Optimal. Leaf size=213 \[ -\frac {(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {b^6 B \log (d+e x)}{e^8} \]
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Rubi [A] time = 0.26, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 43} \[ -\frac {(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {b^6 B \log (d+e x)}{e^8} \]
Antiderivative was successfully verified.
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Rule 43
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx &=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}+\frac {B \int \frac {(a+b x)^6}{(d+e x)^7} \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}+\frac {B \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}+\frac {b^6 B \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [B] time = 0.48, size = 615, normalized size = 2.89 \[ -\frac {10 a^6 e^6 (6 A e+B (d+7 e x))+12 a^5 b e^5 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+60 a b^5 e \left (A e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+b^6 \left (60 A e \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )-B d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )-420 b^6 B (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 939, normalized size = 4.41 \[ \frac {1089 \, B b^{6} d^{7} - 60 \, A a^{6} e^{7} - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - 10 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 420 \, {\left (7 \, B b^{6} d e^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B b^{6} d^{2} e^{5} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} - {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 350 \, {\left (77 \, B b^{6} d^{3} e^{4} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} - 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 175 \, {\left (175 \, B b^{6} d^{4} e^{3} - 12 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} - 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} - 3 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 21 \, {\left (959 \, B b^{6} d^{5} e^{2} - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} - 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B b^{6} d^{6} e - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} - 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} - 10 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x + 420 \, {\left (B b^{6} e^{7} x^{7} + 7 \, B b^{6} d e^{6} x^{6} + 21 \, B b^{6} d^{2} e^{5} x^{5} + 35 \, B b^{6} d^{3} e^{4} x^{4} + 35 \, B b^{6} d^{4} e^{3} x^{3} + 21 \, B b^{6} d^{5} e^{2} x^{2} + 7 \, B b^{6} d^{6} e x + B b^{6} d^{7}\right )} \log \left (e x + d\right )}{420 \, {\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 779, normalized size = 3.66 \[ B b^{6} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (420 \, {\left (7 \, B b^{6} d e^{5} - 6 \, B a b^{5} e^{6} - A b^{6} e^{6}\right )} x^{6} + 630 \, {\left (21 \, B b^{6} d^{2} e^{4} - 12 \, B a b^{5} d e^{5} - 2 \, A b^{6} d e^{5} - 5 \, B a^{2} b^{4} e^{6} - 2 \, A a b^{5} e^{6}\right )} x^{5} + 350 \, {\left (77 \, B b^{6} d^{3} e^{3} - 36 \, B a b^{5} d^{2} e^{4} - 6 \, A b^{6} d^{2} e^{4} - 15 \, B a^{2} b^{4} d e^{5} - 6 \, A a b^{5} d e^{5} - 8 \, B a^{3} b^{3} e^{6} - 6 \, A a^{2} b^{4} e^{6}\right )} x^{4} + 175 \, {\left (175 \, B b^{6} d^{4} e^{2} - 72 \, B a b^{5} d^{3} e^{3} - 12 \, A b^{6} d^{3} e^{3} - 30 \, B a^{2} b^{4} d^{2} e^{4} - 12 \, A a b^{5} d^{2} e^{4} - 16 \, B a^{3} b^{3} d e^{5} - 12 \, A a^{2} b^{4} d e^{5} - 9 \, B a^{4} b^{2} e^{6} - 12 \, A a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (959 \, B b^{6} d^{5} e - 360 \, B a b^{5} d^{4} e^{2} - 60 \, A b^{6} d^{4} e^{2} - 150 \, B a^{2} b^{4} d^{3} e^{3} - 60 \, A a b^{5} d^{3} e^{3} - 80 \, B a^{3} b^{3} d^{2} e^{4} - 60 \, A a^{2} b^{4} d^{2} e^{4} - 45 \, B a^{4} b^{2} d e^{5} - 60 \, A a^{3} b^{3} d e^{5} - 24 \, B a^{5} b e^{6} - 60 \, A a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (1029 \, B b^{6} d^{6} - 360 \, B a b^{5} d^{5} e - 60 \, A b^{6} d^{5} e - 150 \, B a^{2} b^{4} d^{4} e^{2} - 60 \, A a b^{5} d^{4} e^{2} - 80 \, B a^{3} b^{3} d^{3} e^{3} - 60 \, A a^{2} b^{4} d^{3} e^{3} - 45 \, B a^{4} b^{2} d^{2} e^{4} - 60 \, A a^{3} b^{3} d^{2} e^{4} - 24 \, B a^{5} b d e^{5} - 60 \, A a^{4} b^{2} d e^{5} - 10 \, B a^{6} e^{6} - 60 \, A a^{5} b e^{6}\right )} x + {\left (1089 \, B b^{6} d^{7} - 360 \, B a b^{5} d^{6} e - 60 \, A b^{6} d^{6} e - 150 \, B a^{2} b^{4} d^{5} e^{2} - 60 \, A a b^{5} d^{5} e^{2} - 80 \, B a^{3} b^{3} d^{4} e^{3} - 60 \, A a^{2} b^{4} d^{4} e^{3} - 45 \, B a^{4} b^{2} d^{3} e^{4} - 60 \, A a^{3} b^{3} d^{3} e^{4} - 24 \, B a^{5} b d^{2} e^{5} - 60 \, A a^{4} b^{2} d^{2} e^{5} - 10 \, B a^{6} d e^{6} - 60 \, A a^{5} b d e^{6} - 60 \, A a^{6} e^{7}\right )} e^{\left (-1\right )}\right )} e^{\left (-7\right )}}{420 \, {\left (x e + d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 1227, normalized size = 5.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 842, normalized size = 3.95 \[ \frac {1089 \, B b^{6} d^{7} - 60 \, A a^{6} e^{7} - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - 10 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 420 \, {\left (7 \, B b^{6} d e^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B b^{6} d^{2} e^{5} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} - {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 350 \, {\left (77 \, B b^{6} d^{3} e^{4} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} - 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 175 \, {\left (175 \, B b^{6} d^{4} e^{3} - 12 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} - 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} - 3 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 21 \, {\left (959 \, B b^{6} d^{5} e^{2} - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} - 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B b^{6} d^{6} e - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} - 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} - 10 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{420 \, {\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} + \frac {B b^{6} \log \left (e x + d\right )}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 1046, normalized size = 4.91 \[ -\frac {\frac {A\,a^6\,e^7}{7}-\frac {363\,B\,b^6\,d^7}{140}+\frac {A\,b^6\,d^6\,e}{7}+\frac {B\,a^6\,d\,e^6}{42}-B\,b^6\,d^7\,\ln \left (d+e\,x\right )+\frac {B\,a^6\,e^7\,x}{6}+A\,b^6\,e^7\,x^6-\frac {343\,B\,b^6\,d^6\,e\,x}{20}+\frac {A\,a\,b^5\,d^5\,e^2}{7}+\frac {2\,B\,a^5\,b\,d^2\,e^5}{35}+3\,A\,a\,b^5\,e^7\,x^5+\frac {6\,B\,a^5\,b\,e^7\,x^2}{5}+6\,B\,a\,b^5\,e^7\,x^6+A\,b^6\,d^5\,e^2\,x+3\,A\,b^6\,d\,e^6\,x^5-7\,B\,b^6\,d\,e^6\,x^6-B\,b^6\,e^7\,x^7\,\ln \left (d+e\,x\right )+\frac {A\,a^2\,b^4\,d^4\,e^3}{7}+\frac {A\,a^3\,b^3\,d^3\,e^4}{7}+\frac {A\,a^4\,b^2\,d^2\,e^5}{7}+\frac {5\,B\,a^2\,b^4\,d^5\,e^2}{14}+\frac {4\,B\,a^3\,b^3\,d^4\,e^3}{21}+\frac {3\,B\,a^4\,b^2\,d^3\,e^4}{28}+3\,A\,a^4\,b^2\,e^7\,x^2+5\,A\,a^3\,b^3\,e^7\,x^3+5\,A\,a^2\,b^4\,e^7\,x^4+\frac {15\,B\,a^4\,b^2\,e^7\,x^3}{4}+\frac {20\,B\,a^3\,b^3\,e^7\,x^4}{3}+\frac {15\,B\,a^2\,b^4\,e^7\,x^5}{2}+3\,A\,b^6\,d^4\,e^3\,x^2+5\,A\,b^6\,d^3\,e^4\,x^3+5\,A\,b^6\,d^2\,e^5\,x^4-\frac {959\,B\,b^6\,d^5\,e^2\,x^2}{20}-\frac {875\,B\,b^6\,d^4\,e^3\,x^3}{12}-\frac {385\,B\,b^6\,d^3\,e^4\,x^4}{6}-\frac {63\,B\,b^6\,d^2\,e^5\,x^5}{2}+\frac {A\,a^5\,b\,d\,e^6}{7}+\frac {6\,B\,a\,b^5\,d^6\,e}{7}+A\,a^5\,b\,e^7\,x+A\,a^2\,b^4\,d^3\,e^4\,x+A\,a^3\,b^3\,d^2\,e^5\,x+3\,A\,a\,b^5\,d^3\,e^4\,x^2+3\,A\,a^3\,b^3\,d\,e^6\,x^2+5\,A\,a\,b^5\,d^2\,e^5\,x^3+5\,A\,a^2\,b^4\,d\,e^6\,x^3+\frac {5\,B\,a^2\,b^4\,d^4\,e^3\,x}{2}+\frac {4\,B\,a^3\,b^3\,d^3\,e^4\,x}{3}+\frac {3\,B\,a^4\,b^2\,d^2\,e^5\,x}{4}+18\,B\,a\,b^5\,d^4\,e^3\,x^2+\frac {9\,B\,a^4\,b^2\,d\,e^6\,x^2}{4}+30\,B\,a\,b^5\,d^3\,e^4\,x^3+\frac {20\,B\,a^3\,b^3\,d\,e^6\,x^3}{3}+30\,B\,a\,b^5\,d^2\,e^5\,x^4+\frac {25\,B\,a^2\,b^4\,d\,e^6\,x^4}{2}-21\,B\,b^6\,d^5\,e^2\,x^2\,\ln \left (d+e\,x\right )-35\,B\,b^6\,d^4\,e^3\,x^3\,\ln \left (d+e\,x\right )-35\,B\,b^6\,d^3\,e^4\,x^4\,\ln \left (d+e\,x\right )-21\,B\,b^6\,d^2\,e^5\,x^5\,\ln \left (d+e\,x\right )+\frac {2\,B\,a^5\,b\,d\,e^6\,x}{5}-7\,B\,b^6\,d^6\,e\,x\,\ln \left (d+e\,x\right )+3\,A\,a^2\,b^4\,d^2\,e^5\,x^2+\frac {15\,B\,a^2\,b^4\,d^3\,e^4\,x^2}{2}+4\,B\,a^3\,b^3\,d^2\,e^5\,x^2+\frac {25\,B\,a^2\,b^4\,d^2\,e^5\,x^3}{2}+A\,a\,b^5\,d^4\,e^3\,x+A\,a^4\,b^2\,d\,e^6\,x+5\,A\,a\,b^5\,d\,e^6\,x^4+6\,B\,a\,b^5\,d^5\,e^2\,x+18\,B\,a\,b^5\,d\,e^6\,x^5-7\,B\,b^6\,d\,e^6\,x^6\,\ln \left (d+e\,x\right )}{e^8\,{\left (d+e\,x\right )}^7} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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