Optimal. Leaf size=118 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.191122, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {794, 648} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 794
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}}{(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 )^{5/2}}{7 c e^2 (d+e x)^{3/2}}-\frac{\left (2 \left (\frac{5}{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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{7 c e^3}\\ &=-\frac{2 (7 c e f-3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 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 )^{5/2}}{7 c e^2 (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.087304, size = 78, normalized size = 0.66 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+7 e f+5 e g x)-2 b e g)}{35 c^2 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 79, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -5\,cegx+2\,beg-2\,cdg-7\,cef \right ) }{35\,{c}^{2}{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 = 1.20386, size = 266, normalized size = 2.25 \begin{align*} -\frac{2 \,{\left (c^{2} e^{2} x^{2} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \,{\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{5 \, c e} - \frac{2 \,{\left (5 \, c^{3} e^{3} x^{3} + 2 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - 8 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} +{\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{35 \, c^{2} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36727, size = 458, normalized size = 3.88 \begin{align*} -\frac{2 \,{\left (5 \, c^{3} e^{3} g x^{3} +{\left (7 \, c^{3} e^{3} f - 8 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \,{\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + 2 \,{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g -{\left (14 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} f -{\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\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}}{35 \,{\left (c^{2} e^{3} x + c^{2} 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 )}{\left (d + e x\right )^{\frac{3}{2}}}\, 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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