Optimal. Leaf size=80 \[ \frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1150, 414, 527, 12, 377, 208} \[ \frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 377
Rule 414
Rule 527
Rule 1150
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {\int \frac {-5 d e+2 e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{6 d^2 e}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\int \frac {3 d^2 e^2}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{12 d^4 e^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{4 d^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{d-2 d e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{4 d^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}\\ \end {align*}
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Mathematica [C] time = 3.34, size = 345, normalized size = 4.31 \[ \frac {\frac {384 e^4 x^8 \left (d+e x^2\right )^2 \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {2 e x^2}{d-e x^2}\right )}{e x^2-d}+\frac {384 e^4 x^8 \left (4 d^2+7 d e x^2+3 e^2 x^4\right ) \, _2F_1\left (2,2;\frac {9}{2};-\frac {2 e x^2}{d-e x^2}\right )}{e x^2-d}+\frac {35 \sqrt {2} \sqrt {\frac {e x^2}{e x^2-d}} \left (-15 d^3-5 d^2 e x^2+12 d e^2 x^4+8 e^3 x^6\right ) \left (\sqrt {2} \sqrt {\frac {e x^2}{e x^2-d}} \sqrt {\frac {d+e x^2}{d-e x^2}} \left (-3 d^2-2 d e x^2+5 e^2 x^4\right )+3 \left (d+e x^2\right )^2 \sin ^{-1}\left (\sqrt {2} \sqrt {\frac {e x^2}{e x^2-d}}\right )\right )}{\sqrt {\frac {d+e x^2}{d-e x^2}}}}{2520 d^5 e^3 x^5 \sqrt {d+e x^2} \left (1-\frac {e^2 x^4}{d^2}\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.77, size = 279, normalized size = 3.49 \[ \left [\frac {3 \, \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt {2} {\left (3 \, e x^{3} + d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 8 \, {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{96 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}, -\frac {3 \, \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{4 \, {\left (e^{2} x^{3} + d e x\right )}}\right ) - 4 \, {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{48 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 911, normalized size = 11.39 \[ -\frac {\sqrt {2}\, e \ln \left (\frac {4 d +2 \sqrt {2}\, \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, \sqrt {d}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}{x -\frac {\sqrt {d e}}{e}}\right )}{8 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) d^{\frac {3}{2}}}+\frac {\sqrt {2}\, e \ln \left (\frac {4 d +2 \sqrt {2}\, \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, \sqrt {d}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}{x +\frac {\sqrt {d e}}{e}}\right )}{8 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) d^{\frac {3}{2}}}+\frac {e}{4 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, d}-\frac {e}{4 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, d}-\frac {e x}{3 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\, d^{2}}-\frac {e x}{4 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, d^{2}}-\frac {e x}{3 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\, d^{2}}-\frac {e x}{4 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, d^{2}}-\frac {1}{6 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right ) \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\, d}-\frac {1}{6 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right ) \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (e^{2} x^{4} - d^{2}\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (d^2-e^2\,x^4\right )\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{- d^{3} \sqrt {d + e x^{2}} - d^{2} e x^{2} \sqrt {d + e x^{2}} + d e^{2} x^{4} \sqrt {d + e x^{2}} + e^{3} x^{6} \sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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