Optimal. Leaf size=86 \[ \frac {(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx &=-\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{8 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7}\\ \end {align*}
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Mathematica [B] time = 0.29, size = 597, normalized size = 6.94 \begin {gather*} -\frac {a^6 e^6 (7 A e+B (d+8 e x))+2 a^5 b e^5 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a^4 b^2 e^4 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a^2 b^4 e^2 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+2 a b^5 e \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 B \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{56 e^8 (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.39, size = 823, normalized size = 9.57 \begin {gather*} -\frac {56 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 7 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 28 \, {\left (7 \, B b^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 56 \, {\left (7 \, B b^{6} d^{2} e^{5} + {\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} + 70 \, {\left (7 \, B b^{6} d^{3} e^{4} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 56 \, {\left (7 \, B b^{6} d^{4} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 28 \, {\left (7 \, B b^{6} d^{5} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 8 \, {\left (7 \, B b^{6} d^{6} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{56 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 854, normalized size = 9.93 \begin {gather*} -\frac {{\left (56 \, B b^{6} x^{7} e^{7} + 196 \, B b^{6} d x^{6} e^{6} + 392 \, B b^{6} d^{2} x^{5} e^{5} + 490 \, B b^{6} d^{3} x^{4} e^{4} + 392 \, B b^{6} d^{4} x^{3} e^{3} + 196 \, B b^{6} d^{5} x^{2} e^{2} + 56 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 168 \, B a b^{5} x^{6} e^{7} + 28 \, A b^{6} x^{6} e^{7} + 336 \, B a b^{5} d x^{5} e^{6} + 56 \, A b^{6} d x^{5} e^{6} + 420 \, B a b^{5} d^{2} x^{4} e^{5} + 70 \, A b^{6} d^{2} x^{4} e^{5} + 336 \, B a b^{5} d^{3} x^{3} e^{4} + 56 \, A b^{6} d^{3} x^{3} e^{4} + 168 \, B a b^{5} d^{4} x^{2} e^{3} + 28 \, A b^{6} d^{4} x^{2} e^{3} + 48 \, B a b^{5} d^{5} x e^{2} + 8 \, A b^{6} d^{5} x e^{2} + 6 \, B a b^{5} d^{6} e + A b^{6} d^{6} e + 280 \, B a^{2} b^{4} x^{5} e^{7} + 112 \, A a b^{5} x^{5} e^{7} + 350 \, B a^{2} b^{4} d x^{4} e^{6} + 140 \, A a b^{5} d x^{4} e^{6} + 280 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 112 \, A a b^{5} d^{2} x^{3} e^{5} + 140 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 56 \, A a b^{5} d^{3} x^{2} e^{4} + 40 \, B a^{2} b^{4} d^{4} x e^{3} + 16 \, A a b^{5} d^{4} x e^{3} + 5 \, B a^{2} b^{4} d^{5} e^{2} + 2 \, A a b^{5} d^{5} e^{2} + 280 \, B a^{3} b^{3} x^{4} e^{7} + 210 \, A a^{2} b^{4} x^{4} e^{7} + 224 \, B a^{3} b^{3} d x^{3} e^{6} + 168 \, A a^{2} b^{4} d x^{3} e^{6} + 112 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 84 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 32 \, B a^{3} b^{3} d^{3} x e^{4} + 24 \, A a^{2} b^{4} d^{3} x e^{4} + 4 \, B a^{3} b^{3} d^{4} e^{3} + 3 \, A a^{2} b^{4} d^{4} e^{3} + 168 \, B a^{4} b^{2} x^{3} e^{7} + 224 \, A a^{3} b^{3} x^{3} e^{7} + 84 \, B a^{4} b^{2} d x^{2} e^{6} + 112 \, A a^{3} b^{3} d x^{2} e^{6} + 24 \, B a^{4} b^{2} d^{2} x e^{5} + 32 \, A a^{3} b^{3} d^{2} x e^{5} + 3 \, B a^{4} b^{2} d^{3} e^{4} + 4 \, A a^{3} b^{3} d^{3} e^{4} + 56 \, B a^{5} b x^{2} e^{7} + 140 \, A a^{4} b^{2} x^{2} e^{7} + 16 \, B a^{5} b d x e^{6} + 40 \, A a^{4} b^{2} d x e^{6} + 2 \, B a^{5} b d^{2} e^{5} + 5 \, A a^{4} b^{2} d^{2} e^{5} + 8 \, B a^{6} x e^{7} + 48 \, A a^{5} b x e^{7} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} + 7 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{56 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 814, normalized size = 9.47 \begin {gather*} -\frac {B \,b^{6}}{\left (e x +d \right ) e^{8}}-\frac {\left (A b e +6 B a e -7 B b d \right ) b^{5}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {\left (2 A a b \,e^{2}-2 A d \,b^{2} e +5 B \,a^{2} e^{2}-12 B a b d e +7 B \,b^{2} d^{2}\right ) b^{4}}{\left (e x +d \right )^{3} e^{8}}-\frac {5 \left (3 A \,a^{2} b \,e^{3}-6 A d a \,b^{2} e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B d \,a^{2} b \,e^{2}+18 B a \,b^{2} d^{2} e -7 B \,b^{3} d^{3}\right ) b^{3}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {\left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) b^{2}}{\left (e x +d \right )^{5} e^{8}}-\frac {\left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) b}{2 \left (e x +d \right )^{6} e^{8}}-\frac {A \,a^{6} e^{7}-6 A d \,a^{5} b \,e^{6}+15 A \,d^{2} a^{4} b^{2} e^{5}-20 A \,d^{3} a^{3} b^{3} e^{4}+15 A \,d^{4} a^{2} b^{4} e^{3}-6 A \,d^{5} a \,b^{5} e^{2}+A \,d^{6} b^{6} e -B d \,a^{6} e^{6}+6 B \,d^{2} a^{5} b \,e^{5}-15 B \,d^{3} a^{4} b^{2} e^{4}+20 B \,d^{4} a^{3} b^{3} e^{3}-15 B \,d^{5} a^{2} b^{4} e^{2}+6 B \,d^{6} a \,b^{5} e -B \,b^{6} d^{7}}{8 \left (e x +d \right )^{8} e^{8}}-\frac {6 a^{5} b A \,e^{6}-30 A d \,a^{4} b^{2} e^{5}+60 A \,d^{2} a^{3} b^{3} e^{4}-60 A \,d^{3} a^{2} b^{4} e^{3}+30 A \,d^{4} a \,b^{5} e^{2}-6 A \,d^{5} b^{6} e +a^{6} B \,e^{6}-12 B d \,a^{5} b \,e^{5}+45 B \,d^{2} a^{4} b^{2} e^{4}-80 B \,d^{3} a^{3} b^{3} e^{3}+75 B \,d^{4} a^{2} b^{4} e^{2}-36 B \,d^{5} a \,b^{5} e +7 B \,b^{6} d^{6}}{7 \left (e x +d \right )^{7} e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.81, size = 823, normalized size = 9.57 \begin {gather*} -\frac {56 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 7 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 28 \, {\left (7 \, B b^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 56 \, {\left (7 \, B b^{6} d^{2} e^{5} + {\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} + 70 \, {\left (7 \, B b^{6} d^{3} e^{4} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 56 \, {\left (7 \, B b^{6} d^{4} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 28 \, {\left (7 \, B b^{6} d^{5} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 8 \, {\left (7 \, B b^{6} d^{6} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{56 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 854, normalized size = 9.93 \begin {gather*} -\frac {\frac {B\,a^6\,d\,e^6+7\,A\,a^6\,e^7+2\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+3\,B\,a^4\,b^2\,d^3\,e^4+5\,A\,a^4\,b^2\,d^2\,e^5+4\,B\,a^3\,b^3\,d^4\,e^3+4\,A\,a^3\,b^3\,d^3\,e^4+5\,B\,a^2\,b^4\,d^5\,e^2+3\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e+2\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+A\,b^6\,d^6\,e}{56\,e^8}+\frac {x\,\left (B\,a^6\,e^6+2\,B\,a^5\,b\,d\,e^5+6\,A\,a^5\,b\,e^6+3\,B\,a^4\,b^2\,d^2\,e^4+5\,A\,a^4\,b^2\,d\,e^5+4\,B\,a^3\,b^3\,d^3\,e^3+4\,A\,a^3\,b^3\,d^2\,e^4+5\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3+6\,B\,a\,b^5\,d^5\,e+2\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+A\,b^6\,d^5\,e\right )}{7\,e^7}+\frac {5\,b^3\,x^4\,\left (4\,B\,a^3\,e^3+5\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+6\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (A\,b\,e+6\,B\,a\,e+7\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^5\,e^5+3\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5+4\,B\,a^3\,b^2\,d^2\,e^3+4\,A\,a^3\,b^2\,d\,e^4+5\,B\,a^2\,b^3\,d^3\,e^2+3\,A\,a^2\,b^3\,d^2\,e^3+6\,B\,a\,b^4\,d^4\,e+2\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+A\,b^5\,d^4\,e\right )}{2\,e^6}+\frac {b^2\,x^3\,\left (3\,B\,a^4\,e^4+4\,B\,a^3\,b\,d\,e^3+4\,A\,a^3\,b\,e^4+5\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3+6\,B\,a\,b^3\,d^3\,e+2\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{e^5}+\frac {b^4\,x^5\,\left (5\,B\,a^2\,e^2+6\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{e^3}+\frac {B\,b^6\,x^7}{e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \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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