Optimal. Leaf size=291 \[ \frac{2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{16 \sqrt{d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.412777, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{16 \sqrt{d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 788
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(c e f+3 c d g-2 b e g) \int \frac{(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (c e f+3 c d g-2 b e g) (d+e x)^{5/2}}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(4 (c e f+3 c d g-2 b e g)) \int \frac{(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{8 (c e f+3 c d g-2 b e g) (d+e x)^{3/2}}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (c e f+3 c d g-2 b e g) (d+e x)^{5/2}}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(8 (2 c d-b e) (c e f+3 c d g-2 b e g)) \int \frac{(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c^3 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{16 (2 c d-b e) (c e f+3 c d g-2 b e g) \sqrt{d+e x}}{3 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 (c e f+3 c d g-2 b e g) (d+e x)^{3/2}}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (c e f+3 c d g-2 b e g) (d+e x)^{5/2}}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.16805, size = 180, normalized size = 0.62 \[ \frac{2 \sqrt{d+e x} \left (8 b^2 c e^2 (8 d g+e (f-3 g x))-16 b^3 e^3 g-2 b c^2 e \left (41 d^2 g+2 d e (5 f-18 g x)+3 e^2 x (g x-2 f)\right )+c^3 \left (d^2 e (11 f-51 g x)+34 d^3 g+6 d e^2 x (2 g x-3 f)+e^3 x^2 (3 f+g x)\right )\right )}{3 c^4 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 235, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -g{e}^{3}{x}^{3}{c}^{3}+6\,b{c}^{2}{e}^{3}g{x}^{2}-12\,{c}^{3}d{e}^{2}g{x}^{2}-3\,{c}^{3}{e}^{3}f{x}^{2}+24\,{b}^{2}c{e}^{3}gx-72\,b{c}^{2}d{e}^{2}gx-12\,b{c}^{2}{e}^{3}fx+51\,{c}^{3}{d}^{2}egx+18\,{c}^{3}d{e}^{2}fx+16\,{b}^{3}{e}^{3}g-64\,{b}^{2}cd{e}^{2}g-8\,{b}^{2}c{e}^{3}f+82\,b{c}^{2}{d}^{2}eg+20\,b{c}^{2}d{e}^{2}f-34\,{c}^{3}{d}^{3}g-11\,{c}^{3}{d}^{2}ef \right ) }{3\,{c}^{4}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38911, size = 332, normalized size = 1.14 \begin{align*} \frac{2 \,{\left (3 \, c^{2} e^{2} x^{2} + 11 \, c^{2} d^{2} - 20 \, b c d e + 8 \, b^{2} e^{2} - 6 \,{\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \,{\left (c^{4} e^{2} x - c^{4} d e + b c^{3} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (c^{3} e^{3} x^{3} + 34 \, c^{3} d^{3} - 82 \, b c^{2} d^{2} e + 64 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 6 \,{\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \,{\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} g}{3 \,{\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )} \sqrt{-c e x + c d - b e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47718, size = 624, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (c^{3} e^{3} g x^{3} + 3 \,{\left (c^{3} e^{3} f + 2 \,{\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} +{\left (11 \, c^{3} d^{2} e - 20 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \,{\left (17 \, c^{3} d^{3} - 41 \, b c^{2} d^{2} e + 32 \, b^{2} c d e^{2} - 8 \, b^{3} e^{3}\right )} g - 3 \,{\left (2 \,{\left (3 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f +{\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}}{3 \,{\left (c^{6} e^{5} x^{3} + c^{6} d^{3} e^{2} - 2 \, b c^{5} d^{2} e^{3} + b^{2} c^{4} d e^{4} -{\left (c^{6} d e^{4} - 2 \, b c^{5} e^{5}\right )} x^{2} -{\left (c^{6} d^{2} e^{3} - b^{2} c^{4} e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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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