Optimal. Leaf size=71 \[ -\frac {1}{2} \left (a d^2+2 c\right ) \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )-\frac {a \sqrt {1-d^2 x^2}}{2 x^2}-\frac {b \sqrt {1-d^2 x^2}}{x} \]
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Rubi [A] time = 0.19, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1609, 1807, 807, 266, 63, 208} \[ -\frac {1}{2} \left (a d^2+2 c\right ) \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )-\frac {a \sqrt {1-d^2 x^2}}{2 x^2}-\frac {b \sqrt {1-d^2 x^2}}{x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1609
Rule 1807
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{x^3 \sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {a+b x+c x^2}{x^3 \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-d^2 x^2}}{2 x^2}-\frac {1}{2} \int \frac {-2 b-\left (2 c+a d^2\right ) x}{x^2 \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-d^2 x^2}}{2 x^2}-\frac {b \sqrt {1-d^2 x^2}}{x}-\frac {1}{2} \left (-2 c-a d^2\right ) \int \frac {1}{x \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-d^2 x^2}}{2 x^2}-\frac {b \sqrt {1-d^2 x^2}}{x}-\frac {1}{4} \left (-2 c-a d^2\right ) \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}}{2 x^2}-\frac {b \sqrt {1-d^2 x^2}}{x}-\frac {1}{2} \left (a+\frac {2 c}{d^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{d^2}-\frac {x^2}{d^2}} \, dx,x,\sqrt {1-d^2 x^2}\right )\\ &=-\frac {a \sqrt {1-d^2 x^2}}{2 x^2}-\frac {b \sqrt {1-d^2 x^2}}{x}-\frac {1}{2} \left (2 c+a d^2\right ) \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 56, normalized size = 0.79 \[ -\frac {\sqrt {1-d^2 x^2} (a+2 b x)}{2 x^2}-\frac {1}{2} \left (a d^2+2 c\right ) \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 65, normalized size = 0.92 \[ \frac {{\left (a d^{2} + 2 \, c\right )} x^{2} \log \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{x}\right ) - {\left (2 \, b x + a\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{2 \, x^{2}} \]
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 = 108, normalized size = 1.52 \[ -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (a \,d^{2} x^{2} \arctanh \left (\frac {1}{\sqrt {-d^{2} x^{2}+1}}\right )+2 c \,x^{2} \arctanh \left (\frac {1}{\sqrt {-d^{2} x^{2}+1}}\right )+2 \sqrt {-d^{2} x^{2}+1}\, b x +\sqrt {-d^{2} x^{2}+1}\, a \right ) \mathrm {csgn}\relax (d )^{2}}{2 \sqrt {-d^{2} x^{2}+1}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 98, normalized size = 1.38 \[ -\frac {1}{2} \, a d^{2} \log \left (\frac {2 \, \sqrt {-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - c \log \left (\frac {2 \, \sqrt {-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\sqrt {-d^{2} x^{2} + 1} b}{x} - \frac {\sqrt {-d^{2} x^{2} + 1} a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.85, size = 312, normalized size = 4.39 \[ c\,\left (\ln \left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}-1\right )-\ln \left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )\right )-\frac {\frac {a\,d^2\,{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}-\frac {a\,d^2}{2}+\frac {15\,a\,d^2\,{\left (\sqrt {1-d\,x}-1\right )}^4}{2\,{\left (\sqrt {d\,x+1}-1\right )}^4}}{\frac {16\,{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}-\frac {32\,{\left (\sqrt {1-d\,x}-1\right )}^4}{{\left (\sqrt {d\,x+1}-1\right )}^4}+\frac {16\,{\left (\sqrt {1-d\,x}-1\right )}^6}{{\left (\sqrt {d\,x+1}-1\right )}^6}}+\frac {a\,d^2\,\ln \left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}-1\right )}{2}-\frac {a\,d^2\,\ln \left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )}{2}-\frac {b\,\sqrt {1-d\,x}\,\sqrt {d\,x+1}}{x}+\frac {a\,d^2\,{\left (\sqrt {1-d\,x}-1\right )}^2}{32\,{\left (\sqrt {d\,x+1}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 80.29, size = 218, normalized size = 3.07 \[ \frac {i a d^{2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & 2, 2, \frac {5}{2} \\\frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {a d^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, 1 & \\\frac {5}{4}, \frac {7}{4} & 1, \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i b d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {b d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i c {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {c {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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