Optimal. Leaf size=103 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{d e^3 (d+e x)}-\frac{(C d-B e) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}-\frac{C \sqrt{d^2-e^2 x^2}}{e^3} \]
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Rubi [A] time = 0.12093, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1639, 793, 217, 203} \[ -\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{d e^3 (d+e x)}-\frac{(C d-B e) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}-\frac{C \sqrt{d^2-e^2 x^2}}{e^3} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 793
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{C \sqrt{d^2-e^2 x^2}}{e^3}-\frac{\int \frac{-A e^4+e^3 (C d-B e) x}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{e^4}\\ &=-\frac{C \sqrt{d^2-e^2 x^2}}{e^3}-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{d e^3 (d+e x)}-\frac{(C d-B e) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^2}\\ &=-\frac{C \sqrt{d^2-e^2 x^2}}{e^3}-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{d e^3 (d+e x)}-\frac{(C d-B e) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2}\\ &=-\frac{C \sqrt{d^2-e^2 x^2}}{e^3}-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{d e^3 (d+e x)}-\frac{(C d-B e) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.161356, size = 83, normalized size = 0.81 \[ \frac{(B e-C d) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{\sqrt{d^2-e^2 x^2} (e (A e-B d)+C d (2 d+e x))}{d (d+e x)}}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 149, normalized size = 1.5 \begin{align*} -{\frac{C}{{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{B}{e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{Cd}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{A{e}^{2}-Bde+C{d}^{2}}{{e}^{4}d}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84959, size = 315, normalized size = 3.06 \begin{align*} -\frac{2 \, C d^{3} - B d^{2} e + A d e^{2} +{\left (2 \, C d^{2} e - B d e^{2} + A e^{3}\right )} x - 2 \,{\left (C d^{3} - B d^{2} e +{\left (C d^{2} e - B d e^{2}\right )} x\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (C d e x + 2 \, C d^{2} - B d e + A e^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{d e^{4} x + d^{2} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x + C x^{2}}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \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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