Optimal. Leaf size=238 \[ \frac{2 e (f+g x)^{5/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 (f+g x)^{3/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac{2 \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{g^6}-\frac{2 c e^2 (f+g x)^{7/2} (5 e f-3 d g)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.268299, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {898, 1261} \[ \frac{2 e (f+g x)^{5/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 (f+g x)^{3/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac{2 \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{g^6}-\frac{2 c e^2 (f+g x)^{7/2} (5 e f-3 d g)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 898
Rule 1261
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+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+a g^2}{g^2}-\frac{2 c f 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 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{g^5}+\frac{(-e f+d g)^3 \left (c f^2+a g^2\right )}{g^5 x^2}+\frac{(e f-d g) \left (-3 a e^2 g^2-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 (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) x^4}{g^5}-\frac{c e^2 (5 e f-3 d 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+a g^2\right )}{g^6 \sqrt{f+g x}}+\frac{2 (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) \sqrt{f+g x}}{g^6}-\frac{2 (e f-d g) \left (3 a e^2 g^2+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 (a e^2 g^2+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 c e^2 (5 e f-3 d 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 = 0.246201, size = 207, normalized size = 0.87 \[ \frac{2 \left (63 e (f+g x)^3 \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )-105 (f+g x)^2 (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )+315 \left (a g^2+c f^2\right ) (e f-d g)^3+315 (f+g x) (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )-45 c e^2 (f+g x)^4 (5 e f-3 d g)+35 c e^3 (f+g x)^5\right )}{315 g^6 \sqrt{f+g x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 365, normalized size = 1.5 \begin{align*} -{\frac{-70\,{e}^{3}c{x}^{5}{g}^{5}-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\,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}-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+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}+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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03147, size = 451, normalized size = 1.89 \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 - 3 \, c d e^{2} g\right )}{\left (g x + f\right )}^{\frac{7}{2}} + 63 \,{\left (10 \, c e^{3} f^{2} - 12 \, c d e^{2} f g +{\left (3 \, c d^{2} e + a e^{3}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{5}{2}} - 105 \,{\left (10 \, c e^{3} f^{3} - 18 \, c d e^{2} f^{2} g + 3 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f g^{2} -{\left (c d^{3} + 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} - 12 \, c d e^{2} f^{3} g + 3 \, a d^{2} e g^{4} + 3 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{2} - 2 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{3}\right )} \sqrt{g x + f}}{g^{5}} + \frac{315 \,{\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5} +{\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} -{\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3}\right )}}{\sqrt{g x + f} g^{5}}\right )}}{315 \, g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75808, size = 749, normalized size = 3.15 \begin{align*} \frac{2 \,{\left (35 \, c e^{3} g^{5} x^{5} + 1280 \, c e^{3} f^{5} - 3456 \, c d e^{2} f^{4} g + 1890 \, a d^{2} e f g^{4} - 315 \, a d^{3} g^{5} + 1008 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} - 840 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3} - 5 \,{\left (10 \, c e^{3} f g^{4} - 27 \, c d e^{2} g^{5}\right )} x^{4} +{\left (80 \, c e^{3} f^{2} g^{3} - 216 \, c d e^{2} f g^{4} + 63 \,{\left (3 \, c d^{2} e + a e^{3}\right )} g^{5}\right )} x^{3} -{\left (160 \, c e^{3} f^{3} g^{2} - 432 \, c d e^{2} f^{2} g^{3} + 126 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f g^{4} - 105 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} +{\left (640 \, c e^{3} f^{4} g - 1728 \, c d e^{2} f^{3} g^{2} + 945 \, a d^{2} e g^{5} + 504 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{3} - 420 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{4}\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 56.0701, size = 328, normalized size = 1.38 \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 (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 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} + 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} - 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 (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{3}}{g^{6} \sqrt{f + g x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17236, size = 612, normalized size = 2.57 \begin{align*} -\frac{2 \,{\left (c d^{3} f^{2} g^{3} + a d^{3} g^{5} - 3 \, c d^{2} f^{3} g^{2} e - 3 \, a d^{2} f g^{4} e + 3 \, c d f^{4} g e^{2} + 3 \, a d f^{2} g^{3} e^{2} - c f^{5} e^{3} - a f^{3} g^{2} e^{3}\right )}}{\sqrt{g x + f} g^{6}} + \frac{2 \,{\left (105 \,{\left (g x + f\right )}^{\frac{3}{2}} c d^{3} g^{51} - 630 \, \sqrt{g x + f} c d^{3} f g^{51} + 189 \,{\left (g x + f\right )}^{\frac{5}{2}} c d^{2} g^{50} e - 945 \,{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} f g^{50} e + 2835 \, \sqrt{g x + f} c d^{2} f^{2} g^{50} e + 945 \, \sqrt{g x + f} a d^{2} g^{52} e + 135 \,{\left (g x + f\right )}^{\frac{7}{2}} c d g^{49} e^{2} - 756 \,{\left (g x + f\right )}^{\frac{5}{2}} c d f g^{49} e^{2} + 1890 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f^{2} g^{49} e^{2} - 3780 \, \sqrt{g x + f} c d f^{3} g^{49} e^{2} + 315 \,{\left (g x + f\right )}^{\frac{3}{2}} a d g^{51} e^{2} - 1890 \, \sqrt{g x + f} a d f g^{51} e^{2} + 35 \,{\left (g x + f\right )}^{\frac{9}{2}} c g^{48} e^{3} - 225 \,{\left (g x + f\right )}^{\frac{7}{2}} c f g^{48} e^{3} + 630 \,{\left (g x + f\right )}^{\frac{5}{2}} c f^{2} g^{48} e^{3} - 1050 \,{\left (g x + f\right )}^{\frac{3}{2}} c f^{3} g^{48} e^{3} + 1575 \, \sqrt{g x + f} c f^{4} g^{48} e^{3} + 63 \,{\left (g x + f\right )}^{\frac{5}{2}} a g^{50} e^{3} - 315 \,{\left (g x + f\right )}^{\frac{3}{2}} a f g^{50} e^{3} + 945 \, \sqrt{g x + f} a f^{2} g^{50} e^{3}\right )}}{315 \, g^{54}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]