Optimal. Leaf size=180 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (e (2 A e+5 B d)+23 C d^2\right )}{105 d^3 e^3 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (e (2 A e+5 B d)+23 C d^2\right )}{35 d^2 e^3 (d+e x)^4}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{7 d e^3 (d+e x)^5}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4} \]
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Rubi [A] time = 0.210378, 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} \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (e (2 A e+5 B d)+23 C d^2\right )}{105 d^3 e^3 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (e (2 A e+5 B d)+23 C d^2\right )}{35 d^2 e^3 (d+e x)^4}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{7 d e^3 (d+e x)^5}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 793
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{\left (A+B x+C x^2\right ) \sqrt{d^2-e^2 x^2}}{(d+e x)^5} \, dx &=\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}+\frac{\int \frac{\left (e^2 \left (4 C d^2+A e^2\right )+e^3 (3 C d+B e) x\right ) \sqrt{d^2-e^2 x^2}}{(d+e x)^5} \, dx}{e^4}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{7 d e^3 (d+e x)^5}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}+\frac{\left (23 C d^2+e (5 B d+2 A e)\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx}{7 d e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{7 d e^3 (d+e x)^5}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}-\frac{\left (23 C d^2+e (5 B d+2 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 e^3 (d+e x)^4}+\frac{\left (23 C d^2+e (5 B d+2 A e)\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx}{35 d^2 e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{7 d e^3 (d+e x)^5}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}-\frac{\left (23 C d^2+e (5 B d+2 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 e^3 (d+e x)^4}-\frac{\left (23 C d^2+e (5 B d+2 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{105 d^3 e^3 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.206715, size = 109, normalized size = 0.61 \[ -\frac{(d-e x) \sqrt{d^2-e^2 x^2} \left (e \left (A e \left (23 d^2+10 d e x+2 e^2 x^2\right )+5 B d \left (d^2+5 d e x+e^2 x^2\right )\right )+C d^2 \left (2 d^2+10 d e x+23 e^2 x^2\right )\right )}{105 d^3 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 116, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,A{e}^{4}{x}^{2}+5\,Bd{e}^{3}{x}^{2}+23\,C{d}^{2}{e}^{2}{x}^{2}+10\,Ad{e}^{3}x+25\,B{d}^{2}{e}^{2}x+10\,C{d}^{3}ex+23\,A{d}^{2}{e}^{2}+5\,B{d}^{3}e+2\,C{d}^{4} \right ) \left ( -ex+d \right ) }{105\, \left ( ex+d \right ) ^{4}{d}^{3}{e}^{3}}\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 = 1.96755, size = 678, normalized size = 3.77 \begin{align*} -\frac{2 \, C d^{6} + 5 \, B d^{5} e + 23 \, A d^{4} e^{2} +{\left (2 \, C d^{2} e^{4} + 5 \, B d e^{5} + 23 \, A e^{6}\right )} x^{4} + 4 \,{\left (2 \, C d^{3} e^{3} + 5 \, B d^{2} e^{4} + 23 \, A d e^{5}\right )} x^{3} + 6 \,{\left (2 \, C d^{4} e^{2} + 5 \, B d^{3} e^{3} + 23 \, A d^{2} e^{4}\right )} x^{2} + 4 \,{\left (2 \, C d^{5} e + 5 \, B d^{4} e^{2} + 23 \, A d^{3} e^{3}\right )} x +{\left (2 \, C d^{5} + 5 \, B d^{4} e + 23 \, A d^{3} e^{2} -{\left (23 \, C d^{2} e^{3} + 5 \, B d e^{4} + 2 \, A e^{5}\right )} x^{3} +{\left (13 \, C d^{3} e^{2} - 20 \, B d^{2} e^{3} - 8 \, A d e^{4}\right )} x^{2} +{\left (8 \, C d^{4} e + 20 \, B d^{3} e^{2} - 13 \, A d^{2} e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{105 \,{\left (d^{3} e^{7} x^{4} + 4 \, d^{4} e^{6} x^{3} + 6 \, d^{5} e^{5} x^{2} + 4 \, d^{6} e^{4} x + d^{7} 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 )^{5}}\, 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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