Optimal. Leaf size=774 \[ \frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
[Out]
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Rubi [A]
time = 1.39, antiderivative size = 774, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {955, 1667, 857,
732, 435, 430} \begin {gather*} -\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3-\left (c^3 \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2-\left (c^2 \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}+\frac {2 e^2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 857
Rule 955
Rule 1667
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\int \frac {(d+e x) \left (-7 c d^2 f+e (b d f+4 a e f+a d g)-(c d (12 e f+7 d g)-e (5 b e f+2 b d g+5 a e g)) x-e (c e f+11 c d g-6 b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{7 c}\\ &=\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {2 \int \frac {-\frac {1}{2} g \left (6 b^2 e^3 f^2 g+b e f \left (18 a e^2 g^2-c \left (e^2 f^2+11 d e f g+5 d^2 g^2\right )\right )+c g \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e f g+5 d^2 g^2\right )\right )\right )-\frac {1}{2} g \left (6 b e^3 g^2 (5 b f+3 a g)-c^2 \left (2 e^3 f^3+22 d e^2 f^2 g-95 d^2 e f g^2-35 d^3 g^3\right )-c e g \left (a e g (23 e f+63 d g)-b \left (7 e^2 f^2-85 d e f g-10 d^2 g^2\right )\right )\right ) x-\frac {1}{2} e g^2 \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{35 c^2 g^3}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {4 \int \frac {\frac {1}{4} g^3 \left (24 b^3 e^3 f g^2+b^2 e^2 g \left (24 a e g^2-c f (5 e f+84 d g)\right )-b c e \left (6 a e g^2 (11 e f+14 d g)+c f \left (4 e^2 f^2-21 d e f g-105 d^2 g^2\right )\right )-c g \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )\right )+\frac {1}{4} g^3 \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 g^5}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\left (e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 g^3}-\frac {\left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^3 g^3}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{105 c^4 g^3 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 32.25, size = 1402, normalized size = 1.81 \begin {gather*} \frac {\sqrt {f+g x} \left (a+b x+c x^2\right ) \left (-\frac {2 e \left (4 c^2 e^2 f^2-21 c^2 d e f g+5 b c e^2 f g-105 c^2 d^2 g^2+84 b c d e g^2-24 b^2 e^2 g^2+25 a c e^2 g^2\right )}{105 c^3 g^2}-\frac {2 e^2 (-c e f-21 c d g+6 b e g) x}{35 c^2 g}+\frac {2 e^3 x^2}{7 c}\right )}{\sqrt {a+x (b+c x)}}-\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} \left (-\left (\left (-48 b^3 e^3 g^3+8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)+c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )-c^2 e g \left (a e g (19 e f+189 d g)+b \left (-9 e^2 f^2+63 d e f g+210 d^2 g^2\right )\right )\right ) \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )\right )-\frac {i \sqrt {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {1+\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left (\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)+b \left (-9 e^2 f^2+63 d e f g+210 d^2 g^2\right )\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )+\left (48 b^4 e^3 g^4-8 b^3 e^2 g^3 \left (8 c e f+21 c d g+6 e \sqrt {\left (b^2-4 a c\right ) g^2}\right )+b^2 c e g^2 \left (-152 a e^2 g^2+8 e \sqrt {\left (b^2-4 a c\right ) g^2} (2 e f+21 d g)+c \left (e^2 f^2+231 d e f g+210 d^2 g^2\right )\right )-b \left (-104 a c e^3 g^3 \sqrt {\left (b^2-4 a c\right ) g^2}+105 c^3 d^2 g^3 (3 e f+d g)+c^2 e g \left (-a e g^2 (151 e f+357 d g)+3 \sqrt {\left (b^2-4 a c\right ) g^2} \left (-3 e^2 f^2+21 d e f g+70 d^2 g^2\right )\right )\right )+c^2 \left (50 a^2 e^3 g^4-a e g^2 \left (e \sqrt {\left (b^2-4 a c\right ) g^2} (19 e f+189 d g)+c \left (4 e^2 f^2+294 d e f g+210 d^2 g^2\right )\right )+c \left (210 c d^3 f g^3+\sqrt {\left (b^2-4 a c\right ) g^2} \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right )\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {f+g x}}\right )}{105 c^4 g^4 \sqrt {a+x (b+c x)} \sqrt {\frac {(f+g x)^2 \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )}{g^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(14977\) vs.
\(2(704)=1408\).
time = 0.14, size = 14978, normalized size = 19.35
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{7 c}+\frac {2 \left (3 d \,e^{2} g +f \,e^{3}-\frac {2 e^{3} \left (3 b g +3 c f \right )}{7 c}\right ) x \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{5 c g}+\frac {2 \left (3 g \,d^{2} e +3 f d \,e^{2}-\frac {2 e^{3} \left (\frac {5 a g}{2}+\frac {5 b f}{2}\right )}{7 c}-\frac {2 \left (3 d \,e^{2} g +f \,e^{3}-\frac {2 e^{3} \left (3 b g +3 c f \right )}{7 c}\right ) \left (2 b g +2 c f \right )}{5 c g}\right ) \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{3 c g}+\frac {2 \left (d^{3} f -\frac {2 \left (3 d \,e^{2} g +f \,e^{3}-\frac {2 e^{3} \left (3 b g +3 c f \right )}{7 c}\right ) f a}{5 c g}-\frac {2 \left (3 g \,d^{2} e +3 f d \,e^{2}-\frac {2 e^{3} \left (\frac {5 a g}{2}+\frac {5 b f}{2}\right )}{7 c}-\frac {2 \left (3 d \,e^{2} g +f \,e^{3}-\frac {2 e^{3} \left (3 b g +3 c f \right )}{7 c}\right ) \left (2 b g +2 c f \right )}{5 c g}\right ) \left (\frac {a g}{2}+\frac {b f}{2}\right )}{3 c g}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}+\frac {2 \left (g \,d^{3}+3 f \,d^{2} e -\frac {4 f a \,e^{3}}{7 c}-\frac {2 \left (3 d \,e^{2} g +f \,e^{3}-\frac {2 e^{3} \left (3 b g +3 c f \right )}{7 c}\right ) \left (\frac {3 a g}{2}+\frac {3 b f}{2}\right )}{5 c g}-\frac {2 \left (3 g \,d^{2} e +3 f d \,e^{2}-\frac {2 e^{3} \left (\frac {5 a g}{2}+\frac {5 b f}{2}\right )}{7 c}-\frac {2 \left (3 d \,e^{2} g +f \,e^{3}-\frac {2 e^{3} \left (3 b g +3 c f \right )}{7 c}\right ) \left (2 b g +2 c f \right )}{5 c g}\right ) \left (b g +c f \right )}{3 c g}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\) | \(1283\) |
risch | \(\text {Expression too large to display}\) | \(4891\) |
default | \(\text {Expression too large to display}\) | \(14978\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.85, size = 883, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left ({\left (210 \, c^{4} d^{3} f g^{3} - 105 \, b c^{3} d^{3} g^{4} - {\left (8 \, c^{4} f^{4} + 5 \, b c^{3} f^{3} g + {\left (10 \, b^{2} c^{2} - 13 \, a c^{3}\right )} f^{2} g^{2} + {\left (40 \, b^{3} c - 113 \, a b c^{2}\right )} f g^{3} - {\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} g^{4}\right )} e^{3} + 21 \, {\left (2 \, c^{4} d f^{3} g + 2 \, b c^{3} d f^{2} g^{2} + {\left (7 \, b^{2} c^{2} - 12 \, a c^{3}\right )} d f g^{3} - {\left (8 \, b^{3} c - 21 \, a b c^{2}\right )} d g^{4}\right )} e^{2} - 105 \, {\left (c^{4} d^{2} f^{2} g^{2} + 2 \, b c^{3} d^{2} f g^{3} - {\left (2 \, b^{2} c^{2} - 3 \, a c^{3}\right )} d^{2} g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) - 3 \, {\left (105 \, c^{4} d^{3} g^{4} + {\left (8 \, c^{4} f^{3} g + 9 \, b c^{3} f^{2} g^{2} + {\left (16 \, b^{2} c^{2} - 19 \, a c^{3}\right )} f g^{3} - 8 \, {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} g^{4}\right )} e^{3} - 21 \, {\left (2 \, c^{4} d f^{2} g^{2} + 3 \, b c^{3} d f g^{3} - {\left (8 \, b^{2} c^{2} - 9 \, a c^{3}\right )} d g^{4}\right )} e^{2} + 105 \, {\left (c^{4} d^{2} f g^{3} - 2 \, b c^{3} d^{2} g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) + 3 \, {\left (105 \, c^{4} d^{2} g^{4} e + {\left (15 \, c^{4} g^{4} x^{2} - 4 \, c^{4} f^{2} g^{2} - 5 \, b c^{3} f g^{3} + {\left (24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} g^{4} + 3 \, {\left (c^{4} f g^{3} - 6 \, b c^{3} g^{4}\right )} x\right )} e^{3} + 21 \, {\left (3 \, c^{4} d g^{4} x + c^{4} d f g^{3} - 4 \, b c^{3} d g^{4}\right )} e^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{315 \, c^{5} g^{4}} \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\right )^{3} \sqrt {f + g x}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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