Optimal. Leaf size=270 \[ -\frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{35 c^2 e^2}-\frac{8 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac{16 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d+e x}}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \]
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Rubi [A] time = 0.455283, antiderivative size = 270, 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 = {794, 656, 648} \[ -\frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{35 c^2 e^2}-\frac{8 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac{16 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d+e x}}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2} (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}-\frac{\left (2 \left (\frac{1}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{7 c e^3}\\ &=-\frac{2 (7 c e f+5 c d g-6 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}+\frac{(4 (2 c d-b e) (7 c e f+5 c d g-6 b e g)) \int \frac{(d+e x)^{3/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{35 c^2 e}\\ &=-\frac{8 (2 c d-b e) (7 c e f+5 c d g-6 b e g) \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{105 c^3 e^2}-\frac{2 (7 c e f+5 c d g-6 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}+\frac{\left (8 (2 c d-b e)^2 (7 c e f+5 c d g-6 b e g)\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (7 c e f+5 c d g-6 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{105 c^4 e^2 \sqrt{d+e x}}-\frac{8 (2 c d-b e) (7 c e f+5 c d g-6 b e g) \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{105 c^3 e^2}-\frac{2 (7 c e f+5 c d g-6 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\\ \end{align*}
Mathematica [A] time = 0.171183, size = 181, normalized size = 0.67 \[ \frac{2 \sqrt{d+e x} (b e-c d+c e x) \left (8 b^2 c e^2 (32 d g+7 e f+3 e g x)-48 b^3 e^3 g-2 b c^2 e \left (219 d^2 g+2 d e (63 f+26 g x)+e^2 x (14 f+9 g x)\right )+c^3 \left (d^2 e (301 f+115 g x)+230 d^3 g+2 d e^2 x (49 f+30 g x)+3 e^3 x^2 (7 f+5 g x)\right )\right )}{105 c^4 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 235, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -15\,g{e}^{3}{x}^{3}{c}^{3}+18\,b{c}^{2}{e}^{3}g{x}^{2}-60\,{c}^{3}d{e}^{2}g{x}^{2}-21\,{c}^{3}{e}^{3}f{x}^{2}-24\,{b}^{2}c{e}^{3}gx+104\,b{c}^{2}d{e}^{2}gx+28\,b{c}^{2}{e}^{3}fx-115\,{c}^{3}{d}^{2}egx-98\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-256\,{b}^{2}cd{e}^{2}g-56\,{b}^{2}c{e}^{3}f+438\,b{c}^{2}{d}^{2}eg+252\,b{c}^{2}d{e}^{2}f-230\,{c}^{3}{d}^{3}g-301\,{c}^{3}{d}^{2}ef \right ) }{105\,{c}^{4}{e}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14473, size = 431, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (3 \, c^{3} e^{3} x^{3} - 43 \, c^{3} d^{3} + 79 \, b c^{2} d^{2} e - 44 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} +{\left (11 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} +{\left (29 \, c^{3} d^{2} e - 18 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} f}{15 \, \sqrt{-c e x + c d - b e} c^{3} e} + \frac{2 \,{\left (15 \, c^{4} e^{4} x^{4} - 230 \, c^{4} d^{4} + 668 \, b c^{3} d^{3} e - 694 \, b^{2} c^{2} d^{2} e^{2} + 304 \, b^{3} c d e^{3} - 48 \, b^{4} e^{4} + 3 \,{\left (15 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} +{\left (55 \, c^{4} d^{2} e^{2} - 26 \, b c^{3} d e^{3} + 6 \, b^{2} c^{2} e^{4}\right )} x^{2} +{\left (115 \, c^{4} d^{3} e - 219 \, b c^{3} d^{2} e^{2} + 128 \, b^{2} c^{2} d e^{3} - 24 \, b^{3} c e^{4}\right )} x\right )} g}{105 \, \sqrt{-c e x + c d - b e} c^{4} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37693, size = 510, normalized size = 1.89 \begin{align*} -\frac{2 \,{\left (15 \, c^{3} e^{3} g x^{3} + 3 \,{\left (7 \, c^{3} e^{3} f + 2 \,{\left (10 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \,{\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \,{\left (115 \, c^{3} d^{3} - 219 \, b c^{2} d^{2} e + 128 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g +{\left (14 \,{\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f +{\left (115 \, c^{3} d^{2} e - 104 \, b c^{2} d e^{2} + 24 \, 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}}{105 \,{\left (c^{4} e^{3} x + c^{4} d e^{2}\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{\left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, 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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