Optimal. Leaf size=448 \[ -\frac {256 \sqrt {d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.71, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {256 \sqrt {d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
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\begin {align*} \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(7 c e f+13 c d g-10 b e g) \int \frac {(d+e x)^{11/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(8 (7 c e f+13 c d g-10 b e g)) \int \frac {(d+e x)^{9/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{21 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(16 (2 c d-b e) (7 c e f+13 c d g-10 b e g)) \int \frac {(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{35 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (64 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g)\right ) \int \frac {(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 c^4 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (128 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g)\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 c^5 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {256 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g) \sqrt {d+e x}}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 366, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (78 d g+7 e f-15 e g x)-32 b^3 c^2 e^3 \left (953 d^2 g+2 d e (91 f-204 g x)+3 e^2 x (5 g x-14 f)\right )+16 b^2 c^3 e^2 \left (2844 d^3 g+3 d^2 e (287 f-681 g x)+6 d e^2 x (29 g x-77 f)+e^3 x^2 (21 f+5 g x)\right )-2 b c^4 e \left (16563 d^4 g+12 d^3 e (581 f-1482 g x)+6 d^2 e^2 x (449 g x-1106 f)+12 d e^3 x^2 (63 f+16 g x)+e^4 x^3 (28 f+15 g x)\right )+c^5 \left (9414 d^5 g+3 d^4 e (1687 f-4707 g x)+12 d^3 e^2 x (292 g x-637 f)+2 d^2 e^3 x^2 (903 f+257 g x)+2 d e^4 x^3 (98 f+57 g x)+3 e^5 x^4 (7 f+5 g x)\right )\right )}{105 c^6 e^2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.48, size = 598, normalized size = 1.33 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (-1280 b^5 e^5 g-1920 b^4 c e^4 g (d+e x)+11904 b^4 c d e^4 g+896 b^4 c e^5 f-44032 b^3 c^2 d^2 e^3 g+1344 b^3 c^2 e^4 f (d+e x)-7168 b^3 c^2 d e^4 f-480 b^3 c^2 e^3 g (d+e x)^2+14016 b^3 c^2 d e^3 g (d+e x)+80896 b^2 c^3 d^3 e^2 g+21504 b^2 c^3 d^2 e^3 f-38016 b^2 c^3 d^2 e^2 g (d+e x)+336 b^2 c^3 e^3 f (d+e x)^2-8064 b^2 c^3 d e^3 f (d+e x)+80 b^2 c^3 e^2 g (d+e x)^3+2544 b^2 c^3 d e^2 g (d+e x)^2-73728 b c^4 d^4 e g-28672 b c^4 d^3 e^2 f+45312 b c^4 d^3 e g (d+e x)+16128 b c^4 d^2 e^2 f (d+e x)-4416 b c^4 d^2 e g (d+e x)^2-56 b c^4 e^2 f (d+e x)^3-1344 b c^4 d e^2 f (d+e x)^2-30 b c^4 e g (d+e x)^4-264 b c^4 d e g (d+e x)^3+26624 c^5 d^5 g+14336 c^5 d^4 e f-19968 c^5 d^4 g (d+e x)-10752 c^5 d^3 e f (d+e x)+2496 c^5 d^3 g (d+e x)^2+1344 c^5 d^2 e f (d+e x)^2+208 c^5 d^2 g (d+e x)^3+21 c^5 e f (d+e x)^4+112 c^5 d e f (d+e x)^3+15 c^5 g (d+e x)^5+39 c^5 d g (d+e x)^4\right )}{105 c^6 e^2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 578, normalized size = 1.29 \begin {gather*} -\frac {2 \, {\left (15 \, c^{5} e^{5} g x^{5} + 3 \, {\left (7 \, c^{5} e^{5} f + 2 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 2 \, {\left (14 \, {\left (7 \, c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} f + {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 6 \, {\left (7 \, {\left (43 \, c^{5} d^{2} e^{3} - 36 \, b c^{4} d e^{4} + 8 \, b^{2} c^{3} e^{5}\right )} f + 2 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 7 \, {\left (723 \, c^{5} d^{4} e - 1992 \, b c^{4} d^{3} e^{2} + 1968 \, b^{2} c^{3} d^{2} e^{3} - 832 \, b^{3} c^{2} d e^{4} + 128 \, b^{4} c e^{5}\right )} f + 2 \, {\left (4707 \, c^{5} d^{5} - 16563 \, b c^{4} d^{4} e + 22752 \, b^{2} c^{3} d^{3} e^{2} - 15248 \, b^{3} c^{2} d^{2} e^{3} + 4992 \, b^{4} c d e^{4} - 640 \, b^{5} e^{5}\right )} g - 3 \, {\left (28 \, {\left (91 \, c^{5} d^{3} e^{2} - 158 \, b c^{4} d^{2} e^{3} + 88 \, b^{2} c^{3} d e^{4} - 16 \, b^{3} c^{2} e^{5}\right )} f + {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c^{8} e^{5} x^{3} + c^{8} d^{3} e^{2} - 2 \, b c^{7} d^{2} e^{3} + b^{2} c^{6} d e^{4} - {\left (c^{8} d e^{4} - 2 \, b c^{7} e^{5}\right )} x^{2} - {\left (c^{8} d^{2} e^{3} - b^{2} c^{6} e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.05, size = 535, normalized size = 1.19 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 f \,d^{4} c^{5} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{105 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} c^{6} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 511, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 723 \, c^{4} d^{4} - 1992 \, b c^{3} d^{3} e + 1968 \, b^{2} c^{2} d^{2} e^{2} - 832 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + 4 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} x^{3} + 6 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} x^{2} - 12 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} x\right )} f}{15 \, {\left (c^{6} e^{2} x - c^{6} d e + b c^{5} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (15 \, c^{5} e^{5} x^{5} + 9414 \, c^{5} d^{5} - 33126 \, b c^{4} d^{4} e + 45504 \, b^{2} c^{3} d^{3} e^{2} - 30496 \, b^{3} c^{2} d^{2} e^{3} + 9984 \, b^{4} c d e^{4} - 1280 \, b^{5} e^{5} + 6 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} x^{4} + 2 \, {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} x^{3} + 12 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} x^{2} - 3 \, {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} x\right )} g}{105 \, {\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt {-c e x + c d - b e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 596, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (-2560\,g\,b^5\,e^5+19968\,g\,b^4\,c\,d\,e^4+1792\,f\,b^4\,c\,e^5-60992\,g\,b^3\,c^2\,d^2\,e^3-11648\,f\,b^3\,c^2\,d\,e^4+91008\,g\,b^2\,c^3\,d^3\,e^2+27552\,f\,b^2\,c^3\,d^2\,e^3-66252\,g\,b\,c^4\,d^4\,e-27888\,f\,b\,c^4\,d^3\,e^2+18828\,g\,c^5\,d^5+10122\,f\,c^5\,d^4\,e\right )}{105\,c^8\,e^5}+\frac {2\,g\,x^5\,\sqrt {d+e\,x}}{7\,c^3}+\frac {4\,x^3\,\sqrt {d+e\,x}\,\left (40\,g\,b^2\,e^2-192\,g\,b\,c\,d\,e-28\,f\,b\,c\,e^2+257\,g\,c^2\,d^2+98\,f\,c^2\,d\,e\right )}{105\,c^5\,e^2}+\frac {2\,x^4\,\sqrt {d+e\,x}\,\left (38\,c\,d\,g-10\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^4\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (3840\,g\,b^4\,c\,e^5-26112\,g\,b^3\,c^2\,d\,e^4-2688\,f\,b^3\,c^2\,e^5+65376\,g\,b^2\,c^3\,d^2\,e^3+14784\,f\,b^2\,c^3\,d\,e^4-71136\,g\,b\,c^4\,d^3\,e^2-26544\,f\,b\,c^4\,d^2\,e^3+28242\,g\,c^5\,d^4\,e+15288\,f\,c^5\,d^3\,e^2\right )}{105\,c^8\,e^5}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,g\,b^3\,c^2\,e^5+5568\,g\,b^2\,c^3\,d\,e^4+672\,f\,b^2\,c^3\,e^5-10776\,g\,b\,c^4\,d^2\,e^3-3024\,f\,b\,c^4\,d\,e^4+7008\,g\,c^5\,d^3\,e^2+3612\,f\,c^5\,d^2\,e^3\right )}{105\,c^8\,e^5}\right )}{x^3+\frac {x\,\left (105\,b^2\,c^6\,e^5-105\,c^8\,d^2\,e^3\right )}{105\,c^8\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \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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