Optimal. Leaf size=93 \[ -\frac {a \tanh ^{-1}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{d}-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (b e^2+c d^2\right )}{e^4}+\frac {c (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.16, antiderivative size = 151, normalized size of antiderivative = 1.62, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {520, 1251, 897, 1153, 208} \[ -\frac {a \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (b e^2+c d^2\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 520
Rule 897
Rule 1153
Rule 1251
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x \sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \frac {a+b x+c x^2}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \frac {\frac {c d^4+b d^2 e^2+a e^4}{e^4}-\frac {\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac {c x^4}{e^4}}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \left (b+\frac {c d^2}{e^2}-\frac {c x^2}{e^2}+\frac {a}{\frac {d^2}{e^2}-\frac {x^2}{e^2}}\right ) \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (a \sqrt {d^2-e^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [B] time = 0.88, size = 217, normalized size = 2.33 \[ \frac {-\frac {3 a \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d \sqrt {d-e x}}+\frac {\left (e^2 x^2-d^2\right ) \left (3 b e^2+2 c d^2+c e^2 x^2\right )}{e^4 \sqrt {d-e x}}+\frac {6 d \sqrt {d+e x} \left (b e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )}{e^4}-\frac {6 d^{3/2} \sqrt {\frac {e x}{d}+1} \left (b e^2+c d^2\right ) \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right )}{e^4}}{3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 80, normalized size = 0.86 \[ \frac {3 \, a e^{4} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (c d e^{2} x^{2} + 2 \, c d^{3} + 3 \, b d e^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{3 \, d e^{4}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 143, normalized size = 1.54 \[ -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (\sqrt {-e^{2} x^{2}+d^{2}}\, c d \,e^{2} x^{2} \mathrm {csgn}\relax (d )+3 a \,e^{4} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+3 \sqrt {-e^{2} x^{2}+d^{2}}\, b d \,e^{2} \mathrm {csgn}\relax (d )+2 \sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{3} \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{3 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 105, normalized size = 1.13 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{2}}{3 \, e^{2}} - \frac {a \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2}}{3 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 161, normalized size = 1.73 \[ \frac {a\,\left (\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )-\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )\right )}{d}-\frac {\sqrt {d-e\,x}\,\left (\frac {2\,c\,d^3}{3\,e^4}+\frac {c\,x^3}{3\,e}+\frac {c\,d\,x^2}{3\,e^2}+\frac {2\,c\,d^2\,x}{3\,e^3}\right )}{\sqrt {d+e\,x}}-\frac {\left (\frac {b\,d}{e^2}+\frac {b\,x}{e}\right )\,\sqrt {d-e\,x}}{\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 91.28, size = 304, normalized size = 3.27 \[ \frac {i a {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {a {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i b d {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {b d {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {i c d^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} - \frac {c d^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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