Optimal. Leaf size=193 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.339573, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {794, 656, 648} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}} \]
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}}{(d+e x)^{3/2}} \, dx &=-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}-\frac{\left (2 \left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac{3}{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}}{(d+e x)^{3/2}} \, dx}{11 c e^3}\\ &=-\frac{2 (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}+\frac{(2 (2 c d-b e) (11 c e f-3 c d g-4 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)^{5/2}} \, dx}{99 c^2 e}\\ &=-\frac{4 (2 c d-b e) (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{693 c^3 e^2 (d+e x)^{7/2}}-\frac{2 (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.163536, size = 121, normalized size = 0.63 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (19 d g+11 e f+14 e g x)+c^2 \left (30 d^2 g+d e (121 f+105 g x)+7 e^2 x (11 f+9 g x)\right )\right )}{693 c^3 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 139, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 63\,g{x}^{2}{c}^{2}{e}^{2}-28\,bc{e}^{2}gx+105\,{c}^{2}degx+77\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-38\,bcdeg-22\,bc{e}^{2}f+30\,{c}^{2}{d}^{2}g+121\,{c}^{2}def \right ) }{693\,{c}^{3}{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 = 1.27568, size = 628, normalized size = 3.25 \begin{align*} \frac{2 \,{\left (7 \, c^{4} e^{4} x^{4} - 11 \, c^{4} d^{4} + 35 \, b c^{3} d^{3} e - 39 \, b^{2} c^{2} d^{2} e^{2} + 17 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} -{\left (10 \, c^{4} d e^{3} - 19 \, b c^{3} e^{4}\right )} x^{3} - 3 \,{\left (4 \, c^{4} d^{2} e^{2} + b c^{3} d e^{3} - 5 \, b^{2} c^{2} e^{4}\right )} x^{2} +{\left (26 \, c^{4} d^{3} e - 51 \, b c^{3} d^{2} e^{2} + 24 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{63 \, c^{2} e} + \frac{2 \,{\left (63 \, c^{5} e^{5} x^{5} - 30 \, c^{5} d^{5} + 128 \, b c^{4} d^{4} e - 212 \, b^{2} c^{3} d^{3} e^{2} + 168 \, b^{3} c^{2} d^{2} e^{3} - 62 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \,{\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} -{\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \,{\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} -{\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{693 \, c^{3} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50579, size = 1044, normalized size = 5.41 \begin{align*} \frac{2 \,{\left (63 \, c^{5} e^{5} g x^{5} + 7 \,{\left (11 \, c^{5} e^{5} f -{\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} g\right )} x^{4} -{\left (11 \,{\left (10 \, c^{5} d e^{4} - 19 \, b c^{4} e^{5}\right )} f +{\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \,{\left (11 \,{\left (4 \, c^{5} d^{2} e^{3} + b c^{4} d e^{4} - 5 \, b^{2} c^{3} e^{5}\right )} f -{\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \,{\left (11 \, c^{5} d^{4} e - 35 \, b c^{4} d^{3} e^{2} + 39 \, b^{2} c^{3} d^{2} e^{3} - 17 \, b^{3} c^{2} d e^{4} + 2 \, b^{4} c e^{5}\right )} f - 2 \,{\left (15 \, c^{5} d^{5} - 64 \, b c^{4} d^{4} e + 106 \, b^{2} c^{3} d^{3} e^{2} - 84 \, b^{3} c^{2} d^{2} e^{3} + 31 \, b^{4} c d e^{4} - 4 \, b^{5} e^{5}\right )} g +{\left (11 \,{\left (26 \, c^{5} d^{3} e^{2} - 51 \, b c^{4} d^{2} e^{3} + 24 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} f -{\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, 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}}{693 \,{\left (c^{3} e^{3} x + c^{3} 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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