Optimal. Leaf size=265 \[ -\frac{(2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{8 c^3 e^2}-\frac{(d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{12 c^2 e^2}+\frac{(2 c d-b e)^2 (-5 b e g+4 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{7/2} e^2}-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]
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Rubi [A] time = 0.353908, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1638, 12, 670, 640, 621, 204} \[ -\frac{(2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{8 c^3 e^2}-\frac{(d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{12 c^2 e^2}+\frac{(2 c d-b e)^2 (-5 b e g+4 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{7/2} e^2}-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]
Antiderivative was successfully verified.
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Rule 1638
Rule 12
Rule 670
Rule 640
Rule 621
Rule 204
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}-\frac{\int -\frac{e^2 (6 c e f+4 c d g-5 b e g) (d+e x)^2}{2 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{3 c e^3}\\ &=-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac{(6 c e f+4 c d g-5 b e g) \int \frac{(d+e x)^2}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{6 c e}\\ &=-\frac{(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac{((2 c d-b e) (6 c e f+4 c d g-5 b e g)) \int \frac{d+e x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 c^2 e}\\ &=-\frac{(2 c d-b e) (6 c e f+4 c d g-5 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c^3 e^2}-\frac{(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac{\left ((2 c d-b e)^2 (6 c e f+4 c d g-5 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 c^3 e}\\ &=-\frac{(2 c d-b e) (6 c e f+4 c d g-5 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c^3 e^2}-\frac{(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac{\left ((2 c d-b e)^2 (6 c e f+4 c d g-5 b e g)\right ) \operatorname{Subst}\left (\int \frac{1}{-4 c e^2-x^2} \, dx,x,\frac{-b e^2-2 c e^2 x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{8 c^3 e}\\ &=-\frac{(2 c d-b e) (6 c e f+4 c d g-5 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c^3 e^2}-\frac{(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac{g (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac{(2 c d-b e)^2 (6 c e f+4 c d g-5 b e g) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{7/2} e^2}\\ \end{align*}
Mathematica [A] time = 1.42699, size = 251, normalized size = 0.95 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{e^7 (-5 b e g+4 c d g+6 c e f) \left (3 \sqrt{c} \sqrt{e} \sqrt{d+e x} (b e-2 c d)^2 \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )-c (d+e x) \sqrt{e (2 c d-b e)} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}} (-3 b e+8 c d+2 c e x)\right )}{\sqrt{e (2 c d-b e)} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}-8 c^3 e^7 g (d+e x)^3\right )}{24 c^4 e^9 (d+e x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 786, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55719, size = 1266, normalized size = 4.78 \begin{align*} \left [\frac{3 \,{\left (6 \,{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f +{\left (16 \, c^{3} d^{3} - 36 \, b c^{2} d^{2} e + 24 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g\right )} \sqrt{-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e x + b e\right )} \sqrt{-c}\right ) - 4 \,{\left (8 \, c^{3} e^{2} g x^{2} + 6 \,{\left (8 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f +{\left (40 \, c^{3} d^{2} - 52 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + 2 \,{\left (6 \, c^{3} e^{2} f +{\left (12 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{96 \, c^{4} e^{2}}, -\frac{3 \,{\left (6 \,{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f +{\left (16 \, c^{3} d^{3} - 36 \, b c^{2} d^{2} e + 24 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e x + b e\right )} \sqrt{c}}{2 \,{\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \,{\left (8 \, c^{3} e^{2} g x^{2} + 6 \,{\left (8 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f +{\left (40 \, c^{3} d^{2} - 52 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + 2 \,{\left (6 \, c^{3} e^{2} f +{\left (12 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, c^{4} 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 (d + e x\right )^{2} \left (f + g x\right )}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26881, size = 356, normalized size = 1.34 \begin{align*} -\frac{1}{24} \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left (2 \,{\left (\frac{4 \, g x}{c} + \frac{{\left (12 \, c^{2} d g e^{2} + 6 \, c^{2} f e^{3} - 5 \, b c g e^{3}\right )} e^{\left (-3\right )}}{c^{3}}\right )} x + \frac{{\left (40 \, c^{2} d^{2} g e + 48 \, c^{2} d f e^{2} - 52 \, b c d g e^{2} - 18 \, b c f e^{3} + 15 \, b^{2} g e^{3}\right )} e^{\left (-3\right )}}{c^{3}}\right )} + \frac{{\left (16 \, c^{3} d^{3} g + 24 \, c^{3} d^{2} f e - 36 \, b c^{2} d^{2} g e - 24 \, b c^{2} d f e^{2} + 24 \, b^{2} c d g e^{2} + 6 \, b^{2} c f e^{3} - 5 \, b^{3} g e^{3}\right )} \sqrt{-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{16 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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