Optimal. Leaf size=180 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (e (2 A e+3 B d)+7 C d^2\right )}{15 d^3 e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2} \left (e (2 A e+3 B d)+7 C d^2\right )}{15 d^2 e^3 (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{5 d e^3 (d+e x)^3}+\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2} \]
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Rubi [A] time = 0.205141, antiderivative size = 180, 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, 659, 651} \[ -\frac{\sqrt{d^2-e^2 x^2} \left (e (2 A e+3 B d)+7 C d^2\right )}{15 d^3 e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2} \left (e (2 A e+3 B d)+7 C d^2\right )}{15 d^2 e^3 (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2} \left (A e^2-B d e+C d^2\right )}{5 d e^3 (d+e x)^3}+\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 793
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx &=\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2}+\frac{\int \frac{e^2 \left (2 C d^2+A e^2\right )+e^3 (C d+B e) x}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx}{e^4}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{5 d e^3 (d+e x)^3}+\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2}+\frac{\left (7 C d^2+e (3 B d+2 A e)\right ) \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{5 d e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{5 d e^3 (d+e x)^3}+\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2}-\frac{\left (7 C d^2+e (3 B d+2 A e)\right ) \sqrt{d^2-e^2 x^2}}{15 d^2 e^3 (d+e x)^2}+\frac{\left (7 C d^2+e (3 B d+2 A e)\right ) \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{15 d^2 e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \sqrt{d^2-e^2 x^2}}{5 d e^3 (d+e x)^3}+\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2}-\frac{\left (7 C d^2+e (3 B d+2 A e)\right ) \sqrt{d^2-e^2 x^2}}{15 d^2 e^3 (d+e x)^2}-\frac{\left (7 C d^2+e (3 B d+2 A e)\right ) \sqrt{d^2-e^2 x^2}}{15 d^3 e^3 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.19948, size = 103, normalized size = 0.57 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (e \left (A e \left (7 d^2+6 d e x+2 e^2 x^2\right )+3 B d \left (d^2+3 d e x+e^2 x^2\right )\right )+C d^2 \left (2 d^2+6 d e x+7 e^2 x^2\right )\right )}{15 d^3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 116, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 2\,A{e}^{4}{x}^{2}+3\,Bd{e}^{3}{x}^{2}+7\,C{d}^{2}{e}^{2}{x}^{2}+6\,Ad{e}^{3}x+9\,B{d}^{2}{e}^{2}x+6\,C{d}^{3}ex+7\,A{d}^{2}{e}^{2}+3\,B{d}^{3}e+2\,C{d}^{4} \right ) }{15\,{e}^{3}{d}^{3} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{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 = 1.78098, size = 510, normalized size = 2.83 \begin{align*} -\frac{2 \, C d^{5} + 3 \, B d^{4} e + 7 \, A d^{3} e^{2} +{\left (2 \, C d^{2} e^{3} + 3 \, B d e^{4} + 7 \, A e^{5}\right )} x^{3} + 3 \,{\left (2 \, C d^{3} e^{2} + 3 \, B d^{2} e^{3} + 7 \, A d e^{4}\right )} x^{2} + 3 \,{\left (2 \, C d^{4} e + 3 \, B d^{3} e^{2} + 7 \, A d^{2} e^{3}\right )} x +{\left (2 \, C d^{4} + 3 \, B d^{3} e + 7 \, A d^{2} e^{2} +{\left (7 \, C d^{2} e^{2} + 3 \, B d e^{3} + 2 \, A e^{4}\right )} x^{2} + 3 \,{\left (2 \, C d^{3} e + 3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{6} x^{3} + 3 \, d^{4} e^{5} x^{2} + 3 \, d^{5} e^{4} x + d^{6} 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 )^{3}}\, 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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