Optimal. Leaf size=679 \[ \frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 \sqrt {c} f^3}-\frac {\sqrt {a+b x+c x^2} (-5 b f+4 c e-2 c f x)}{4 f^2} \]
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Rubi [A] time = 11.03, antiderivative size = 678, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {977, 1076, 621, 206, 1032, 724} \begin {gather*} \frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac {\left (-2 f^3 \left (b^2 d-a^2 f\right )-\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+4 c d f^2 (b e-a f)-2 c^2 d f \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 \sqrt {c} f^3}-\frac {\sqrt {a+b x+c x^2} (-5 b f+4 c e-2 c f x)}{4 f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 977
Rule 1032
Rule 1076
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}-\frac {\int \frac {\frac {1}{4} \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} \left (8 c^2 d e-4 a c e f-b f (5 b e-16 a f)+4 b c \left (e^2-4 d f\right )\right ) x-\frac {1}{4} \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) x^2}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 f^2}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}-\frac {\int \frac {\frac {1}{4} f \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} d \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )+\left (\frac {1}{4} f \left (-8 c^2 d e+4 a c e f+b f (5 b e-16 a f)-4 b c \left (e^2-4 d f\right )\right )-\frac {1}{4} e \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right ) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 f^3}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 f^3}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\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 )}{4 f^3}-\frac {\left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{f^3 \sqrt {e^2-4 d f}}+\frac {\left (2 f \left (\frac {1}{4} f \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} d \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (\frac {1}{4} f \left (-8 c^2 d e+4 a c e f+b f (5 b e-16 a f)-4 b c \left (e^2-4 d f\right )\right )-\frac {1}{4} e \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f^3 \sqrt {e^2-4 d f}}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} f^3}+\frac {\left (2 \left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e-\sqrt {e^2-4 d f}\right )+4 c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f^3 \sqrt {e^2-4 d f}}-\frac {\left (2 f \left (\frac {1}{4} f \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} d \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (\frac {1}{4} f \left (-8 c^2 d e+4 a c e f+b f (5 b e-16 a f)-4 b c \left (e^2-4 d f\right )\right )-\frac {1}{4} e \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e+\sqrt {e^2-4 d f}\right )+4 c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f^3 \sqrt {e^2-4 d f}}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} f^3}+\frac {\left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left ((c e-b f) \left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}}\\ \end {align*}
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Mathematica [A] time = 4.58, size = 1232, normalized size = 1.81 \begin {gather*} \frac {\sqrt {a+x (b+c x)} \left (-4 \left (e+\sqrt {e^2-4 d f}\right )^2 c^2+4 f \left (e+\sqrt {e^2-4 d f}\right ) x c^2-16 a f^2 c+10 b f \left (e+\sqrt {e^2-4 d f}\right ) c-4 b f^2 x c-2 b^2 f^2\right )+2 \sqrt {a+x (b+c x)} \left (-2 \left (-2 e^2+f x e+2 \sqrt {e^2-4 d f} e+4 d f-f \sqrt {e^2-4 d f} x\right ) c^2+f \left (8 a f+b \left (-5 e+2 f x+5 \sqrt {e^2-4 d f}\right )\right ) c+b^2 f^2\right )-\frac {\left (b f+c \left (\sqrt {e^2-4 d f}-e\right )\right ) \left (4 \left (-e^2+\sqrt {e^2-4 d f} e+2 d f\right ) c^2-4 f \left (3 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right ) c+b^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c} f}+\frac {\left (c \left (e+\sqrt {e^2-4 d f}\right )-b f\right ) \left (4 \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) c^2-4 f \left (b \left (e+\sqrt {e^2-4 d f}\right )-3 a f\right ) c-b^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c} f}+\frac {8 \sqrt {2} c \left (\left (e^4+\sqrt {e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt {e^2-4 d f} e+2 d^2 f^2\right ) c^2+2 f \left (a f \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )-b \left (e^3+\sqrt {e^2-4 d f} e^2-3 d f e-d f \sqrt {e^2-4 d f}\right )\right ) c+f^2 \left (\left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) b^2-2 a f \left (e+\sqrt {e^2-4 d f}\right ) b+2 a^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {4 a f-2 c \left (e+\sqrt {e^2-4 d f}\right ) x-b \left (e-2 f x+\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+x (b+c x)}}\right )}{f \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}}+\frac {8 \sqrt {2} c \left (\left (-e^4+\sqrt {e^2-4 d f} e^3+4 d f e^2-2 d f \sqrt {e^2-4 d f} e-2 d^2 f^2\right ) c^2+2 f \left (a f \left (-e^2+\sqrt {e^2-4 d f} e+2 d f\right )+b \left (e^3-\sqrt {e^2-4 d f} e^2-3 d f e+d f \sqrt {e^2-4 d f}\right )\right ) c+f^2 \left (\left (-e^2+\sqrt {e^2-4 d f} e+2 d f\right ) b^2+2 a f \left (e-\sqrt {e^2-4 d f}\right ) b-2 a^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 c \left (\sqrt {e^2-4 d f}-e\right ) x+b \left (-e+2 f x+\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right )} \sqrt {a+x (b+c x)}}\right )}{f \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right )}}}{16 c f^2 \sqrt {e^2-4 d f}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 1.67, size = 1160, normalized size = 1.71 \begin {gather*} \frac {\sqrt {c x^2+b x+a} (-4 c e+5 b f+2 c f x)}{4 f^2}+\frac {\left (-8 e^2 c^2+8 d f c^2-12 a f^2 c+12 b e f c-3 b^2 f^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{8 \sqrt {c} f^3}+\frac {\text {RootSum}\left [f \text {$\#$1}^4-2 \sqrt {c} e \text {$\#$1}^3+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+b^2 d-a b e+a^2 f\&,\frac {d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b^3+e f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 b^2-a e f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b^2-2 c d e f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b^2-2 \sqrt {c} d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} b^2-2 a f^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 b+2 c d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 b-2 c e^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 b+a^2 f^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b+c^2 d e^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b-c^2 d^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b+2 a c e^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) b+4 c^{3/2} d e f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} b+c^2 e^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2+2 a c e f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2-2 c^2 d e f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2-a c^2 e^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )-2 a^2 c e f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )+2 a c^2 d e f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )+2 a^2 \sqrt {c} f^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}-2 c^{5/2} d e^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}-4 a c^{3/2} d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}+2 c^{5/2} d^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}}{-2 f \text {$\#$1}^3+3 \sqrt {c} e \text {$\#$1}^2-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+2 b \sqrt {c} d-a \sqrt {c} e}\&\right ]}{f^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 22523, normalized size = 33.17 \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 (c\,x^2+b\,x+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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