Optimal. Leaf size=118 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt{d+e x}} \]
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Rubi [A] time = 0.174452, 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} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^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 )^{3/2}}{5 c e^2 \sqrt{d+e x}}-\frac{\left (2 \left (\frac{3}{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{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}} \, dx}{5 c e^3}\\ &=-\frac{2 (5 c e f-c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 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 )^{3/2}}{5 c e^2 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0698958, size = 76, normalized size = 0.64 \[ \frac{2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+5 e f+3 e g x)-2 b e g)}{15 c^2 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 79, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -3\,cegx+2\,beg-2\,cdg-5\,cef \right ) }{15\,{c}^{2}{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 = 1.16521, size = 151, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (c e x - c d + b e\right )} \sqrt{-c e x + c d - b e} f}{3 \, c e} + \frac{2 \,{\left (3 \, c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} -{\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{15 \, c^{2} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7, size = 285, normalized size = 2.42 \begin{align*} \frac{2 \,{\left (3 \, c^{2} e^{2} g x^{2} - 5 \,{\left (c^{2} d e - b c e^{2}\right )} f - 2 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} g +{\left (5 \, c^{2} e^{2} f -{\left (c^{2} d e - b c e^{2}\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}}{15 \,{\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{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \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(-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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