Optimal. Leaf size=375 \[ \frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt {x} (13 b c-5 a d) (b c-a d)}{2 d^4}-\frac {x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{9/2}}{9 d^2} \]
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Rubi [A] time = 0.44, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {463, 459, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac {\sqrt {x} (13 b c-5 a d) (b c-a d)}{2 d^4}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{17/4}}+\frac {2 b^2 x^{9/2}}{9 d^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 321
Rule 329
Rule 459
Rule 463
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{7/2} \left (\frac {1}{2} (3 b c-5 a d) (3 b c-a d)-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((13 b c-5 a d) (b c-a d)) \int \frac {x^{7/2}}{c+d x^2} \, dx}{4 c d^2}\\ &=-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((13 b c-5 a d) (b c-a d)) \int \frac {x^{3/2}}{c+d x^2} \, dx}{4 d^3}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(c (13 b c-5 a d) (b c-a d)) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{4 d^4}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(c (13 b c-5 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^4}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^4}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^4}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^{9/2}}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^{9/2}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} d^{17/4}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} d^{17/4}}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 372, normalized size = 0.99 \[ \frac {1440 \sqrt [4]{d} \sqrt {x} \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )+45 \sqrt {2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-45 \sqrt {2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+90 \sqrt {2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )-90 \sqrt {2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )-576 b d^{5/4} x^{5/2} (b c-a d)+\frac {360 c \sqrt [4]{d} \sqrt {x} (b c-a d)^2}{c+d x^2}+160 b^2 d^{9/4} x^{9/2}}{720 d^{17/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 1373, normalized size = 3.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 440, normalized size = 1.17 \[ -\frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {b^{2} c^{3} \sqrt {x} - 2 \, a b c^{2} d \sqrt {x} + a^{2} c d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{16} x^{\frac {9}{2}} - 18 \, b^{2} c d^{15} x^{\frac {5}{2}} + 18 \, a b d^{16} x^{\frac {5}{2}} + 135 \, b^{2} c^{2} d^{14} \sqrt {x} - 180 \, a b c d^{15} \sqrt {x} + 45 \, a^{2} d^{16} \sqrt {x}\right )}}{45 \, d^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 563, normalized size = 1.50 \[ \frac {2 b^{2} x^{\frac {9}{2}}}{9 d^{2}}+\frac {4 a b \,x^{\frac {5}{2}}}{5 d^{2}}-\frac {4 b^{2} c \,x^{\frac {5}{2}}}{5 d^{3}}+\frac {a^{2} c \sqrt {x}}{2 \left (d \,x^{2}+c \right ) d^{2}}-\frac {a b \,c^{2} \sqrt {x}}{\left (d \,x^{2}+c \right ) d^{3}}+\frac {b^{2} c^{3} \sqrt {x}}{2 \left (d \,x^{2}+c \right ) d^{4}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 d^{2}}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{4 d^{3}}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{4 d^{3}}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b c \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{8 d^{3}}-\frac {13 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 d^{4}}-\frac {13 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 d^{4}}-\frac {13 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 d^{4}}+\frac {2 a^{2} \sqrt {x}}{d^{2}}-\frac {8 a b c \sqrt {x}}{d^{3}}+\frac {6 b^{2} c^{2} \sqrt {x}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 377, normalized size = 1.01 \[ \frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}}{2 \, {\left (d^{5} x^{2} + c d^{4}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{16 \, d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{2} x^{\frac {9}{2}} - 18 \, {\left (b^{2} c d - a b d^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right )}}{45 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 1367, normalized size = 3.65 \[ \text {result too large to display} \]
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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