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 = 0.184015, antiderivative size = 71, normalized size of antiderivative = 1., 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 1609
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
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*}
Mathematica [A] time = 0.0466222, 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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Maple [C] time = 0., size = 108, normalized size = 1.5 \begin{align*} -{\frac{ \left ({\it csgn} \left ( d \right ) \right ) ^{2}}{2\,{x}^{2}}\sqrt{-dx+1}\sqrt{dx+1} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}a{d}^{2}+2\,{\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}c+2\,\sqrt{-{d}^{2}{x}^{2}+1}xb+\sqrt{-{d}^{2}{x}^{2}+1}a \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.97374, size = 132, normalized size = 1.86 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01834, size = 154, normalized size = 2.17 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 34.2892, size = 218, normalized size = 3.07 \begin{align*} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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