Integrand size = 24, antiderivative size = 101 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) x}{3 d^2 e^2 \sqrt {d+e x^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1171, 393, 223, 212} \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {x \left (4 c d^2-e (2 a e+b d)\right )}{3 d^2 e^2 \sqrt {d+e x^2}}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{5/2}} \]
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Rule 212
Rule 223
Rule 393
Rule 1171
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^2\right )^{3/2}}-\frac {\int \frac {-2 a+\frac {d (c d-b e)}{e^2}-\frac {3 c d x^2}{e}}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d} \\ & = \frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^2\right )^{3/2}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) x}{3 d^2 e^2 \sqrt {d+e x^2}}+\frac {c \int \frac {1}{\sqrt {d+e x^2}} \, dx}{e^2} \\ & = \frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^2\right )^{3/2}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) x}{3 d^2 e^2 \sqrt {d+e x^2}}+\frac {c \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{e^2} \\ & = \frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^2\right )^{3/2}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) x}{3 d^2 e^2 \sqrt {d+e x^2}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{5/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-c d^2 x \left (3 d+4 e x^2\right )+e^2 x \left (3 a d+b d x^2+2 a e x^2\right )}{3 d^2 e^2 \left (d+e x^2\right )^{3/2}}-\frac {c \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{e^{5/2}} \]
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Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} c \,d^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+x \left (d \left (\frac {b \,x^{2}}{3}+a \right ) e^{\frac {5}{2}}-\frac {4 c \,d^{2} e^{\frac {3}{2}} x^{2}}{3}-c \,d^{3} \sqrt {e}+\frac {2 a \,e^{\frac {7}{2}} x^{2}}{3}\right )}{e^{\frac {5}{2}} \left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{2}}\) | \(95\) |
default | \(a \left (\frac {x}{3 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {e \,x^{2}+d}}\right )+c \left (-\frac {x^{3}}{3 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {-\frac {x}{e \sqrt {e \,x^{2}+d}}+\frac {\ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{e^{\frac {3}{2}}}}{e}\right )+b \left (-\frac {x}{2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {d \left (\frac {x}{3 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {e \,x^{2}+d}}\right )}{2 e}\right )\) | \(150\) |
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Time = 0.31 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.86 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (c d^{2} e^{2} x^{4} + 2 \, c d^{3} e x^{2} + c d^{4}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, {\left ({\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{3} + 3 \, {\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{6 \, {\left (d^{2} e^{5} x^{4} + 2 \, d^{3} e^{4} x^{2} + d^{4} e^{3}\right )}}, -\frac {3 \, {\left (c d^{2} e^{2} x^{4} + 2 \, c d^{3} e x^{2} + c d^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left ({\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{3} + 3 \, {\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{5} x^{4} + 2 \, d^{3} e^{4} x^{2} + d^{4} e^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (94) = 188\).
Time = 6.51 (sec) , antiderivative size = 450, normalized size of antiderivative = 4.46 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=a \left (\frac {3 d x}{3 d^{\frac {7}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {5}{2}} e x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {2 e x^{3}}{3 d^{\frac {7}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {5}{2}} e x^{2} \sqrt {1 + \frac {e x^{2}}{d}}}\right ) + \frac {b x^{3}}{3 d^{\frac {5}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {3}{2}} e x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} + c \left (\frac {3 d^{\frac {39}{2}} e^{11} \sqrt {1 + \frac {e x^{2}}{d}} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 d^{\frac {37}{2}} e^{12} x^{2} \sqrt {1 + \frac {e x^{2}}{d}} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {3 d^{19} e^{\frac {23}{2}} x}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {4 d^{18} e^{\frac {25}{2}} x^{3}}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}}\right ) \]
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Exception generated. \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {x {\left (\frac {{\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{2}}{d^{2} e^{3}} + \frac {3 \, {\left (c d^{3} e - a d e^{3}\right )}}{d^{2} e^{3}}\right )}}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} - \frac {c \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{e^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {c\,x^4+b\,x^2+a}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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