Optimal. Leaf size=163 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{3 d^2 e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{3 d e^3 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2} (2 C d-B e)}{d e^3 (d+e x)}+\frac{C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
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Rubi [A] time = 0.169427, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {1637, 217, 203, 659, 651} \[ -\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{3 d^2 e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{3 d e^3 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2} (2 C d-B e)}{d e^3 (d+e x)}+\frac{C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
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Rule 1637
Rule 217
Rule 203
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx &=\int \left (\frac{C}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{C d^2-B d e+A e^2}{e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}+\frac{-2 C d+B e}{e^2 (d+e x) \sqrt{d^2-e^2 x^2}}\right ) \, dx\\ &=\frac{C \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^2}-\frac{(2 C d-B e) \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{e^2}+\frac{\left (C d^2-B d e+A e^2\right ) \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{3 d e^3 (d+e x)^2}+\frac{(2 C d-B e) \sqrt{d^2-e^2 x^2}}{d e^3 (d+e x)}+\frac{C \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{\left (C d^2-B d e+A e^2\right ) \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{3 d e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{3 d e^3 (d+e x)^2}+\frac{(2 C d-B e) \sqrt{d^2-e^2 x^2}}{d e^3 (d+e x)}-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{3 d^2 e^3 (d+e x)}+\frac{C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.221091, size = 95, normalized size = 0.58 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (C d^2 (4 d+5 e x)-e (A e (2 d+e x)+B d (d+2 e x))\right )}{d^2 (d+e x)^2}+3 C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{3 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 355, normalized size = 2.2 \begin{align*}{\frac{C}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{Be-2\,Cd}{{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}}-{\frac{A}{3\,d{e}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-2}}+{\frac{B}{3\,{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{Cd}{3\,{e}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{A}{3\,{d}^{2}{e}^{2}}\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}}+{\frac{B}{3\,d{e}^{3}}\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}}-{\frac{C}{3\,{e}^{4}}\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 = 1.82777, size = 451, normalized size = 2.77 \begin{align*} \frac{4 \, C d^{4} - B d^{3} e - 2 \, A d^{2} e^{2} +{\left (4 \, C d^{2} e^{2} - B d e^{3} - 2 \, A e^{4}\right )} x^{2} + 2 \,{\left (4 \, C d^{3} e - B d^{2} e^{2} - 2 \, A d e^{3}\right )} x - 6 \,{\left (C d^{2} e^{2} x^{2} + 2 \, C d^{3} e x + C d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (4 \, C d^{3} - B d^{2} e - 2 \, A d e^{2} +{\left (5 \, C d^{2} e - 2 \, B d e^{2} - A e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{2} e^{5} x^{2} + 2 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \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 )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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