Optimal. Leaf size=270 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}} \]
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Rubi [A] time = 0.439427, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {794, 656, 648} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}}-\frac{\left (2 \left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{1}{2} \left (c e^3 f-\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx}{13 c e^3}\\ &=-\frac{2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}}+\frac{(4 (2 c d-b e) (13 c e f-c d g-6 b e g)) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{143 c^2 e}\\ &=-\frac{8 (2 c d-b e) (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac{2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}}+\frac{\left (8 (2 c d-b e)^2 (13 c e f-c d g-6 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{1287 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac{8 (2 c d-b e) (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac{2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.247809, size = 183, normalized size = 0.68 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 c e^2 (44 d g+13 e f+21 e g x)-48 b^3 e^3 g-2 b c^2 e \left (423 d^2 g+d e (390 f+532 g x)+7 e^2 x (26 f+27 g x)\right )+c^3 \left (d^2 e (1963 f+1897 g x)+542 d^3 g+14 d e^2 x (169 f+144 g x)+63 e^3 x^2 (13 f+11 g x)\right )\right )}{9009 c^4 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 235, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -693\,g{e}^{3}{x}^{3}{c}^{3}+378\,b{c}^{2}{e}^{3}g{x}^{2}-2016\,{c}^{3}d{e}^{2}g{x}^{2}-819\,{c}^{3}{e}^{3}f{x}^{2}-168\,{b}^{2}c{e}^{3}gx+1064\,b{c}^{2}d{e}^{2}gx+364\,b{c}^{2}{e}^{3}fx-1897\,{c}^{3}{d}^{2}egx-2366\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-352\,{b}^{2}cd{e}^{2}g-104\,{b}^{2}c{e}^{3}f+846\,b{c}^{2}{d}^{2}eg+780\,b{c}^{2}d{e}^{2}f-542\,{c}^{3}{d}^{3}g-1963\,f{d}^{2}{c}^{3}e \right ) }{9009\,{c}^{4}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21438, size = 861, normalized size = 3.19 \begin{align*} \frac{2 \,{\left (63 \, c^{5} e^{5} x^{5} - 151 \, c^{5} d^{5} + 513 \, b c^{4} d^{4} e - 641 \, b^{2} c^{3} d^{3} e^{2} + 355 \, b^{3} c^{2} d^{2} e^{3} - 84 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \,{\left (c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} -{\left (206 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \,{\left (10 \, c^{5} d^{3} e^{2} - 118 \, b c^{4} d^{2} e^{3} + 107 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} +{\left (271 \, c^{5} d^{4} e - 512 \, b c^{4} d^{3} e^{2} + 207 \, b^{2} c^{3} d^{2} e^{3} + 38 \, b^{3} c^{2} d e^{4} - 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{693 \, c^{3} e} + \frac{2 \,{\left (693 \, c^{6} e^{6} x^{6} - 542 \, c^{6} d^{6} + 2472 \, b c^{5} d^{5} e - 4516 \, b^{2} c^{4} d^{4} e^{2} + 4184 \, b^{3} c^{3} d^{3} e^{3} - 2046 \, b^{4} c^{2} d^{2} e^{4} + 496 \, b^{5} c d e^{5} - 48 \, b^{6} e^{6} - 63 \,{\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} x^{5} - 7 \,{\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} x^{4} +{\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} x^{3} + 3 \,{\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} x^{2} -{\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{9009 \, c^{4} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.489, size = 1462, normalized size = 5.41 \begin{align*} \frac{2 \,{\left (693 \, c^{6} e^{6} g x^{6} + 63 \,{\left (13 \, c^{6} e^{6} f -{\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} g\right )} x^{5} - 7 \,{\left (13 \,{\left (c^{6} d e^{5} - 23 \, b c^{5} e^{6}\right )} f +{\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} g\right )} x^{4} -{\left (13 \,{\left (206 \, c^{6} d^{2} e^{4} - 192 \, b c^{5} d e^{5} - 113 \, b^{2} c^{4} e^{6}\right )} f -{\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} + 3 \,{\left (13 \,{\left (10 \, c^{6} d^{3} e^{3} - 118 \, b c^{5} d^{2} e^{4} + 107 \, b^{2} c^{4} d e^{5} + b^{3} c^{3} e^{6}\right )} f +{\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} - 13 \,{\left (151 \, c^{6} d^{5} e - 513 \, b c^{5} d^{4} e^{2} + 641 \, b^{2} c^{4} d^{3} e^{3} - 355 \, b^{3} c^{3} d^{2} e^{4} + 84 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\right )} f - 2 \,{\left (271 \, c^{6} d^{6} - 1236 \, b c^{5} d^{5} e + 2258 \, b^{2} c^{4} d^{4} e^{2} - 2092 \, b^{3} c^{3} d^{3} e^{3} + 1023 \, b^{4} c^{2} d^{2} e^{4} - 248 \, b^{5} c d e^{5} + 24 \, b^{6} e^{6}\right )} g +{\left (13 \,{\left (271 \, c^{6} d^{4} e^{2} - 512 \, b c^{5} d^{3} e^{3} + 207 \, b^{2} c^{4} d^{2} e^{4} + 38 \, b^{3} c^{3} d e^{5} - 4 \, b^{4} c^{2} e^{6}\right )} f -{\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}}{9009 \,{\left (c^{4} e^{3} x + c^{4} d e^{2}\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(-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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