Optimal. Leaf size=347 \[ -\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{1155 c^4 e^2 \sqrt{d+e x}}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{231 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{99 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2} \]
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Rubi [A] time = 0.624455, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {794, 656, 648} \[ -\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{1155 c^4 e^2 \sqrt{d+e x}}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{231 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{99 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int (d+e x)^{5/2} (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}-\frac{\left (2 \left (\frac{3}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{5/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{11 c e^3}\\ &=-\frac{2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}+\frac{(2 (2 c d-b e) (11 c e f+5 c d g-8 b e g)) \int (d+e x)^{3/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{33 c^2 e}\\ &=-\frac{4 (2 c d-b e) (11 c e f+5 c d g-8 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 c^3 e^2}-\frac{2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}+\frac{\left (8 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g)\right ) \int \sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{231 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) (11 c e f+5 c d g-8 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 c^3 e^2}-\frac{2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}+\frac{\left (16 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}} \, dx}{1155 c^4 e}\\ &=-\frac{32 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) (11 c e f+5 c d g-8 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 c^3 e^2}-\frac{2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\\ \end{align*}
Mathematica [A] time = 0.25097, size = 262, normalized size = 0.76 \[ \frac{2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} \left (24 b^2 c^2 e^2 \left (131 d^2 g+d e (55 f+57 g x)+e^2 x (11 f+10 g x)\right )-16 b^3 c e^3 (65 d g+11 e f+12 e g x)+128 b^4 e^4 g-2 b c^3 e \left (3 d^2 e (583 f+558 g x)+2071 d^3 g+3 d e^2 x (286 f+245 g x)+5 e^3 x^2 (33 f+28 g x)\right )+c^4 \left (3 d^2 e^2 x (1177 f+905 g x)+d^3 e (3509 f+2865 g x)+1910 d^4 g+5 d e^3 x^2 (363 f+287 g x)+35 e^4 x^3 (11 f+9 g x)\right )\right )}{3465 c^5 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 367, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 315\,g{e}^{4}{x}^{4}{c}^{4}-280\,b{c}^{3}{e}^{4}g{x}^{3}+1435\,{c}^{4}d{e}^{3}g{x}^{3}+385\,{c}^{4}{e}^{4}f{x}^{3}+240\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}-1470\,b{c}^{3}d{e}^{3}g{x}^{2}-330\,b{c}^{3}{e}^{4}f{x}^{2}+2715\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}+1815\,{c}^{4}d{e}^{3}f{x}^{2}-192\,{b}^{3}c{e}^{4}gx+1368\,{b}^{2}{c}^{2}d{e}^{3}gx+264\,{b}^{2}{c}^{2}{e}^{4}fx-3348\,b{c}^{3}{d}^{2}{e}^{2}gx-1716\,b{c}^{3}d{e}^{3}fx+2865\,{c}^{4}{d}^{3}egx+3531\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-1040\,{b}^{3}cd{e}^{3}g-176\,{b}^{3}c{e}^{4}f+3144\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+1320\,{b}^{2}{c}^{2}d{e}^{3}f-4142\,b{c}^{3}{d}^{3}eg-3498\,b{c}^{3}{d}^{2}{e}^{2}f+1910\,{c}^{4}{d}^{4}g+3509\,f{d}^{3}{c}^{4}e \right ) }{3465\,{c}^{5}{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28503, size = 676, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} - 319 \, c^{4} d^{4} + 637 \, b c^{3} d^{3} e - 438 \, b^{2} c^{2} d^{2} e^{2} + 136 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \,{\left (26 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (52 \, c^{4} d^{2} e^{2} + 13 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} -{\left (2 \, c^{4} d^{3} e - 159 \, b c^{3} d^{2} e^{2} + 60 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{315 \,{\left (c^{4} e^{2} x + c^{4} d e\right )}} + \frac{2 \,{\left (315 \, c^{5} e^{5} x^{5} - 1910 \, c^{5} d^{5} + 6052 \, b c^{4} d^{4} e - 7286 \, b^{2} c^{3} d^{3} e^{2} + 4184 \, b^{3} c^{2} d^{2} e^{3} - 1168 \, b^{4} c d e^{4} + 128 \, b^{5} e^{5} + 35 \,{\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} x^{4} + 5 \,{\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \,{\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} x^{2} -{\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{3465 \,{\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99778, size = 1081, normalized size = 3.12 \begin{align*} \frac{2 \,{\left (315 \, c^{5} e^{5} g x^{5} + 35 \,{\left (11 \, c^{5} e^{5} f +{\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \,{\left (11 \,{\left (26 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} f +{\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 3 \,{\left (11 \,{\left (52 \, c^{5} d^{2} e^{3} + 13 \, b c^{4} d e^{4} - 2 \, b^{2} c^{3} e^{5}\right )} f +{\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \,{\left (319 \, c^{5} d^{4} e - 637 \, b c^{4} d^{3} e^{2} + 438 \, b^{2} c^{3} d^{2} e^{3} - 136 \, b^{3} c^{2} d e^{4} + 16 \, b^{4} c e^{5}\right )} f - 2 \,{\left (955 \, c^{5} d^{5} - 3026 \, b c^{4} d^{4} e + 3643 \, b^{2} c^{3} d^{3} e^{2} - 2092 \, b^{3} c^{2} d^{2} e^{3} + 584 \, b^{4} c d e^{4} - 64 \, b^{5} e^{5}\right )} g -{\left (11 \,{\left (2 \, c^{5} d^{3} e^{2} - 159 \, b c^{4} d^{2} e^{3} + 60 \, b^{2} c^{3} d e^{4} - 8 \, b^{3} c^{2} e^{5}\right )} f +{\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\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}}{3465 \,{\left (c^{5} e^{3} x + c^{5} 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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