Optimal. Leaf size=153 \[ -\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac{(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \]
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Rubi [A] time = 0.228531, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {792, 660, 208} \[ -\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac{(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 660
Rule 208
Rubi steps
\begin{align*} \int \frac{f+g x}{(d+e x)^{3/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{(c e f+3 c d g-2 b e 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}{2 e (2 c d-b e)}\\ &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{(c e f+3 c d g-2 b e 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 )}{2 c d-b e}\\ &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac{(c e f+3 c d g-2 b e 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 (2 c d-b e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.183669, size = 153, normalized size = 1. \[ \frac{(e f-d g) (b e-c d+c e x)-\frac{(d+e x) \sqrt{c (d-e x)-b e} (-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{\sqrt{2 c d-b e}}}{e^2 \sqrt{d+e x} (2 c d-b e) \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 328, normalized size = 2.1 \begin{align*}{\frac{1}{{e}^{2}} \left ( 2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xb{e}^{2}g-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xcdeg-\arctan \left ({\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ) xc{e}^{2}f+2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bdeg-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) c{d}^{2}g-\arctan \left ({\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ) cdef-\sqrt{be-2\,cd}\sqrt{-cex-be+cd}dg+\sqrt{be-2\,cd}\sqrt{-cex-be+cd}ef \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( be-2\,cd \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \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{g x + f}{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\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 [B] time = 1.51393, size = 1462, normalized size = 9.56 \begin{align*} \left [\frac{{\left (c d^{2} e f +{\left (c e^{3} f +{\left (3 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} +{\left (3 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \,{\left (c d e^{2} f +{\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} 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 ({\left (2 \, c d e - b e^{2}\right )} f -{\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt{e x + d}}{2 \,{\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4} +{\left (4 \, c^{2} d^{2} e^{4} - 4 \, b c d e^{5} + b^{2} e^{6}\right )} x^{2} + 2 \,{\left (4 \, c^{2} d^{3} e^{3} - 4 \, b c d^{2} e^{4} + b^{2} d e^{5}\right )} x\right )}}, -\frac{{\left (c d^{2} e f +{\left (c e^{3} f +{\left (3 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} +{\left (3 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \,{\left (c d e^{2} f +{\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} 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 ({\left (2 \, c d e - b e^{2}\right )} f -{\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt{e x + d}}{4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4} +{\left (4 \, c^{2} d^{2} e^{4} - 4 \, b c d e^{5} + b^{2} e^{6}\right )} x^{2} + 2 \,{\left (4 \, c^{2} d^{3} e^{3} - 4 \, b c d^{2} e^{4} + b^{2} d e^{5}\right )} x}\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{f + g x}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e 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] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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