Integrand size = 27, antiderivative size = 212 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^5}-\frac {2 (e f-d g) (2 c f (2 e f-d g)-g (3 b e f-b d g-2 a e g)) (f+g x)^{3/2}}{3 g^5}-\frac {2 \left (e g (3 b e f-2 b d g-a e g)-c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^5}-\frac {2 e (4 c e f-2 c d g-b e g) (f+g x)^{7/2}}{7 g^5}+\frac {2 c e^2 (f+g x)^{9/2}}{9 g^5} \]
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Time = 0.20 (sec) , antiderivative size = 212, 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, 1167} \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=-\frac {2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac {2 \sqrt {f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac {2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac {2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac {2 c e^2 (f+g x)^{9/2}}{9 g^5} \]
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Rule 911
Rule 1167
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2 \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 ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {(-e f+d g)^2 \left (c f^2-b f g+a g^2\right )}{g^4}+\frac {(e f-d g) (-2 c f (2 e f-d g)+g (3 b e f-b d g-2 a e g)) x^2}{g^4}+\frac {\left (-e g (3 b e f-2 b d g-a e g)+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) x^4}{g^4}+\frac {e (-4 c e f+2 c d g+b e g) x^6}{g^4}+\frac {c e^2 x^8}{g^4}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^5}-\frac {2 (e f-d g) (2 c f (2 e f-d g)-g (3 b e f-b d g-2 a e g)) (f+g x)^{3/2}}{3 g^5}-\frac {2 \left (e g (3 b e f-2 b d g-a e g)-c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^5}-\frac {2 e (4 c e f-2 c d g-b e g) (f+g x)^{7/2}}{7 g^5}+\frac {2 c e^2 (f+g x)^{9/2}}{9 g^5} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (c \left (21 d^2 g^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+18 d e g \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+e^2 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )\right )+3 g \left (7 a g \left (15 d^2 g^2+10 d e g (-2 f+g x)+e^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )+b \left (35 d^2 g^2 (-2 f+g x)+14 d e g \left (8 f^2-4 f g x+3 g^2 x^2\right )-3 e^2 \left (16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right )\right )\right )\right )}{315 g^5} \]
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Time = 0.53 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {g x +f}\, \left (\left (\frac {\left (\frac {5}{9} c \,x^{2}+\frac {5}{7} b x +a \right ) x^{2} e^{2}}{5}+\frac {2 d \left (\frac {3}{7} c \,x^{2}+\frac {3}{5} b x +a \right ) x e}{3}+d^{2} \left (\frac {1}{5} c \,x^{2}+\frac {1}{3} b x +a \right )\right ) g^{4}-\frac {4 \left (\frac {\left (\frac {10}{21} c \,x^{2}+\frac {9}{14} b x +a \right ) x \,e^{2}}{5}+d \left (\frac {9}{35} c \,x^{2}+\frac {2}{5} b x +a \right ) e +\frac {d^{2} \left (\frac {2 c x}{5}+b \right )}{2}\right ) f \,g^{3}}{3}+\frac {8 \left (\left (\frac {2}{7} c \,x^{2}+\frac {3}{7} b x +a \right ) e^{2}+2 d \left (\frac {3 c x}{7}+b \right ) e +c \,d^{2}\right ) f^{2} g^{2}}{15}-\frac {16 e \left (\left (\frac {4 c x}{9}+b \right ) e +2 c d \right ) f^{3} g}{35}+\frac {128 c \,e^{2} f^{4}}{315}\right )}{g^{5}}\) | \(192\) |
derivativedivides | \(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (d g -e f \right ) e c +e^{2} \left (b g -2 c f \right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right )^{2} c +2 \left (d g -e f \right ) e \left (b g -2 c f \right )+e^{2} \left (a \,g^{2}-b f g +c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (d g -e f \right )^{2} \left (b g -2 c f \right )+2 \left (d g -e f \right ) e \left (a \,g^{2}-b f g +c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right )^{2} \left (a \,g^{2}-b f g +c \,f^{2}\right ) \sqrt {g x +f}}{g^{5}}\) | \(205\) |
default | \(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (d g -e f \right ) e c +e^{2} \left (b g -2 c f \right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right )^{2} c +2 \left (d g -e f \right ) e \left (b g -2 c f \right )+e^{2} \left (a \,g^{2}-b f g +c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (d g -e f \right )^{2} \left (b g -2 c f \right )+2 \left (d g -e f \right ) e \left (a \,g^{2}-b f g +c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right )^{2} \left (a \,g^{2}-b f g +c \,f^{2}\right ) \sqrt {g x +f}}{g^{5}}\) | \(205\) |
gosper | \(\frac {2 \sqrt {g x +f}\, \left (35 c \,e^{2} x^{4} g^{4}+45 b \,e^{2} g^{4} x^{3}+90 c d e \,g^{4} x^{3}-40 c \,e^{2} f \,g^{3} x^{3}+63 a \,e^{2} g^{4} x^{2}+126 b d e \,g^{4} x^{2}-54 b \,e^{2} f \,g^{3} x^{2}+63 c \,d^{2} g^{4} x^{2}-108 c d e f \,g^{3} x^{2}+48 c \,e^{2} f^{2} g^{2} x^{2}+210 a d e \,g^{4} x -84 a \,e^{2} f \,g^{3} x +105 b \,d^{2} g^{4} x -168 b d e f \,g^{3} x +72 b \,e^{2} f^{2} g^{2} x -84 c \,d^{2} f \,g^{3} x +144 c d e \,f^{2} g^{2} x -64 c \,e^{2} f^{3} g x +315 a \,d^{2} g^{4}-420 a d e f \,g^{3}+168 a \,e^{2} f^{2} g^{2}-210 b \,d^{2} f \,g^{3}+336 b d e \,f^{2} g^{2}-144 b \,e^{2} f^{3} g +168 c \,d^{2} f^{2} g^{2}-288 c d e \,f^{3} g +128 c \,e^{2} f^{4}\right )}{315 g^{5}}\) | \(315\) |
trager | \(\frac {2 \sqrt {g x +f}\, \left (35 c \,e^{2} x^{4} g^{4}+45 b \,e^{2} g^{4} x^{3}+90 c d e \,g^{4} x^{3}-40 c \,e^{2} f \,g^{3} x^{3}+63 a \,e^{2} g^{4} x^{2}+126 b d e \,g^{4} x^{2}-54 b \,e^{2} f \,g^{3} x^{2}+63 c \,d^{2} g^{4} x^{2}-108 c d e f \,g^{3} x^{2}+48 c \,e^{2} f^{2} g^{2} x^{2}+210 a d e \,g^{4} x -84 a \,e^{2} f \,g^{3} x +105 b \,d^{2} g^{4} x -168 b d e f \,g^{3} x +72 b \,e^{2} f^{2} g^{2} x -84 c \,d^{2} f \,g^{3} x +144 c d e \,f^{2} g^{2} x -64 c \,e^{2} f^{3} g x +315 a \,d^{2} g^{4}-420 a d e f \,g^{3}+168 a \,e^{2} f^{2} g^{2}-210 b \,d^{2} f \,g^{3}+336 b d e \,f^{2} g^{2}-144 b \,e^{2} f^{3} g +168 c \,d^{2} f^{2} g^{2}-288 c d e \,f^{3} g +128 c \,e^{2} f^{4}\right )}{315 g^{5}}\) | \(315\) |
risch | \(\frac {2 \sqrt {g x +f}\, \left (35 c \,e^{2} x^{4} g^{4}+45 b \,e^{2} g^{4} x^{3}+90 c d e \,g^{4} x^{3}-40 c \,e^{2} f \,g^{3} x^{3}+63 a \,e^{2} g^{4} x^{2}+126 b d e \,g^{4} x^{2}-54 b \,e^{2} f \,g^{3} x^{2}+63 c \,d^{2} g^{4} x^{2}-108 c d e f \,g^{3} x^{2}+48 c \,e^{2} f^{2} g^{2} x^{2}+210 a d e \,g^{4} x -84 a \,e^{2} f \,g^{3} x +105 b \,d^{2} g^{4} x -168 b d e f \,g^{3} x +72 b \,e^{2} f^{2} g^{2} x -84 c \,d^{2} f \,g^{3} x +144 c d e \,f^{2} g^{2} x -64 c \,e^{2} f^{3} g x +315 a \,d^{2} g^{4}-420 a d e f \,g^{3}+168 a \,e^{2} f^{2} g^{2}-210 b \,d^{2} f \,g^{3}+336 b d e \,f^{2} g^{2}-144 b \,e^{2} f^{3} g +168 c \,d^{2} f^{2} g^{2}-288 c d e \,f^{3} g +128 c \,e^{2} f^{4}\right )}{315 g^{5}}\) | \(315\) |
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Time = 0.47 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (35 \, c e^{2} g^{4} x^{4} + 128 \, c e^{2} f^{4} + 315 \, a d^{2} g^{4} - 144 \, {\left (2 \, c d e + b e^{2}\right )} f^{3} g + 168 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - 210 \, {\left (b d^{2} + 2 \, a d e\right )} f g^{3} - 5 \, {\left (8 \, c e^{2} f g^{3} - 9 \, {\left (2 \, c d e + b e^{2}\right )} g^{4}\right )} x^{3} + 3 \, {\left (16 \, c e^{2} f^{2} g^{2} - 18 \, {\left (2 \, c d e + b e^{2}\right )} f g^{3} + 21 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{4}\right )} x^{2} - {\left (64 \, c e^{2} f^{3} g - 72 \, {\left (2 \, c d e + b e^{2}\right )} f^{2} g^{2} + 84 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{3} - 105 \, {\left (b d^{2} + 2 \, a d e\right )} g^{4}\right )} x\right )} \sqrt {g x + f}}{315 \, g^{5}} \]
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Time = 1.05 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.01 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (\frac {c e^{2} \left (f + g x\right )^{\frac {9}{2}}}{9 g^{4}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \left (b e^{2} g + 2 c d e g - 4 c e^{2} f\right )}{7 g^{4}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (a e^{2} g^{2} + 2 b d e g^{2} - 3 b e^{2} f g + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{5 g^{4}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \cdot \left (2 a d e g^{3} - 2 a e^{2} f g^{2} + b d^{2} g^{3} - 4 b d e f g^{2} + 3 b e^{2} f^{2} g - 2 c d^{2} f g^{2} + 6 c d e f^{2} g - 4 c e^{2} f^{3}\right )}{3 g^{4}} + \frac {\sqrt {f + g x} \left (a d^{2} g^{4} - 2 a d e f g^{3} + a e^{2} f^{2} g^{2} - b d^{2} f g^{3} + 2 b d e f^{2} g^{2} - b e^{2} f^{3} g + c d^{2} f^{2} g^{2} - 2 c d e f^{3} g + c e^{2} f^{4}\right )}{g^{4}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d^{2} x + \frac {c e^{2} x^{5}}{5} + \frac {x^{4} \left (b e^{2} + 2 c d e\right )}{4} + \frac {x^{3} \left (a e^{2} + 2 b d e + c d^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 a d e + b d^{2}\right )}{2}}{\sqrt {f}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{2} - 45 \, {\left (4 \, c e^{2} f - {\left (2 \, c d e + b e^{2}\right )} g\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 63 \, {\left (6 \, c e^{2} f^{2} - 3 \, {\left (2 \, c d e + b e^{2}\right )} f g + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, c e^{2} f^{3} - 3 \, {\left (2 \, c d e + b e^{2}\right )} f^{2} g + 2 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{2} - {\left (b d^{2} + 2 \, a d e\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 315 \, {\left (c e^{2} f^{4} + a d^{2} g^{4} - {\left (2 \, c d e + b e^{2}\right )} f^{3} g + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - {\left (b d^{2} + 2 \, a d e\right )} f g^{3}\right )} \sqrt {g x + f}\right )}}{315 \, g^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.71 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {g x + f} a d^{2} + \frac {105 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} b d^{2}}{g} + \frac {210 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a d e}{g} + \frac {21 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c d^{2}}{g^{2}} + \frac {42 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} b d e}{g^{2}} + \frac {21 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a e^{2}}{g^{2}} + \frac {18 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} c d e}{g^{3}} + \frac {9 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} b e^{2}}{g^{3}} + \frac {{\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c e^{2}}{g^{4}}\right )}}{315 \, g} \]
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Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {{\left (f+g\,x\right )}^{7/2}\,\left (2\,b\,e^2\,g-8\,c\,e^2\,f+4\,c\,d\,e\,g\right )}{7\,g^5}+\frac {{\left (f+g\,x\right )}^{5/2}\,\left (2\,c\,d^2\,g^2-12\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+12\,c\,e^2\,f^2-6\,b\,e^2\,f\,g+2\,a\,e^2\,g^2\right )}{5\,g^5}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (d\,g-e\,f\right )\,\left (2\,a\,e\,g^2+b\,d\,g^2+4\,c\,e\,f^2-3\,b\,e\,f\,g-2\,c\,d\,f\,g\right )}{3\,g^5}+\frac {2\,\sqrt {f+g\,x}\,{\left (d\,g-e\,f\right )}^2\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{g^5}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{9/2}}{9\,g^5} \]
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