Optimal. Leaf size=148 \[ \frac{\sqrt{d^2-e^2 x^2} \left (C d^2-e (B d-2 A e)\right )}{2 e^3}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (C d^2-e (B d-2 A e)\right )}{2 e^3}+\frac{\left (d^2-e^2 x^2\right )^{3/2} (C d-B e)}{2 e^3 (d+e x)}-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3} \]
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Rubi [A] time = 0.1763, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {1639, 795, 665, 217, 203} \[ \frac{\sqrt{d^2-e^2 x^2} \left (C d^2-e (B d-2 A e)\right )}{2 e^3}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (C d^2-e (B d-2 A e)\right )}{2 e^3}+\frac{\left (d^2-e^2 x^2\right )^{3/2} (C d-B e)}{2 e^3 (d+e x)}-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 795
Rule 665
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (A+B x+C x^2\right ) \sqrt{d^2-e^2 x^2}}{d+e x} \, dx &=-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac{\int \frac{\left (-3 A e^4+3 e^3 (C d-B e) x\right ) \sqrt{d^2-e^2 x^2}}{d+e x} \, dx}{3 e^4}\\ &=-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac{\left (C d^2-e (B d-2 A e)\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{d+e x} \, dx}{2 e^2}\\ &=\frac{\left (C d^2-e (B d-2 A e)\right ) \sqrt{d^2-e^2 x^2}}{2 e^3}-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac{\left (d \left (C d^2-e (B d-2 A e)\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=\frac{\left (C d^2-e (B d-2 A e)\right ) \sqrt{d^2-e^2 x^2}}{2 e^3}-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac{\left (d \left (C d^2-e (B d-2 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^2}\\ &=\frac{\left (C d^2-e (B d-2 A e)\right ) \sqrt{d^2-e^2 x^2}}{2 e^3}-\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac{d \left (C d^2-e (B d-2 A e)\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3}\\ \end{align*}
Mathematica [A] time = 0.226047, size = 103, normalized size = 0.7 \[ \frac{\sqrt{d^2-e^2 x^2} \left (3 e (2 A e-2 B d+B e x)+C \left (4 d^2-3 d e x+2 e^2 x^2\right )\right )+3 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (e (2 A e-B d)+C d^2\right )}{6 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 384, normalized size = 2.6 \begin{align*} -{\frac{C}{3\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{Bx}{2\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{B{d}^{2}}{2\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{Cdx}{2\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{C{d}^{3}}{2\,{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}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{Bd}{{e}^{2}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{C{d}^{2}}{{e}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{Ad\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{B{d}^{2}}{e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{C{d}^{3}}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \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 = 1.87252, size = 236, normalized size = 1.59 \begin{align*} -\frac{6 \,{\left (C d^{3} - B d^{2} e + 2 \, A d e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (2 \, C e^{2} x^{2} + 4 \, C d^{2} - 6 \, B d e + 6 \, A e^{2} - 3 \,{\left (C d e - B e^{2}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \, 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{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (A + B x + C x^{2}\right )}{d + e x}\, 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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