Optimal. Leaf size=200 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-4 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{(b e g-4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt{c} e^2} \]
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Rubi [A] time = 0.338962, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {792, 664, 621, 204} \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-4 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{(b e g-4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt{c} e^2} \]
Antiderivative was successfully verified.
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Rule 792
Rule 664
Rule 621
Rule 204
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}}{(d+e x)^2} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{(2 c e f-4 c d g+b e g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{e (2 c d-b e)}\\ &=-\frac{(2 c e f-4 c d g+b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^2}+\frac{((-2 c d+b e) (2 c e f-4 c d g+b e g)) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac{(2 c e f-4 c d g+b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^2}+\frac{((-2 c d+b e) (2 c e f-4 c d g+b e g)) \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 )}{e (2 c d-b e)}\\ &=-\frac{(2 c e f-4 c d g+b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{(2 c e f-4 c d g+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 )}{2 \sqrt{c} e^2}\\ \end{align*}
Mathematica [A] time = 0.790027, size = 185, normalized size = 0.92 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{\sqrt{e} (b e-2 c d) (3 d g-2 e f+e g x)}{d+e x}-\frac{\sqrt{e (2 c d-b e)} (b e g-4 c d g+2 c e f) \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )}{\sqrt{c} \sqrt{d+e x} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}\right )}{e^{5/2} (2 c d-b e)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 880, normalized size = 4.4 \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 = 5.87423, size = 861, normalized size = 4.3 \begin{align*} \left [-\frac{{\left (2 \, c d e f -{\left (4 \, c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\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 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e g x - 2 \, c e f + 3 \, c d g\right )}}{4 \,{\left (c e^{3} x + c d e^{2}\right )}}, \frac{{\left (2 \, c d e f -{\left (4 \, c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\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 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e g x - 2 \, c e f + 3 \, c d g\right )}}{2 \,{\left (c e^{3} x + c d e^{2}\right )}}\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 )}{\left (d + e x\right )^{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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