Optimal. Leaf size=196 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{15 d^2 e^3 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{5 d e^3 (d+e x)^4}+\frac{\left (d^2-e^2 x^2\right )^{3/2} (2 C d-B e)}{3 d e^3 (d+e x)^3}-\frac{2 C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
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Rubi [A] time = 0.183946, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1637, 659, 651, 663, 217, 203} \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{15 d^2 e^3 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{5 d e^3 (d+e x)^4}+\frac{\left (d^2-e^2 x^2\right )^{3/2} (2 C d-B e)}{3 d e^3 (d+e x)^3}-\frac{2 C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
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Rule 1637
Rule 659
Rule 651
Rule 663
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)^4} \, dx &=\int \left (\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{e^2 (d+e x)^4}+\frac{(-2 C d+B e) \sqrt{d^2-e^2 x^2}}{e^2 (d+e x)^3}+\frac{C \sqrt{d^2-e^2 x^2}}{e^2 (d+e x)^2}\right ) \, dx\\ &=\frac{C \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^2} \, dx}{e^2}-\frac{(2 C d-B e) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx}{e^2}+\frac{\left (C d^2-B d e+A e^2\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx}{e^2}\\ &=-\frac{2 C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{5 d e^3 (d+e x)^4}+\frac{(2 C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac{C \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^2}+\frac{\left (C d^2-B d e+A e^2\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx}{5 d e^2}\\ &=-\frac{2 C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{5 d e^3 (d+e x)^4}+\frac{(2 C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e^3 (d+e x)^3}-\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{2 C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{5 d e^3 (d+e x)^4}+\frac{(2 C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e^3 (d+e x)^3}-\frac{C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.295722, size = 112, normalized size = 0.57 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (e (d-e x) (A e (4 d+e x)+B d (d+4 e x))+3 C d^2 \left (8 d^2+19 d e x+13 e^2 x^2\right )\right )}{d^2 (d+e x)^3}+15 C \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 453, normalized size = 2.3 \begin{align*} -{\frac{A}{5\,{e}^{5}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}+{\frac{B}{5\,{e}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{Cd}{5\,{e}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{A}{15\,{d}^{2}{e}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+{\frac{B}{15\,{e}^{5}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{C}{15\,{e}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{C}{{e}^{5}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{C}{d{e}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{C}{{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}}}}}-{\frac{Be-2\,Cd}{3\,{e}^{6}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}} \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 = 1.91587, size = 637, normalized size = 3.25 \begin{align*} -\frac{24 \, C d^{5} + B d^{4} e + 4 \, A d^{3} e^{2} +{\left (24 \, C d^{2} e^{3} + B d e^{4} + 4 \, A e^{5}\right )} x^{3} + 3 \,{\left (24 \, C d^{3} e^{2} + B d^{2} e^{3} + 4 \, A d e^{4}\right )} x^{2} + 3 \,{\left (24 \, C d^{4} e + B d^{3} e^{2} + 4 \, A d^{2} e^{3}\right )} x - 30 \,{\left (C d^{2} e^{3} x^{3} + 3 \, C d^{3} e^{2} x^{2} + 3 \, C d^{4} e x + C d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (24 \, C d^{4} + B d^{3} e + 4 \, A d^{2} e^{2} +{\left (39 \, C d^{2} e^{2} - 4 \, B d e^{3} - A e^{4}\right )} x^{2} + 3 \,{\left (19 \, C d^{3} e + B d^{2} e^{2} - A d e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{6} x^{3} + 3 \, d^{3} e^{5} x^{2} + 3 \, d^{4} e^{4} x + d^{5} 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{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (A + B x + C x^{2}\right )}{\left (d + e x\right )^{4}}\, 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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