Optimal. Leaf size=285 \[ -\frac{2 e (f+g x)^{5/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}+\frac{2 (f+g x)^{3/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac{2 (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac{2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6} \]
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Rubi [A] time = 0.405313, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {897, 1261} \[ -\frac{2 e (f+g x)^{5/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}+\frac{2 (f+g x)^{3/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac{2 (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac{2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6} \]
Antiderivative was successfully verified.
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Rule 897
Rule 1261
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^3 \left (\frac{c f^2-b f g+a g^2}{g^2}-\frac{(2 c f-b g) x^2}{g^2}+\frac{c x^4}{g^2}\right )}{x^2} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{(e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g))}{g^5}+\frac{(-e f+d g)^3 \left (c f^2-b f g+a g^2\right )}{g^5 x^2}+\frac{(e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^2}{g^5}+\frac{e \left (-e g (4 b e f-3 b d g-a e g)+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) x^4}{g^5}+\frac{e^2 (-5 c e f+3 c d g+b e g) x^6}{g^5}+\frac{c e^3 x^8}{g^5}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 (e f-d g)^3 \left (c f^2-b f g+a g^2\right )}{g^6 \sqrt{f+g x}}+\frac{2 (e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g)) \sqrt{f+g x}}{g^6}+\frac{2 (e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^6}-\frac{2 e \left (e g (4 b e f-3 b d g-a e g)-c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac{2 e^2 (5 c e f-3 c d g-b e g) (f+g x)^{7/2}}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6}\\ \end{align*}
Mathematica [A] time = 0.747241, size = 249, normalized size = 0.87 \[ \frac{2 \left (-63 e (f+g x)^3 \left (c \left (-3 d^2 g^2+12 d e f g-10 e^2 f^2\right )-e g (a e g+3 b d g-4 b e f)\right )+105 (f+g x)^2 (e f-d g) \left (-3 e g (a e g+b d g-2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )+315 (e f-d g)^3 \left (g (a g-b f)+c f^2\right )+315 (f+g x) (e f-d g)^2 (g (3 a e g+b d g-4 b e f)+c f (5 e f-2 d g))-45 e^2 (f+g x)^4 (-b e g-3 c d g+5 c e f)+35 c e^3 (f+g x)^5\right )}{315 g^6 \sqrt{f+g x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 540, normalized size = 1.9 \begin{align*} -{\frac{-70\,{e}^{3}c{x}^{5}{g}^{5}-90\,b{e}^{3}{g}^{5}{x}^{4}-270\,cd{e}^{2}{g}^{5}{x}^{4}+100\,c{e}^{3}f{g}^{4}{x}^{4}-126\,a{e}^{3}{g}^{5}{x}^{3}-378\,bd{e}^{2}{g}^{5}{x}^{3}+144\,b{e}^{3}f{g}^{4}{x}^{3}-378\,c{d}^{2}e{g}^{5}{x}^{3}+432\,cd{e}^{2}f{g}^{4}{x}^{3}-160\,c{e}^{3}{f}^{2}{g}^{3}{x}^{3}-630\,ad{e}^{2}{g}^{5}{x}^{2}+252\,a{e}^{3}f{g}^{4}{x}^{2}-630\,b{d}^{2}e{g}^{5}{x}^{2}+756\,bd{e}^{2}f{g}^{4}{x}^{2}-288\,b{e}^{3}{f}^{2}{g}^{3}{x}^{2}-210\,c{d}^{3}{g}^{5}{x}^{2}+756\,c{d}^{2}ef{g}^{4}{x}^{2}-864\,cd{e}^{2}{f}^{2}{g}^{3}{x}^{2}+320\,c{e}^{3}{f}^{3}{g}^{2}{x}^{2}-1890\,a{d}^{2}e{g}^{5}x+2520\,ad{e}^{2}f{g}^{4}x-1008\,a{e}^{3}{f}^{2}{g}^{3}x-630\,b{d}^{3}{g}^{5}x+2520\,b{d}^{2}ef{g}^{4}x-3024\,bd{e}^{2}{f}^{2}{g}^{3}x+1152\,b{e}^{3}{f}^{3}{g}^{2}x+840\,c{d}^{3}f{g}^{4}x-3024\,c{d}^{2}e{f}^{2}{g}^{3}x+3456\,cd{e}^{2}{f}^{3}{g}^{2}x-1280\,c{e}^{3}{f}^{4}gx+630\,{d}^{3}a{g}^{5}-3780\,a{d}^{2}ef{g}^{4}+5040\,ad{e}^{2}{f}^{2}{g}^{3}-2016\,a{e}^{3}{f}^{3}{g}^{2}-1260\,b{d}^{3}f{g}^{4}+5040\,b{d}^{2}e{f}^{2}{g}^{3}-6048\,bd{e}^{2}{f}^{3}{g}^{2}+2304\,b{e}^{3}{f}^{4}g+1680\,c{d}^{3}{f}^{2}{g}^{3}-6048\,c{d}^{2}e{f}^{3}{g}^{2}+6912\,cd{e}^{2}{f}^{4}g-2560\,c{e}^{3}{f}^{5}}{315\,{g}^{6}}{\frac{1}{\sqrt{gx+f}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978479, size = 590, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (g x + f\right )}^{\frac{9}{2}} c e^{3} - 45 \,{\left (5 \, c e^{3} f -{\left (3 \, c d e^{2} + b e^{3}\right )} g\right )}{\left (g x + f\right )}^{\frac{7}{2}} + 63 \,{\left (10 \, c e^{3} f^{2} - 4 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f g +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{5}{2}} - 105 \,{\left (10 \, c e^{3} f^{3} - 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g + 3 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{2} -{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{3}\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, c e^{3} f^{4} - 4 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g + 3 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{2} - 2 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{3} +{\left (b d^{3} + 3 \, a d^{2} e\right )} g^{4}\right )} \sqrt{g x + f}}{g^{5}} + \frac{315 \,{\left (c e^{3} f^{5} - a d^{3} g^{5} -{\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} -{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} +{\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4}\right )}}{\sqrt{g x + f} g^{5}}\right )}}{315 \, g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55055, size = 971, normalized size = 3.41 \begin{align*} \frac{2 \,{\left (35 \, c e^{3} g^{5} x^{5} + 1280 \, c e^{3} f^{5} - 315 \, a d^{3} g^{5} - 1152 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g + 1008 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} - 840 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} + 630 \,{\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4} - 5 \,{\left (10 \, c e^{3} f g^{4} - 9 \,{\left (3 \, c d e^{2} + b e^{3}\right )} g^{5}\right )} x^{4} +{\left (80 \, c e^{3} f^{2} g^{3} - 72 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f g^{4} + 63 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{5}\right )} x^{3} -{\left (160 \, c e^{3} f^{3} g^{2} - 144 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g^{3} + 126 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{4} - 105 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} +{\left (640 \, c e^{3} f^{4} g - 576 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g^{2} + 504 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{3} - 420 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{4} + 315 \,{\left (b d^{3} + 3 \, a d^{2} e\right )} g^{5}\right )} x\right )} \sqrt{g x + f}}{315 \,{\left (g^{7} x + f g^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 159.296, size = 452, normalized size = 1.59 \begin{align*} \frac{2 c e^{3} \left (f + g x\right )^{\frac{9}{2}}}{9 g^{6}} + \frac{\left (f + g x\right )^{\frac{7}{2}} \left (2 b e^{3} g + 6 c d e^{2} g - 10 c e^{3} f\right )}{7 g^{6}} + \frac{\left (f + g x\right )^{\frac{5}{2}} \left (2 a e^{3} g^{2} + 6 b d e^{2} g^{2} - 8 b e^{3} f g + 6 c d^{2} e g^{2} - 24 c d e^{2} f g + 20 c e^{3} f^{2}\right )}{5 g^{6}} + \frac{\left (f + g x\right )^{\frac{3}{2}} \left (6 a d e^{2} g^{3} - 6 a e^{3} f g^{2} + 6 b d^{2} e g^{3} - 18 b d e^{2} f g^{2} + 12 b e^{3} f^{2} g + 2 c d^{3} g^{3} - 18 c d^{2} e f g^{2} + 36 c d e^{2} f^{2} g - 20 c e^{3} f^{3}\right )}{3 g^{6}} + \frac{\sqrt{f + g x} \left (6 a d^{2} e g^{4} - 12 a d e^{2} f g^{3} + 6 a e^{3} f^{2} g^{2} + 2 b d^{3} g^{4} - 12 b d^{2} e f g^{3} + 18 b d e^{2} f^{2} g^{2} - 8 b e^{3} f^{3} g - 4 c d^{3} f g^{3} + 18 c d^{2} e f^{2} g^{2} - 24 c d e^{2} f^{3} g + 10 c e^{3} f^{4}\right )}{g^{6}} - \frac{2 \left (d g - e f\right )^{3} \left (a g^{2} - b f g + c f^{2}\right )}{g^{6} \sqrt{f + g x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18099, size = 903, normalized size = 3.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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