Integrand size = 27, antiderivative size = 285 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 (e f-d g)^3 \left (c f^2-b f g+a g^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 (e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{g^6}+\frac {2 (e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^6}-\frac {2 e \left (e g (4 b e f-3 b d g-a e g)-c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac {2 e^2 (5 c e f-3 c d g-b e g) (f+g x)^{7/2}}{7 g^6}+\frac {2 c e^3 (f+g x)^{9/2}}{9 g^6} \]
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Time = 0.24 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {911, 1275} \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 e (f+g x)^{5/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}+\frac {2 (f+g x)^{3/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac {2 (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 \sqrt {f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac {2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{7 g^6}+\frac {2 c e^3 (f+g x)^{9/2}}{9 g^6} \]
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Rule 911
Rule 1275
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {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 \text {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*}
Time = 0.33 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \left (c \left (105 d^3 g^3 \left (-8 f^2-4 f g x+g^2 x^2\right )+189 d^2 e g^2 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+27 d e^2 g \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )+5 e^3 \left (256 f^5+128 f^4 g x-32 f^3 g^2 x^2+16 f^2 g^3 x^3-10 f g^4 x^4+7 g^5 x^5\right )\right )+9 g \left (7 a g \left (-5 d^3 g^3+15 d^2 e g^2 (2 f+g x)+5 d e^2 g \left (-8 f^2-4 f g x+g^2 x^2\right )+e^3 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )+b \left (35 d^3 g^3 (2 f+g x)+35 d^2 e g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+21 d e^2 g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+e^3 \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )\right )\right )\right )}{315 g^6 \sqrt {f+g x}} \]
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Time = 0.57 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {\left (\frac {5}{9} c \,x^{2}+\frac {5}{7} b x +a \right ) x^{3} e^{3}}{5}-d \left (\frac {3}{7} c \,x^{2}+\frac {3}{5} b x +a \right ) x^{2} e^{2}-3 d^{2} \left (\frac {1}{5} c \,x^{2}+\frac {1}{3} b x +a \right ) x e +d^{3} \left (a -\frac {1}{3} c \,x^{2}-b x \right )\right ) g^{5}-6 \left (-\frac {\left (\frac {25}{63} c \,x^{2}+\frac {4}{7} b x +a \right ) x^{2} e^{3}}{15}-\frac {2 d \left (\frac {6}{35} c \,x^{2}+\frac {3}{10} b x +a \right ) x \,e^{2}}{3}+d^{2} \left (-\frac {1}{5} c \,x^{2}-\frac {2}{3} b x +a \right ) e +\frac {d^{3} \left (-\frac {2 c x}{3}+b \right )}{3}\right ) f \,g^{4}+8 \left (-\frac {x \left (\frac {10}{63} c \,x^{2}+\frac {2}{7} b x +a \right ) e^{3}}{5}+d \left (-\frac {6}{35} c \,x^{2}-\frac {3}{5} b x +a \right ) e^{2}+d^{2} \left (-\frac {3 c x}{5}+b \right ) e +\frac {c \,d^{3}}{3}\right ) f^{2} g^{3}-\frac {16 e \,f^{3} \left (\left (-\frac {10}{63} c \,x^{2}-\frac {4}{7} b x +a \right ) e^{2}+3 d \left (-\frac {4 c x}{7}+b \right ) e +3 c \,d^{2}\right ) g^{2}}{5}+\frac {128 \left (\left (-\frac {5 c x}{9}+b \right ) e +3 c d \right ) e^{2} f^{4} g}{35}-\frac {256 c \,e^{3} f^{5}}{63}\right )}{\sqrt {g x +f}\, g^{6}}\) | \(303\) |
risch | \(\frac {2 \left (35 c \,e^{3} x^{4} g^{4}+45 b \,e^{3} g^{4} x^{3}+135 c d \,e^{2} g^{4} x^{3}-85 c \,e^{3} f \,g^{3} x^{3}+63 a \,e^{3} g^{4} x^{2}+189 b d \,e^{2} g^{4} x^{2}-117 b \,e^{3} f \,g^{3} x^{2}+189 c \,d^{2} e \,g^{4} x^{2}-351 c d \,e^{2} f \,g^{3} x^{2}+165 c \,e^{3} f^{2} g^{2} x^{2}+315 a d \,e^{2} g^{4} x -189 a \,e^{3} f \,g^{3} x +315 b \,d^{2} e \,g^{4} x -567 b d \,e^{2} f \,g^{3} x +261 b \,e^{3} f^{2} g^{2} x +105 c \,d^{3} g^{4} x -567 c \,d^{2} e f \,g^{3} x +783 c d \,e^{2} f^{2} g^{2} x -325 c \,e^{3} f^{3} g x +945 a \,d^{2} e \,g^{4}-1575 a d \,e^{2} f \,g^{3}+693 a \,e^{3} f^{2} g^{2}+315 b \,d^{3} g^{4}-1575 b \,d^{2} e f \,g^{3}+2079 b d \,e^{2} f^{2} g^{2}-837 b \,e^{3} f^{3} g -525 c \,d^{3} f \,g^{3}+2079 c \,d^{2} e \,f^{2} g^{2}-2511 c d \,e^{2} f^{3} g +965 c \,e^{3} f^{4}\right ) \sqrt {g x +f}}{315 g^{6}}-\frac {2 \left (a \,d^{3} g^{5}-3 a \,d^{2} e f \,g^{4}+3 a d \,e^{2} f^{2} g^{3}-a \,e^{3} f^{3} g^{2}-b \,d^{3} f \,g^{4}+3 b \,d^{2} e \,f^{2} g^{3}-3 b d \,e^{2} f^{3} g^{2}+b \,e^{3} f^{4} g +c \,d^{3} f^{2} g^{3}-3 c \,d^{2} e \,f^{3} g^{2}+3 c d \,e^{2} f^{4} g -c \,e^{3} f^{5}\right )}{g^{6} \sqrt {g x +f}}\) | \(515\) |
gosper | \(-\frac {2 \left (-35 c \,e^{3} x^{5} g^{5}-45 b \,e^{3} g^{5} x^{4}-135 c d \,e^{2} g^{5} x^{4}+50 c \,e^{3} f \,g^{4} x^{4}-63 a \,e^{3} g^{5} x^{3}-189 b d \,e^{2} g^{5} x^{3}+72 b \,e^{3} f \,g^{4} x^{3}-189 c \,d^{2} e \,g^{5} x^{3}+216 c d \,e^{2} f \,g^{4} x^{3}-80 c \,e^{3} f^{2} g^{3} x^{3}-315 a d \,e^{2} g^{5} x^{2}+126 a \,e^{3} f \,g^{4} x^{2}-315 b \,d^{2} e \,g^{5} x^{2}+378 b d \,e^{2} f \,g^{4} x^{2}-144 b \,e^{3} f^{2} g^{3} x^{2}-105 c \,d^{3} g^{5} x^{2}+378 c \,d^{2} e f \,g^{4} x^{2}-432 c d \,e^{2} f^{2} g^{3} x^{2}+160 c \,e^{3} f^{3} g^{2} x^{2}-945 a \,d^{2} e \,g^{5} x +1260 a d \,e^{2} f \,g^{4} x -504 a \,e^{3} f^{2} g^{3} x -315 b \,d^{3} g^{5} x +1260 b \,d^{2} e f \,g^{4} x -1512 b d \,e^{2} f^{2} g^{3} x +576 b \,e^{3} f^{3} g^{2} x +420 c \,d^{3} f \,g^{4} x -1512 c \,d^{2} e \,f^{2} g^{3} x +1728 c d \,e^{2} f^{3} g^{2} x -640 c \,e^{3} f^{4} g x +315 a \,d^{3} g^{5}-1890 a \,d^{2} e f \,g^{4}+2520 a d \,e^{2} f^{2} g^{3}-1008 a \,e^{3} f^{3} g^{2}-630 b \,d^{3} f \,g^{4}+2520 b \,d^{2} e \,f^{2} g^{3}-3024 b d \,e^{2} f^{3} g^{2}+1152 b \,e^{3} f^{4} g +840 c \,d^{3} f^{2} g^{3}-3024 c \,d^{2} e \,f^{3} g^{2}+3456 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{315 \sqrt {g x +f}\, g^{6}}\) | \(540\) |
trager | \(-\frac {2 \left (-35 c \,e^{3} x^{5} g^{5}-45 b \,e^{3} g^{5} x^{4}-135 c d \,e^{2} g^{5} x^{4}+50 c \,e^{3} f \,g^{4} x^{4}-63 a \,e^{3} g^{5} x^{3}-189 b d \,e^{2} g^{5} x^{3}+72 b \,e^{3} f \,g^{4} x^{3}-189 c \,d^{2} e \,g^{5} x^{3}+216 c d \,e^{2} f \,g^{4} x^{3}-80 c \,e^{3} f^{2} g^{3} x^{3}-315 a d \,e^{2} g^{5} x^{2}+126 a \,e^{3} f \,g^{4} x^{2}-315 b \,d^{2} e \,g^{5} x^{2}+378 b d \,e^{2} f \,g^{4} x^{2}-144 b \,e^{3} f^{2} g^{3} x^{2}-105 c \,d^{3} g^{5} x^{2}+378 c \,d^{2} e f \,g^{4} x^{2}-432 c d \,e^{2} f^{2} g^{3} x^{2}+160 c \,e^{3} f^{3} g^{2} x^{2}-945 a \,d^{2} e \,g^{5} x +1260 a d \,e^{2} f \,g^{4} x -504 a \,e^{3} f^{2} g^{3} x -315 b \,d^{3} g^{5} x +1260 b \,d^{2} e f \,g^{4} x -1512 b d \,e^{2} f^{2} g^{3} x +576 b \,e^{3} f^{3} g^{2} x +420 c \,d^{3} f \,g^{4} x -1512 c \,d^{2} e \,f^{2} g^{3} x +1728 c d \,e^{2} f^{3} g^{2} x -640 c \,e^{3} f^{4} g x +315 a \,d^{3} g^{5}-1890 a \,d^{2} e f \,g^{4}+2520 a d \,e^{2} f^{2} g^{3}-1008 a \,e^{3} f^{3} g^{2}-630 b \,d^{3} f \,g^{4}+2520 b \,d^{2} e \,f^{2} g^{3}-3024 b d \,e^{2} f^{3} g^{2}+1152 b \,e^{3} f^{4} g +840 c \,d^{3} f^{2} g^{3}-3024 c \,d^{2} e \,f^{3} g^{2}+3456 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{315 \sqrt {g x +f}\, g^{6}}\) | \(540\) |
derivativedivides | \(\frac {\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {9}{2}}}{9}-\frac {24 c d \,e^{2} f g \left (g x +f \right )^{\frac {5}{2}}}{5}-12 a d \,e^{2} f \,g^{3} \sqrt {g x +f}-12 b \,d^{2} e f \,g^{3} \sqrt {g x +f}-6 b d \,e^{2} f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}-6 c \,d^{2} e f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}+12 c d \,e^{2} f^{2} g \left (g x +f \right )^{\frac {3}{2}}-24 c d \,e^{2} f^{3} g \sqrt {g x +f}+18 c \,d^{2} e \,f^{2} g^{2} \sqrt {g x +f}+18 b d \,e^{2} f^{2} g^{2} \sqrt {g x +f}-\frac {10 c \,e^{3} f \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {20 c \,e^{3} f^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+\frac {2 b \,e^{3} g \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 c \,d^{3} g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+10 c \,e^{3} f^{4} \sqrt {g x +f}+\frac {2 a \,e^{3} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+4 c \,e^{3} f^{2} \left (g x +f \right )^{\frac {5}{2}}+2 b \,d^{3} g^{4} \sqrt {g x +f}+2 a d \,e^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}-4 c \,d^{3} f \,g^{3} \sqrt {g x +f}+2 b \,d^{2} e \,g^{3} \left (g x +f \right )^{\frac {3}{2}}+\frac {6 b d \,e^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {2 \left (a \,d^{3} g^{5}-3 a \,d^{2} e f \,g^{4}+3 a d \,e^{2} f^{2} g^{3}-a \,e^{3} f^{3} g^{2}-b \,d^{3} f \,g^{4}+3 b \,d^{2} e \,f^{2} g^{3}-3 b d \,e^{2} f^{3} g^{2}+b \,e^{3} f^{4} g +c \,d^{3} f^{2} g^{3}-3 c \,d^{2} e \,f^{3} g^{2}+3 c d \,e^{2} f^{4} g -c \,e^{3} f^{5}\right )}{\sqrt {g x +f}}+4 b \,e^{3} f^{2} g \left (g x +f \right )^{\frac {3}{2}}-2 a \,e^{3} f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}-\frac {8 b \,e^{3} f g \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {6 c d \,e^{2} g \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {6 c \,d^{2} e \,g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+6 a \,d^{2} e \,g^{4} \sqrt {g x +f}+6 a \,e^{3} f^{2} g^{2} \sqrt {g x +f}-8 b \,e^{3} f^{3} g \sqrt {g x +f}}{g^{6}}\) | \(650\) |
default | \(\frac {\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {9}{2}}}{9}-\frac {24 c d \,e^{2} f g \left (g x +f \right )^{\frac {5}{2}}}{5}-12 a d \,e^{2} f \,g^{3} \sqrt {g x +f}-12 b \,d^{2} e f \,g^{3} \sqrt {g x +f}-6 b d \,e^{2} f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}-6 c \,d^{2} e f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}+12 c d \,e^{2} f^{2} g \left (g x +f \right )^{\frac {3}{2}}-24 c d \,e^{2} f^{3} g \sqrt {g x +f}+18 c \,d^{2} e \,f^{2} g^{2} \sqrt {g x +f}+18 b d \,e^{2} f^{2} g^{2} \sqrt {g x +f}-\frac {10 c \,e^{3} f \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {20 c \,e^{3} f^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+\frac {2 b \,e^{3} g \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 c \,d^{3} g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+10 c \,e^{3} f^{4} \sqrt {g x +f}+\frac {2 a \,e^{3} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+4 c \,e^{3} f^{2} \left (g x +f \right )^{\frac {5}{2}}+2 b \,d^{3} g^{4} \sqrt {g x +f}+2 a d \,e^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}-4 c \,d^{3} f \,g^{3} \sqrt {g x +f}+2 b \,d^{2} e \,g^{3} \left (g x +f \right )^{\frac {3}{2}}+\frac {6 b d \,e^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {2 \left (a \,d^{3} g^{5}-3 a \,d^{2} e f \,g^{4}+3 a d \,e^{2} f^{2} g^{3}-a \,e^{3} f^{3} g^{2}-b \,d^{3} f \,g^{4}+3 b \,d^{2} e \,f^{2} g^{3}-3 b d \,e^{2} f^{3} g^{2}+b \,e^{3} f^{4} g +c \,d^{3} f^{2} g^{3}-3 c \,d^{2} e \,f^{3} g^{2}+3 c d \,e^{2} f^{4} g -c \,e^{3} f^{5}\right )}{\sqrt {g x +f}}+4 b \,e^{3} f^{2} g \left (g x +f \right )^{\frac {3}{2}}-2 a \,e^{3} f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}-\frac {8 b \,e^{3} f g \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {6 c d \,e^{2} g \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {6 c \,d^{2} e \,g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+6 a \,d^{2} e \,g^{4} \sqrt {g x +f}+6 a \,e^{3} f^{2} g^{2} \sqrt {g x +f}-8 b \,e^{3} f^{3} g \sqrt {g x +f}}{g^{6}}\) | \(650\) |
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Time = 0.42 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.54 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, c e^{3} g^{5} x^{5} + 1280 \, c e^{3} f^{5} - 315 \, a d^{3} g^{5} - 1152 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g + 1008 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} - 840 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} + 630 \, {\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4} - 5 \, {\left (10 \, c e^{3} f g^{4} - 9 \, {\left (3 \, c d e^{2} + b e^{3}\right )} g^{5}\right )} x^{4} + {\left (80 \, c e^{3} f^{2} g^{3} - 72 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f g^{4} + 63 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{5}\right )} x^{3} - {\left (160 \, c e^{3} f^{3} g^{2} - 144 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g^{3} + 126 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{4} - 105 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} + {\left (640 \, c e^{3} f^{4} g - 576 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g^{2} + 504 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{3} - 420 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{4} + 315 \, {\left (b d^{3} + 3 \, a d^{2} e\right )} g^{5}\right )} x\right )} \sqrt {g x + f}}{315 \, {\left (g^{7} x + f g^{6}\right )}} \]
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Time = 32.42 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.99 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c e^{3} \left (f + g x\right )^{\frac {9}{2}}}{9 g^{5}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \left (b e^{3} g + 3 c d e^{2} g - 5 c e^{3} f\right )}{7 g^{5}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (a e^{3} g^{2} + 3 b d e^{2} g^{2} - 4 b e^{3} f g + 3 c d^{2} e g^{2} - 12 c d e^{2} f g + 10 c e^{3} f^{2}\right )}{5 g^{5}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \cdot \left (3 a d e^{2} g^{3} - 3 a e^{3} f g^{2} + 3 b d^{2} e g^{3} - 9 b d e^{2} f g^{2} + 6 b e^{3} f^{2} g + c d^{3} g^{3} - 9 c d^{2} e f g^{2} + 18 c d e^{2} f^{2} g - 10 c e^{3} f^{3}\right )}{3 g^{5}} + \frac {\sqrt {f + g x} \left (3 a d^{2} e g^{4} - 6 a d e^{2} f g^{3} + 3 a e^{3} f^{2} g^{2} + b d^{3} g^{4} - 6 b d^{2} e f g^{3} + 9 b d e^{2} f^{2} g^{2} - 4 b e^{3} f^{3} g - 2 c d^{3} f g^{3} + 9 c d^{2} e f^{2} g^{2} - 12 c d e^{2} f^{3} g + 5 c e^{3} f^{4}\right )}{g^{5}} - \frac {\left (d g - e f\right )^{3} \left (a g^{2} - b f g + c f^{2}\right )}{g^{5} \sqrt {f + g x}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d^{3} x + \frac {c e^{3} x^{6}}{6} + \frac {x^{5} \left (b e^{3} + 3 c d e^{2}\right )}{5} + \frac {x^{4} \left (a e^{3} + 3 b d e^{2} + 3 c d^{2} e\right )}{4} + \frac {x^{3} \cdot \left (3 a d e^{2} + 3 b d^{2} e + c d^{3}\right )}{3} + \frac {x^{2} \cdot \left (3 a d^{2} e + b d^{3}\right )}{2}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} - 45 \, {\left (5 \, c e^{3} f - {\left (3 \, c d e^{2} + b e^{3}\right )} g\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, c e^{3} f^{2} - 4 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f g + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, c e^{3} f^{3} - 6 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g + 3 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{2} - {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, c e^{3} f^{4} - 4 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g + 3 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{2} - 2 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{3} + {\left (b d^{3} + 3 \, a d^{2} e\right )} g^{4}\right )} \sqrt {g x + f}}{g^{5}} + \frac {315 \, {\left (c e^{3} f^{5} - a d^{3} g^{5} - {\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} - {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} + {\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4}\right )}}{\sqrt {g x + f} g^{5}}\right )}}{315 \, g} \]
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Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (265) = 530\).
Time = 0.31 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.41 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g - b e^{3} f^{4} g + 3 \, c d^{2} e f^{3} g^{2} + 3 \, b d e^{2} f^{3} g^{2} + a e^{3} f^{3} g^{2} - c d^{3} f^{2} g^{3} - 3 \, b d^{2} e f^{2} g^{3} - 3 \, a d e^{2} f^{2} g^{3} + b d^{3} f g^{4} + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5}\right )}}{\sqrt {g x + f} g^{6}} + \frac {2 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} g^{48} - 225 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{3} f g^{48} + 630 \, {\left (g x + f\right )}^{\frac {5}{2}} c e^{3} f^{2} g^{48} - 1050 \, {\left (g x + f\right )}^{\frac {3}{2}} c e^{3} f^{3} g^{48} + 1575 \, \sqrt {g x + f} c e^{3} f^{4} g^{48} + 135 \, {\left (g x + f\right )}^{\frac {7}{2}} c d e^{2} g^{49} + 45 \, {\left (g x + f\right )}^{\frac {7}{2}} b e^{3} g^{49} - 756 \, {\left (g x + f\right )}^{\frac {5}{2}} c d e^{2} f g^{49} - 252 \, {\left (g x + f\right )}^{\frac {5}{2}} b e^{3} f g^{49} + 1890 \, {\left (g x + f\right )}^{\frac {3}{2}} c d e^{2} f^{2} g^{49} + 630 \, {\left (g x + f\right )}^{\frac {3}{2}} b e^{3} f^{2} g^{49} - 3780 \, \sqrt {g x + f} c d e^{2} f^{3} g^{49} - 1260 \, \sqrt {g x + f} b e^{3} f^{3} g^{49} + 189 \, {\left (g x + f\right )}^{\frac {5}{2}} c d^{2} e g^{50} + 189 \, {\left (g x + f\right )}^{\frac {5}{2}} b d e^{2} g^{50} + 63 \, {\left (g x + f\right )}^{\frac {5}{2}} a e^{3} g^{50} - 945 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} e f g^{50} - 945 \, {\left (g x + f\right )}^{\frac {3}{2}} b d e^{2} f g^{50} - 315 \, {\left (g x + f\right )}^{\frac {3}{2}} a e^{3} f g^{50} + 2835 \, \sqrt {g x + f} c d^{2} e f^{2} g^{50} + 2835 \, \sqrt {g x + f} b d e^{2} f^{2} g^{50} + 945 \, \sqrt {g x + f} a e^{3} f^{2} g^{50} + 105 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{3} g^{51} + 315 \, {\left (g x + f\right )}^{\frac {3}{2}} b d^{2} e g^{51} + 315 \, {\left (g x + f\right )}^{\frac {3}{2}} a d e^{2} g^{51} - 630 \, \sqrt {g x + f} c d^{3} f g^{51} - 1890 \, \sqrt {g x + f} b d^{2} e f g^{51} - 1890 \, \sqrt {g x + f} a d e^{2} f g^{51} + 315 \, \sqrt {g x + f} b d^{3} g^{52} + 945 \, \sqrt {g x + f} a d^{2} e g^{52}\right )}}{315 \, g^{54}} \]
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Time = 0.12 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {{\left (f+g\,x\right )}^{7/2}\,\left (2\,b\,e^3\,g-10\,c\,e^3\,f+6\,c\,d\,e^2\,g\right )}{7\,g^6}-\frac {2\,c\,d^3\,f^2\,g^3-2\,b\,d^3\,f\,g^4+2\,a\,d^3\,g^5-6\,c\,d^2\,e\,f^3\,g^2+6\,b\,d^2\,e\,f^2\,g^3-6\,a\,d^2\,e\,f\,g^4+6\,c\,d\,e^2\,f^4\,g-6\,b\,d\,e^2\,f^3\,g^2+6\,a\,d\,e^2\,f^2\,g^3-2\,c\,e^3\,f^5+2\,b\,e^3\,f^4\,g-2\,a\,e^3\,f^3\,g^2}{g^6\,\sqrt {f+g\,x}}+\frac {{\left (f+g\,x\right )}^{5/2}\,\left (6\,c\,d^2\,e\,g^2-24\,c\,d\,e^2\,f\,g+6\,b\,d\,e^2\,g^2+20\,c\,e^3\,f^2-8\,b\,e^3\,f\,g+2\,a\,e^3\,g^2\right )}{5\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (d\,g-e\,f\right )\,\left (c\,d^2\,g^2-8\,c\,d\,e\,f\,g+3\,b\,d\,e\,g^2+10\,c\,e^2\,f^2-6\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{3\,g^6}+\frac {2\,\sqrt {f+g\,x}\,{\left (d\,g-e\,f\right )}^2\,\left (3\,a\,e\,g^2+b\,d\,g^2+5\,c\,e\,f^2-4\,b\,e\,f\,g-2\,c\,d\,f\,g\right )}{g^6}+\frac {2\,c\,e^3\,{\left (f+g\,x\right )}^{9/2}}{9\,g^6} \]
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