Optimal. Leaf size=106 \[ -\frac {x}{d \sqrt {d+e x^2} (2 c d-b e)}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1149, 382, 377, 208} \begin {gather*} -\frac {x}{d \sqrt {d+e x^2} (2 c d-b e)}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 377
Rule 382
Rule 1149
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d+e x^2\right )^{3/2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx\\ &=-\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}+\frac {c \int \frac {1}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{2 c d-b e}\\ &=-\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}+\frac {c \operatorname {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 )}{2 c d-b e}\\ &=-\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.05, size = 418, normalized size = 3.94 \begin {gather*} -\frac {x \left (-\frac {2 c e x^2 \left (\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )}{c d-b e}+2 \left (\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )+\frac {10 c e x^2 \sqrt {\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}}}{c d-b e}-15 \sqrt {\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}}-\frac {10 c e x^2 \tanh ^{-1}\left (\sqrt {\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}}\right )}{c d-b e}+15 \tanh ^{-1}\left (\sqrt {\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}}\right )\right )}{5 \left (d+e x^2\right )^{3/2} (c d-b e) \left (\frac {e x^2 (b e-2 c d)}{\left (d+e x^2\right ) (b e-c d)}\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [B] time = 0.37, size = 221, normalized size = 2.08 \begin {gather*} \frac {c \sqrt {b^2 e^2-3 b c d e+2 c^2 d^2} \tanh ^{-1}\left (-\frac {c e x^2}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}+\frac {c \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}+\frac {c d}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}-\frac {b e}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}\right )}{\sqrt {e} (b e-2 c d)^2 (b e-c d)}-\frac {x}{d \sqrt {d+e x^2} (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.27, size = 701, normalized size = 6.61 \begin {gather*} \left [-\frac {4 \, {\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {e x^{2} + d} x + \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left (c d e x^{2} + c d^{2}\right )} \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 \, \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left ({\left (3 \, c d e - 2 \, b e^{2}\right )} x^{3} + {\left (c d^{2} - b d e\right )} x\right )} \sqrt {e x^{2} + d}}{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 \, {\left (4 \, c^{3} d^{5} e - 8 \, b c^{2} d^{4} e^{2} + 5 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4} + {\left (4 \, c^{3} d^{4} e^{2} - 8 \, b c^{2} d^{3} e^{3} + 5 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2}\right )}}, -\frac {2 \, {\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {e x^{2} + d} x + \sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\left (c d e x^{2} + c d^{2}\right )} \arctan \left (-\frac {\sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\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}}{2 \, {\left ({\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )}}\right )}{2 \, {\left (4 \, c^{3} d^{5} e - 8 \, b c^{2} d^{4} e^{2} + 5 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4} + {\left (4 \, c^{3} d^{4} e^{2} - 8 \, b c^{2} d^{3} e^{3} + 5 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 771, normalized size = 7.27 \begin {gather*} -\frac {c^{2} e \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}+\frac {2 \sqrt {-\left (b e -c d \right ) c e}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}\right )^{2} e +\frac {2 \sqrt {-\left (b e -c d \right ) c e}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}\right )}{c}-\frac {b e -2 c d}{c}}}{x -\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}}\right )}{2 \left (\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \left (-\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \sqrt {-\left (b e -c d \right ) c e}\, \sqrt {-\frac {b e -2 c d}{c}}}+\frac {c^{2} e \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}-\frac {2 \sqrt {-\left (b e -c d \right ) c e}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}\right )^{2} e -\frac {2 \sqrt {-\left (b e -c d \right ) c e}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}\right )}{c}-\frac {b e -2 c d}{c}}}{x +\frac {\sqrt {-\left (b e -c d \right ) c e}}{c e}}\right )}{2 \left (\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \left (-\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \sqrt {-\left (b e -c d \right ) c e}\, \sqrt {-\frac {b e -2 c d}{c}}}-\frac {\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\, c}{2 \left (\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \left (-\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right ) d}-\frac {\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\, c}{2 \left (\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \left (-\sqrt {-d e}\, c +\sqrt {-\left (b e -c d \right ) c e}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right ) d} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e\right )} \sqrt {e x^{2} + d}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {e\,x^2+d}\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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