Optimal. Leaf size=48 \[ -\frac {a \sqrt {1-d^2 x^2}}{x}-b \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )+\frac {c \sin ^{-1}(d x)}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {1609, 1807, 844, 216, 266, 63, 208} \[ -\frac {a \sqrt {1-d^2 x^2}}{x}-b \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )+\frac {c \sin ^{-1}(d x)}{d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1609
Rule 1807
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{x^2 \sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {a+b x+c x^2}{x^2 \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-d^2 x^2}}{x}-\int \frac {-b-c x}{x \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-d^2 x^2}}{x}+b \int \frac {1}{x \sqrt {1-d^2 x^2}} \, dx+c \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-d^2 x^2}}{x}+\frac {c \sin ^{-1}(d x)}{d}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-d^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {1-d^2 x^2}}{x}+\frac {c \sin ^{-1}(d x)}{d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {1}{d^2}-\frac {x^2}{d^2}} \, dx,x,\sqrt {1-d^2 x^2}\right )}{d^2}\\ &=-\frac {a \sqrt {1-d^2 x^2}}{x}+\frac {c \sin ^{-1}(d x)}{d}-b \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 48, normalized size = 1.00 \[ -\frac {a \sqrt {1-d^2 x^2}}{x}-b \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )+\frac {c \sin ^{-1}(d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 84, normalized size = 1.75 \[ \frac {b d x \log \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{x}\right ) - \sqrt {d x + 1} \sqrt {-d x + 1} a d - 2 \, c x \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d x} \]
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 = 0.02, size = 97, normalized size = 2.02 \[ \frac {\left (-b d x \arctanh \left (\frac {1}{\sqrt {-d^{2} x^{2}+1}}\right ) \mathrm {csgn}\relax (d )-\sqrt {-d^{2} x^{2}+1}\, a d \,\mathrm {csgn}\relax (d )+c x \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )\right ) \sqrt {-d x +1}\, \sqrt {d x +1}\, \mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}\, d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 57, normalized size = 1.19 \[ -b \log \left (\frac {2 \, \sqrt {-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {c \arcsin \left (d x\right )}{d} - \frac {\sqrt {-d^{2} x^{2} + 1} a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 114, normalized size = 2.38 \[ b\,\left (\ln \left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}-1\right )-\ln \left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )\right )-\frac {4\,c\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {1-d\,x}-1\right )}{\left (\sqrt {d\,x+1}-1\right )\,\sqrt {d^2}}\right )}{\sqrt {d^2}}-\frac {a\,\sqrt {1-d\,x}\,\sqrt {d\,x+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 49.85, size = 221, normalized size = 4.60 \[ \frac {i a d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {a d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i b {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {b {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i c {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} + \frac {c {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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