Optimal. Leaf size=186 \[ \frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 \sqrt{2 c d-b e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.351664, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {794, 664, 660, 208} \[ \frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 \sqrt{2 c d-b e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 664
Rule 660
Rule 208
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)^{3/2}} \, dx &=-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}-\frac{\left (2 \left (\frac{3}{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{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{3 c e^3}\\ &=\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}+\frac{((2 c d-b e) (e f-d g)) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e}\\ &=\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}+(2 (2 c d-b e) (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}}\right )\\ &=\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}-\frac{2 \sqrt{2 c d-b e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{2 c d-b e} \sqrt{d+e x}}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.195818, size = 154, normalized size = 0.83 \[ \frac{2 \sqrt{d+e x} \sqrt{c (d-e x)-b e} \left (\sqrt{c (d-e x)-b e} (b e g+c (-4 d g+3 e f+e g x))+3 c \sqrt{2 c d-b e} (d g-e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )\right )}{3 c e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 330, normalized size = 1.8 \begin{align*}{\frac{2}{3\,c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bcdeg-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{e}^{2}f-6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}g+6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}def+xceg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+beg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-4\,cdg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+3\,fce\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59348, size = 907, normalized size = 4.88 \begin{align*} \left [-\frac{3 \,{\left (c d e f - c d^{2} g +{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{2 \, c d - b e} \log \left (-\frac{c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \,{\left (c d e - b e^{2}\right )} x - 2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{2 \, c d - b e} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e g x + 3 \, c e f -{\left (4 \, c d - b e\right )} g\right )} \sqrt{e x + d}}{3 \,{\left (c e^{3} x + c d e^{2}\right )}}, -\frac{2 \,{\left (3 \,{\left (c d e f - c d^{2} g +{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{-2 \, c d + b e} \arctan \left (\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-2 \, c d + b e} \sqrt{e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e g x + 3 \, c e f -{\left (4 \, c d - b e\right )} g\right )} \sqrt{e x + d}\right )}}{3 \,{\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 )^{\frac{3}{2}}}\, 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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