Optimal. Leaf size=223 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-5 c d g+c e f)}{e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{(2 b e g-5 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 \sqrt{2 c d-b e}} \]
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Rubi [A] time = 0.358471, antiderivative size = 223, 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 = {792, 664, 660, 208} \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-5 c d g+c e f)}{e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{(2 b e g-5 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 \sqrt{2 c d-b e}} \]
Antiderivative was successfully verified.
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Rule 792
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)^{5/2}} \, dx &=-\frac{(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)^{5/2}}-\frac{(c e f-5 c d g+2 b e g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac{(c e f-5 c d g+2 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) \sqrt{d+e x}}-\frac{(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)^{5/2}}+\frac{((-2 c d+b e) (c e f-5 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{(c e f-5 c d g+2 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) \sqrt{d+e x}}-\frac{(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)^{5/2}}+(-c e f+5 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 )\\ &=-\frac{(c e f-5 c d g+2 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) \sqrt{d+e x}}-\frac{(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)^{5/2}}+\frac{(c e f-5 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 \sqrt{2 c d-b e}}\\ \end{align*}
Mathematica [A] time = 0.258885, size = 173, normalized size = 0.78 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left ((d+e x) \sqrt{2 c d-b e} (-2 b e g+5 c d g-c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )-(2 c d-b e) \sqrt{c (d-e x)-b e} (3 d g-e f+2 e g x)\right )}{e^2 (d+e x)^{3/2} (b e-2 c d) \sqrt{c (d-e x)-b e}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 359, normalized size = 1.6 \begin{align*}{\frac{1}{{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( -2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xb{e}^{2}g+5\,\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\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}xeg-2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bdeg+5\,\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+3\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}dg-\sqrt{be-2\,cd}\sqrt{-cex-be+cd}ef \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4976, size = 1374, normalized size = 6.16 \begin{align*} \left [\frac{{\left (c d^{2} e f +{\left (c e^{3} f -{\left (5 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} -{\left (5 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \,{\left (c d e^{2} f -{\left (5 \, 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 (2 \,{\left (2 \, c d e - b e^{2}\right )} g x -{\left (2 \, c d e - b e^{2}\right )} f + 3 \,{\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt{e x + d}}{2 \,{\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} +{\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \,{\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, \frac{{\left (c d^{2} e f +{\left (c e^{3} f -{\left (5 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} -{\left (5 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \,{\left (c d e^{2} f -{\left (5 \, 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 (2 \,{\left (2 \, c d e - b e^{2}\right )} g x -{\left (2 \, c d e - b e^{2}\right )} f + 3 \,{\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt{e x + d}}{2 \, c d^{3} e^{2} - b d^{2} e^{3} +{\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \,{\left (2 \, c d^{2} e^{3} - b d e^{4}\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{\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{5}{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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