Optimal. Leaf size=193 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt{d+e x}} \]
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Rubi [A] time = 0.307681, 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} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt{d+e x}} \]
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 )^{3/2}}{\sqrt{d+e x}} \, dx &=-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt{d+e x}}-\frac{\left (2 \left (\frac{5}{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 )^{3/2}}{\sqrt{d+e x}} \, dx}{9 c e^3}\\ &=-\frac{2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 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 )^{5/2}}{9 c e^2 \sqrt{d+e x}}+\frac{(2 (2 c d-b e) (9 c e f-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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{63 c^2 e}\\ &=-\frac{4 (2 c d-b e) (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{315 c^3 e^2 (d+e x)^{5/2}}-\frac{2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 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 )^{5/2}}{9 c e^2 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.128084, size = 121, normalized size = 0.63 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (17 d g+9 e f+10 e g x)+c^2 \left (26 d^2 g+d e (81 f+65 g x)+5 e^2 x (9 f+7 g x)\right )\right )}{315 c^3 e^2 \sqrt{d+e x}} \]
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 ( 35\,g{x}^{2}{c}^{2}{e}^{2}-20\,bc{e}^{2}gx+65\,{c}^{2}degx+45\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-34\,bcdeg-18\,bc{e}^{2}f+26\,{c}^{2}{d}^{2}g+81\,{c}^{2}def \right ) }{315\,{c}^{3}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17709, size = 432, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (5 \, c^{3} e^{3} x^{3} + 9 \, c^{3} d^{3} - 20 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} -{\left (c^{3} d e^{2} - 8 \, b c^{2} e^{3}\right )} x^{2} -{\left (13 \, c^{3} d^{2} e - 12 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{35 \, c^{2} e} - \frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} + 26 \, c^{4} d^{4} - 86 \, b c^{3} d^{3} e + 102 \, b^{2} c^{2} d^{2} e^{2} - 50 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} - 5 \,{\left (c^{4} d e^{3} - 10 \, b c^{3} e^{4}\right )} x^{3} - 3 \,{\left (23 \, c^{4} d^{2} e^{2} - 22 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} +{\left (13 \, c^{4} d^{3} e - 30 \, b c^{3} d^{2} e^{2} + 21 \, b^{2} c^{2} d e^{3} - 4 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{315 \, c^{3} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.40656, size = 729, normalized size = 3.78 \begin{align*} -\frac{2 \,{\left (35 \, c^{4} e^{4} g x^{4} + 5 \,{\left (9 \, c^{4} e^{4} f -{\left (c^{4} d e^{3} - 10 \, b c^{3} e^{4}\right )} g\right )} x^{3} - 3 \,{\left (3 \,{\left (c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} f +{\left (23 \, c^{4} d^{2} e^{2} - 22 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} g\right )} x^{2} + 9 \,{\left (9 \, c^{4} d^{3} e - 20 \, b c^{3} d^{2} e^{2} + 13 \, b^{2} c^{2} d e^{3} - 2 \, b^{3} c e^{4}\right )} f + 2 \,{\left (13 \, c^{4} d^{4} - 43 \, b c^{3} d^{3} e + 51 \, b^{2} c^{2} d^{2} e^{2} - 25 \, b^{3} c d e^{3} + 4 \, b^{4} e^{4}\right )} g -{\left (9 \,{\left (13 \, c^{4} d^{2} e^{2} - 12 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} f -{\left (13 \, c^{4} d^{3} e - 30 \, b c^{3} d^{2} e^{2} + 21 \, b^{2} c^{2} d e^{3} - 4 \, b^{3} c e^{4}\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}}{315 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (f + g x\right )}{\sqrt{d + e x}}\, dx \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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