Optimal. Leaf size=234 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+11 C d^2\right )}{315 d^4 e^3 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+11 C d^2\right )}{105 d^3 e^3 (d+e x)^4}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+11 C d^2\right )}{42 d^2 e^3 (d+e x)^5}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{9 d e^3 (d+e x)^6}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5} \]
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Rubi [A] time = 0.264216, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1639, 793, 659, 651} \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+11 C d^2\right )}{315 d^4 e^3 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+11 C d^2\right )}{105 d^3 e^3 (d+e x)^4}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+11 C d^2\right )}{42 d^2 e^3 (d+e x)^5}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{9 d e^3 (d+e x)^6}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5} \]
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)^6} \, dx &=\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5}+\frac{\int \frac{\left (e^2 \left (5 C d^2+2 A e^2\right )+e^3 (3 C d+2 B e) x\right ) \sqrt{d^2-e^2 x^2}}{(d+e x)^6} \, dx}{2 e^4}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{9 d e^3 (d+e x)^6}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5}+\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^5} \, dx}{6 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}}{9 d e^3 (d+e x)^6}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5}-\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{42 d^2 e^3 (d+e x)^5}+\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx}{21 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}}{9 d e^3 (d+e x)^6}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5}-\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{42 d^2 e^3 (d+e x)^5}-\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{105 d^3 e^3 (d+e x)^4}+\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx}{105 d^3 e^2}\\ &=-\frac{\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{9 d e^3 (d+e x)^6}+\frac{C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)^5}-\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{42 d^2 e^3 (d+e x)^5}-\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{105 d^3 e^3 (d+e x)^4}-\frac{\left (11 C d^2+2 e (2 B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{315 d^4 e^3 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.227784, size = 144, normalized size = 0.62 \[ -\frac{(d-e x) \sqrt{d^2-e^2 x^2} \left (e \left (A e \left (33 d^2 e x+58 d^3+12 d e^2 x^2+2 e^3 x^3\right )+B d \left (66 d^2 e x+11 d^3+24 d e^2 x^2+4 e^3 x^3\right )\right )+C d^2 \left (24 d^2 e x+4 d^3+66 d e^2 x^2+11 e^3 x^3\right )\right )}{315 d^4 e^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 152, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,A{e}^{5}{x}^{3}+4\,Bd{e}^{4}{x}^{3}+11\,C{d}^{2}{e}^{3}{x}^{3}+12\,Ad{e}^{4}{x}^{2}+24\,B{d}^{2}{e}^{3}{x}^{2}+66\,C{d}^{3}{e}^{2}{x}^{2}+33\,A{d}^{2}{e}^{3}x+66\,B{d}^{3}{e}^{2}x+24\,C{d}^{4}ex+58\,A{d}^{3}{e}^{2}+11\,B{d}^{4}e+4\,C{d}^{5} \right ) \left ( -ex+d \right ) }{315\, \left ( ex+d \right ) ^{5}{d}^{4}{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 = 2.5496, size = 852, normalized size = 3.64 \begin{align*} -\frac{4 \, C d^{7} + 11 \, B d^{6} e + 58 \, A d^{5} e^{2} +{\left (4 \, C d^{2} e^{5} + 11 \, B d e^{6} + 58 \, A e^{7}\right )} x^{5} + 5 \,{\left (4 \, C d^{3} e^{4} + 11 \, B d^{2} e^{5} + 58 \, A d e^{6}\right )} x^{4} + 10 \,{\left (4 \, C d^{4} e^{3} + 11 \, B d^{3} e^{4} + 58 \, A d^{2} e^{5}\right )} x^{3} + 10 \,{\left (4 \, C d^{5} e^{2} + 11 \, B d^{4} e^{3} + 58 \, A d^{3} e^{4}\right )} x^{2} + 5 \,{\left (4 \, C d^{6} e + 11 \, B d^{5} e^{2} + 58 \, A d^{4} e^{3}\right )} x +{\left (4 \, C d^{6} + 11 \, B d^{5} e + 58 \, A d^{4} e^{2} -{\left (11 \, C d^{2} e^{4} + 4 \, B d e^{5} + 2 \, A e^{6}\right )} x^{4} - 5 \,{\left (11 \, C d^{3} e^{3} + 4 \, B d^{2} e^{4} + 2 \, A d e^{5}\right )} x^{3} + 21 \,{\left (2 \, C d^{4} e^{2} - 2 \, B d^{3} e^{3} - A d^{2} e^{4}\right )} x^{2} + 5 \,{\left (4 \, C d^{5} e + 11 \, B d^{4} e^{2} - 5 \, A d^{3} e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{315 \,{\left (d^{4} e^{8} x^{5} + 5 \, d^{5} e^{7} x^{4} + 10 \, d^{6} e^{6} x^{3} + 10 \, d^{7} e^{5} x^{2} + 5 \, d^{8} e^{4} x + d^{9} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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