Integrand size = 41, antiderivative size = 139 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}}-\frac {(2 c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}} \]
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Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1163, 427, 537, 223, 212, 385, 214} \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {(2 c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (5 c d-2 b e)}{2 c^2 \sqrt {e}}+\frac {x \sqrt {d+e x^2}}{2 c} \]
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Rule 212
Rule 214
Rule 223
Rule 385
Rule 427
Rule 537
Rule 1163
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (d+e x^2\right )^{3/2}}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx \\ & = \frac {x \sqrt {d+e x^2}}{2 c}+\frac {\int \frac {d e (3 c d-b e)+e^2 (5 c d-2 b e) x^2}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{2 c e} \\ & = \frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{2 c^2}+\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{c^2} \\ & = \frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{2 c^2}+\frac {(2 c d-b e)^2 \text {Subst}\left (\int \frac {1}{\frac {-c d^2+b d e}{d}-\left (-c d e+\frac {e \left (-c d^2+b d e\right )}{d}\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c^2} \\ & = \frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}}-\frac {(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {-c x \sqrt {d+e x^2}+\frac {2 (2 c d-b e) \sqrt {2 c^2 d^2-3 b c d e+b^2 e^2} \text {arctanh}\left (\frac {-b e+c \left (d-e x^2+\sqrt {e} x \sqrt {d+e x^2}\right )}{\sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}}\right )}{\sqrt {e} (c d-b e)}+\frac {(5 c d-2 b e) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{\sqrt {e}}}{2 c^2} \]
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Time = 1.00 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(-\frac {-\frac {2 \left (b e -2 c d \right )^{2} \operatorname {arctanh}\left (\frac {\left (b e -c d \right ) \sqrt {e \,x^{2}+d}}{x \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}\right )}{\sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}-\frac {\sqrt {e \,x^{2}+d}\, c x \sqrt {e}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) b e +5 \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) c d}{\sqrt {e}}}{2 c^{2}}\) | \(144\) |
risch | \(\frac {x \sqrt {e \,x^{2}+d}}{2 c}-\frac {\frac {\left (2 b e -5 c d \right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{c \sqrt {e}}-\frac {\left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}-\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )^{2} e -\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}-\frac {b e -2 c d}{c}}}{x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}}\right )}{\sqrt {-\left (b e -c d \right ) e c}\, c \sqrt {-\frac {b e -2 c d}{c}}}+\frac {\left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}+\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )^{2} e +\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}-\frac {b e -2 c d}{c}}}{x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}}\right )}{\sqrt {-\left (b e -c d \right ) e c}\, c \sqrt {-\frac {b e -2 c d}{c}}}}{2 c}\) | \(540\) |
default | \(\text {Expression too large to display}\) | \(3792\) |
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Time = 0.54 (sec) , antiderivative size = 1079, normalized size of antiderivative = 7.76 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\left [\frac {2 \, \sqrt {e x^{2} + d} c e x - {\left (5 \, c d - 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, c d e - b e^{2}\right )} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, {\left ({\left (3 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, c^{2} e}, \frac {2 \, \sqrt {e x^{2} + d} c e x - 2 \, {\left (5 \, c d - 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (2 \, c d e - b e^{2}\right )} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, {\left ({\left (3 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, c^{2} e}, \frac {2 \, \sqrt {e x^{2} + d} c e x + 2 \, {\left (2 \, c d e - b e^{2}\right )} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}} \arctan \left (\frac {{\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}}}{2 \, {\left ({\left (2 \, c d e - b e^{2}\right )} x^{3} + {\left (2 \, c d^{2} - b d e\right )} x\right )}}\right ) - {\left (5 \, c d - 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right )}{4 \, c^{2} e}, \frac {\sqrt {e x^{2} + d} c e x - {\left (5 \, c d - 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, c d e - b e^{2}\right )} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}} \arctan \left (\frac {{\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}}}{2 \, {\left ({\left (2 \, c d e - b e^{2}\right )} x^{3} + {\left (2 \, c d^{2} - b d e\right )} x\right )}}\right )}{2 \, c^{2} e}\right ] \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2} \,d x \]
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