Optimal. Leaf size=649 \[ \frac {3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )}{64 c^5}+\frac {f x \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{32 c^4}+\frac {f^2 x^2 \sqrt {a+b x+c x^2} (8 c e-5 b f)}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2} \]
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Rubi [A] time = 2.11, antiderivative size = 649, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 621, 206} \begin {gather*} \frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^2 c^2 d f+b^2 c^2 e^2-2 b^3 c e f+b^4 f^2-b c^3 d e+c^4 d^2\right )-b c^2 \left (-9 a^2 c f \left (d f+e^2\right )+5 a^3 f^3+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )+3 a b^4 c e f^2-a b^5 f^3\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac {f x \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{32 c^4}-\frac {\sqrt {a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )}{64 c^5}+\frac {f^2 x^2 \sqrt {a+b x+c x^2} (8 c e-5 b f)}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 640
Rule 1660
Rule 1661
Rubi steps
\begin {align*} \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {-\frac {\left (b^2-4 a c\right ) \left (b^4 f^3-3 b^2 c f^2 (b e+a f)+3 c^4 d \left (e^2+d f\right )+c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )}{2 c^5}+\frac {\left (b^2-4 a c\right ) \left (b^3 f^3-b c f^2 (3 b e+2 a f)-c^3 \left (e^3+6 d e f\right )+3 c^2 f \left (a e f+b \left (e^2+d f\right )\right )\right ) x}{2 c^4}-\frac {\left (b^2-4 a c\right ) f \left (b^2 f^2-c f (3 b e+a f)+3 c^2 \left (e^2+d f\right )\right ) x^2}{2 c^3}-\frac {\left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{2 c^2}-\frac {\left (b^2-4 a c\right ) f^3 x^4}{2 c}}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}-\frac {\int \frac {-\frac {2 \left (b^2-4 a c\right ) \left (b^4 f^3-3 b^2 c f^2 (b e+a f)+3 c^4 d \left (e^2+d f\right )+c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )}{c^4}+\frac {2 \left (b^2-4 a c\right ) \left (b^3 f^3-b c f^2 (3 b e+2 a f)-c^3 \left (e^3+6 d e f\right )+3 c^2 f \left (a e f+b \left (e^2+d f\right )\right )\right ) x}{c^3}-\frac {\left (b^2-4 a c\right ) f \left (4 b^2 f^2-c f (12 b e+7 a f)+12 c^2 \left (e^2+d f\right )\right ) x^2}{2 c^2}-\frac {3 \left (b^2-4 a c\right ) f^2 (8 c e-5 b f) x^3}{4 c}}{\sqrt {a+b x+c x^2}} \, dx}{2 c \left (b^2-4 a c\right )}\\ &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f^2 (8 c e-5 b f) x^2 \sqrt {a+b x+c x^2}}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}-\frac {\int \frac {-\frac {6 \left (b^2-4 a c\right ) \left (b^4 f^3-3 b^2 c f^2 (b e+a f)+3 c^4 d \left (e^2+d f\right )+c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )}{c^3}+\frac {3 \left (b^2-4 a c\right ) \left (4 b^3 f^3-b c f^2 (12 b e+13 a f)-4 c^3 \left (e^3+6 d e f\right )+4 c^2 f \left (5 a e f+3 b \left (e^2+d f\right )\right )\right ) x}{2 c^2}-\frac {3 \left (b^2-4 a c\right ) f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x^2}{8 c}}{\sqrt {a+b x+c x^2}} \, dx}{6 c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt {a+b x+c x^2}}{32 c^4}+\frac {f^2 (8 c e-5 b f) x^2 \sqrt {a+b x+c x^2}}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}-\frac {\int \frac {-\frac {3 \left (b^2-4 a c\right ) \left (32 b^4 f^3-b^2 c f^2 (96 b e+137 a f)+96 c^4 d \left (e^2+d f\right )+4 c^2 f \left (70 a b e f+15 a^2 f^2+24 b^2 \left (e^2+d f\right )\right )-16 c^3 \left (9 a f \left (e^2+d f\right )+2 b \left (e^3+6 d e f\right )\right )\right )}{8 c^2}+\frac {3 \left (b^2-4 a c\right ) \left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) x}{16 c}}{\sqrt {a+b x+c x^2}} \, dx}{12 c^3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt {a+b x+c x^2}}{64 c^5}+\frac {f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt {a+b x+c x^2}}{32 c^4}+\frac {f^2 (8 c e-5 b f) x^2 \sqrt {a+b x+c x^2}}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^5}\\ &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt {a+b x+c x^2}}{64 c^5}+\frac {f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt {a+b x+c x^2}}{32 c^4}+\frac {f^2 (8 c e-5 b f) x^2 \sqrt {a+b x+c x^2}}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{64 c^5}\\ &=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt {a+b x+c x^2}}{64 c^5}+\frac {f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt {a+b x+c x^2}}{32 c^4}+\frac {f^2 (8 c e-5 b f) x^2 \sqrt {a+b x+c x^2}}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}+\frac {3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 1.66, size = 745, normalized size = 1.15 \begin {gather*} \frac {3 \log \left (2 \sqrt {c} \sqrt {a+x (b+c x)}+b+2 c x\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac {-8 b^3 c \left (210 a^2 f^3+a c f \left (f \left (77 f x^2-90 d\right )-90 e^2-530 e f x\right )-c^2 x \left (2 e f \left (7 f x^2-72 d\right )+3 f^2 x \left (10 d+f x^2\right )-24 e^3+30 e^2 f x\right )\right )-16 b^2 c^2 \left (-a^2 f^2 (230 e+169 f x)+a c \left (2 e f \left (36 d-43 f x^2\right )+f^2 x \left (186 d-13 f x^2\right )+12 e^3+186 e^2 f x\right )+c^2 x \left (-24 d^2 f+6 d \left (-4 e^2+4 e f x+f^2 x^2\right )+x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )\right )+16 b c^2 \left (113 a^3 f^3+a^2 c f \left (f \left (49 f x^2-156 d\right )-156 e^2-244 e f x\right )+2 a c^2 \left (12 d^2 f+6 d \left (2 e^2+20 e f x-5 f^2 x^2\right )-x \left (-20 e^3+30 e^2 f x+14 e f^2 x^2+3 f^3 x^3\right )\right )+8 c^3 d^2 (d-3 e x)\right )+32 c^3 \left (a^3 \left (-f^2\right ) (64 e+15 f x)+a^2 c \left (-32 e f \left (f x^2-3 d\right )+f^2 x \left (36 d-5 f x^2\right )+16 e^3+36 e^2 f x\right )+2 a c^2 \left (-12 d^2 (e+f x)+6 d x \left (-2 e^2+4 e f x+f^2 x^2\right )+x^2 \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )+8 c^3 d^3 x\right )+105 b^5 f^2 (3 a f+c x (f x-8 e))-2 b^4 c f \left (105 a f (4 e+9 f x)+c x \left (-360 d f-360 e^2+140 e f x+21 f^2 x^2\right )\right )+315 b^6 f^3 x}{64 c^5 \left (4 a c-b^2\right ) \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
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IntegrateAlgebraic [A] time = 5.98, size = 1100, normalized size = 1.69 \begin {gather*} -\frac {-315 f^3 x b^6-315 a f^3 b^5-105 c f^3 x^2 b^5+840 c e f^2 x b^5+42 c^2 f^3 x^3 b^4+840 a c e f^2 b^4+280 c^2 e f^2 x^2 b^4+1890 a c f^3 x b^4-720 c^2 d f^2 x b^4-720 c^2 e^2 f x b^4-24 c^3 f^3 x^4 b^3+1680 a^2 c f^3 b^3-112 c^3 e f^2 x^3 b^3-720 a c^2 d f^2 b^3+616 a c^2 f^3 x^2 b^3-240 c^3 d f^2 x^2 b^3-240 c^3 e^2 f x^2 b^3-720 a c^2 e^2 f b^3+192 c^3 e^3 x b^3-4240 a c^2 e f^2 x b^3+1152 c^3 d e f x b^3+16 c^4 f^3 x^5 b^2+64 c^4 e f^2 x^4 b^2+192 a c^3 e^3 b^2-208 a c^3 f^3 x^3 b^2+96 c^4 d f^2 x^3 b^2+96 c^4 e^2 f x^3 b^2-3680 a^2 c^2 e f^2 b^2+64 c^4 e^3 x^2 b^2-1376 a c^3 e f^2 x^2 b^2+384 c^4 d e f x^2 b^2+1152 a c^3 d e f b^2-2704 a^2 c^2 f^3 x b^2-384 c^4 d e^2 x b^2+2976 a c^3 d f^2 x b^2-384 c^4 d^2 f x b^2+2976 a c^3 e^2 f x b^2+96 a c^4 f^3 x^4 b-128 c^5 d^3 b-1808 a^3 c^2 f^3 b+448 a c^4 e f^2 x^3 b-384 a c^4 d e^2 b+2496 a^2 c^3 d f^2 b-784 a^2 c^3 f^3 x^2 b+960 a c^4 d f^2 x^2 b+960 a c^4 e^2 f x^2 b-384 a c^4 d^2 f b+2496 a^2 c^3 e^2 f b-640 a c^4 e^3 x b+3904 a^2 c^3 e f^2 x b+384 c^5 d^2 e x b-3840 a c^4 d e f x b-64 a c^5 f^3 x^5-256 a c^5 e f^2 x^4-512 a^2 c^4 e^3+160 a^2 c^4 f^3 x^3-384 a c^5 d f^2 x^3-384 a c^5 e^2 f x^3+2048 a^3 c^3 e f^2-256 a c^5 e^3 x^2+1024 a^2 c^4 e f^2 x^2-1536 a c^5 d e f x^2+768 a c^5 d^2 e-3072 a^2 c^4 d e f-256 c^6 d^3 x+480 a^3 c^3 f^3 x+768 a c^5 d e^2 x-1152 a^2 c^4 d f^2 x+768 a c^5 d^2 f x-1152 a^2 c^4 e^2 f x}{64 c^5 \left (4 a c-b^2\right ) \sqrt {c x^2+b x+a}}-\frac {3 \left (105 f^3 b^4-280 c e f^2 b^3-280 a c f^3 b^2+240 c^2 d f^2 b^2+240 c^2 e^2 f b^2-64 c^3 e^3 b+480 a c^2 e f^2 b-384 c^3 d e f b+80 a^2 c^2 f^3+128 c^4 d e^2-192 a c^3 d f^2+128 c^4 d^2 f-192 a c^3 e^2 f\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{128 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.95, size = 3143, normalized size = 4.84
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 1099, normalized size = 1.69 \begin {gather*} \frac {{\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (b^{2} c^{4} f^{3} - 4 \, a c^{5} f^{3}\right )} x}{b^{2} c^{5} - 4 \, a c^{6}} - \frac {3 \, b^{3} c^{3} f^{3} - 12 \, a b c^{4} f^{3} - 8 \, b^{2} c^{4} f^{2} e + 32 \, a c^{5} f^{2} e}{b^{2} c^{5} - 4 \, a c^{6}}\right )} x + \frac {48 \, b^{2} c^{4} d f^{2} - 192 \, a c^{5} d f^{2} + 21 \, b^{4} c^{2} f^{3} - 104 \, a b^{2} c^{3} f^{3} + 80 \, a^{2} c^{4} f^{3} - 56 \, b^{3} c^{3} f^{2} e + 224 \, a b c^{4} f^{2} e + 48 \, b^{2} c^{4} f e^{2} - 192 \, a c^{5} f e^{2}}{b^{2} c^{5} - 4 \, a c^{6}}\right )} x - \frac {240 \, b^{3} c^{3} d f^{2} - 960 \, a b c^{4} d f^{2} + 105 \, b^{5} c f^{3} - 616 \, a b^{3} c^{2} f^{3} + 784 \, a^{2} b c^{3} f^{3} - 384 \, b^{2} c^{4} d f e + 1536 \, a c^{5} d f e - 280 \, b^{4} c^{2} f^{2} e + 1376 \, a b^{2} c^{3} f^{2} e - 1024 \, a^{2} c^{4} f^{2} e + 240 \, b^{3} c^{3} f e^{2} - 960 \, a b c^{4} f e^{2} - 64 \, b^{2} c^{4} e^{3} + 256 \, a c^{5} e^{3}}{b^{2} c^{5} - 4 \, a c^{6}}\right )} x - \frac {256 \, c^{6} d^{3} + 384 \, b^{2} c^{4} d^{2} f - 768 \, a c^{5} d^{2} f + 720 \, b^{4} c^{2} d f^{2} - 2976 \, a b^{2} c^{3} d f^{2} + 1152 \, a^{2} c^{4} d f^{2} + 315 \, b^{6} f^{3} - 1890 \, a b^{4} c f^{3} + 2704 \, a^{2} b^{2} c^{2} f^{3} - 480 \, a^{3} c^{3} f^{3} - 384 \, b c^{5} d^{2} e - 1152 \, b^{3} c^{3} d f e + 3840 \, a b c^{4} d f e - 840 \, b^{5} c f^{2} e + 4240 \, a b^{3} c^{2} f^{2} e - 3904 \, a^{2} b c^{3} f^{2} e + 384 \, b^{2} c^{4} d e^{2} - 768 \, a c^{5} d e^{2} + 720 \, b^{4} c^{2} f e^{2} - 2976 \, a b^{2} c^{3} f e^{2} + 1152 \, a^{2} c^{4} f e^{2} - 192 \, b^{3} c^{3} e^{3} + 640 \, a b c^{4} e^{3}}{b^{2} c^{5} - 4 \, a c^{6}}\right )} x - \frac {128 \, b c^{5} d^{3} + 384 \, a b c^{4} d^{2} f + 720 \, a b^{3} c^{2} d f^{2} - 2496 \, a^{2} b c^{3} d f^{2} + 315 \, a b^{5} f^{3} - 1680 \, a^{2} b^{3} c f^{3} + 1808 \, a^{3} b c^{2} f^{3} - 768 \, a c^{5} d^{2} e - 1152 \, a b^{2} c^{3} d f e + 3072 \, a^{2} c^{4} d f e - 840 \, a b^{4} c f^{2} e + 3680 \, a^{2} b^{2} c^{2} f^{2} e - 2048 \, a^{3} c^{3} f^{2} e + 384 \, a b c^{4} d e^{2} + 720 \, a b^{3} c^{2} f e^{2} - 2496 \, a^{2} b c^{3} f e^{2} - 192 \, a b^{2} c^{3} e^{3} + 512 \, a^{2} c^{4} e^{3}}{b^{2} c^{5} - 4 \, a c^{6}}}{64 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (128 \, c^{4} d^{2} f + 240 \, b^{2} c^{2} d f^{2} - 192 \, a c^{3} d f^{2} + 105 \, b^{4} f^{3} - 280 \, a b^{2} c f^{3} + 80 \, a^{2} c^{2} f^{3} - 384 \, b c^{3} d f e - 280 \, b^{3} c f^{2} e + 480 \, a b c^{2} f^{2} e + 128 \, c^{4} d e^{2} + 240 \, b^{2} c^{2} f e^{2} - 192 \, a c^{3} f e^{2} - 64 \, b c^{3} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 2827, normalized size = 4.36 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f\,x^2+e\,x+d\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x + f x^{2}\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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