} x^{5}}{5 b}-\frac {d^{3} x^{3} a}{3 b^{2}}+\frac {d^{2} x^{3} c}{b}+\frac {d^{3} a^{2} x}{b^{3}}-\frac {3 d^{2} a c x}{b^{2}}+\frac {3 d \,c^{2} x}{b}-\frac {\ln \left (b x -\sqrt {-a b}\right ) a^{3} d^{3}}{2 b^{3} \sqrt {-a b}}+\frac {3 \ln \left (b x -\sqrt {-a b}\right ) a^{2} c \,d^{2}}{2 b^{2} \sqrt {-a b}}-\frac {3 \ln \left (b x -\sqrt {-a b}\right ) a \,c^{2} d}{2 b \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c^{3}}{2 \sqrt {-a b}}+\frac {\ln \left (-b x -\sqrt {-a b}\right ) a^{3} d^{3}}{2 b^{3} \sqrt {-a b}}-\frac {3 \ln \left (-b x -\sqrt {-a b}\right ) a^{2} c \,d^{2}}{2 b^{2} \sqrt {-a b}}+\frac {3 \ln \left (-b x -\sqrt {-a b}\right ) a \,c^{2} d}{2 b \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c^{3}}{2 \sqrt {-a b}}\) | \(300\) |
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Time = 0.25 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.98 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=\left [\frac {6 \, a b^{3} d^{3} x^{5} + 10 \, {\left (3 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (3 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x}{30 \, a b^{4}}, \frac {3 \, a b^{3} d^{3} x^{5} + 5 \, {\left (3 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (3 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x}{15 \, a b^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (92) = 184\).
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.43 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{3} \left (- \frac {a d^{3}}{3 b^{2}} + \frac {c d^{2}}{b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \frac {\sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a b^{3} \sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3} \log {\left (\frac {a b^{3} \sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{5}}{5 b} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{2} d^{3} x^{5} + 5 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{3} + 15 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{15 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{4} d^{3} x^{5} + 15 \, b^{4} c d^{2} x^{3} - 5 \, a b^{3} d^{3} x^{3} + 45 \, b^{4} c^{2} d x - 45 \, a b^{3} c d^{2} x + 15 \, a^{2} b^{2} d^{3} x}{15 \, b^{5}} \]
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Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.49 \[ \int \frac {\left (c+d x^2\right )^3}{a+b x^2} \, dx=x\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,d^3}{3\,b^2}-\frac {c\,d^2}{b}\right )+\frac {d^3\,x^5}{5\,b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,b^{7/2}} \]
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