Optimal. Leaf size=711 \[ -\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b c-14 a f) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {b (b d-4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}-\frac {3 b^{4/3} \sqrt {a+b x^3} (5 b c-14 a f)}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 b \sqrt {a+b x^3} (5 b c-14 a f)}{112 a^2 x}-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2} \]
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Rubi [A] time = 1.12, antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ -\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 b^{4/3} \sqrt {a+b x^3} (5 b c-14 a f)}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b c-14 a f) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 b \sqrt {a+b x^3} (5 b c-14 a f)}{112 a^2 x}+\frac {b (b d-4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 208
Rule 218
Rule 266
Rule 1825
Rule 1832
Rule 1835
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {1}{2} (3 b) \int \frac {-\frac {c}{7}-\frac {d x}{6}-\frac {e x^2}{5}-\frac {f x^3}{4}-\frac {g x^4}{3}}{x^5 \sqrt {a+b x^3}} \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}+\frac {(3 b) \int \frac {\frac {4 a d}{3}+\frac {8 a e x}{5}-\frac {1}{7} (5 b c-14 a f) x^2+\frac {8}{3} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx}{16 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {b \int \frac {-\frac {48 a^2 e}{5}+\frac {6}{7} a (5 b c-14 a f) x+4 a (b d-4 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx}{32 a^2}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {b \int \frac {-\frac {24}{7} a^2 (5 b c-14 a f)-16 a^2 (b d-4 a g) x-\frac {48}{5} a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{128 a^3}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {b \int \frac {32 a^3 (b d-4 a g)+\frac {96}{5} a^3 b e x+\frac {24}{7} a^2 b (5 b c-14 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{256 a^4}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {b \int \frac {\frac {96}{5} a^3 b e+\frac {24}{7} a^2 b (5 b c-14 a f) x}{\sqrt {a+b x^3}} \, dx}{256 a^4}-\frac {(b (b d-4 a g)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{8 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {\left (3 b^{5/3} (5 b c-14 a f)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{224 a^2}-\frac {\left (3 b^{5/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1120 a^{5/3}}-\frac {(b (b d-4 a g)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 b c-14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {(b d-4 a g) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{12 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {b (b d-4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 b c-14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 213, normalized size = 0.30 \[ -\frac {\sqrt {a+b x^3} \left (180 a^3 c \, _2F_1\left (-\frac {7}{3},-\frac {1}{2};-\frac {4}{3};-\frac {b x^3}{a}\right )+7 x^2 \left (36 a^3 e \, _2F_1\left (-\frac {5}{3},-\frac {1}{2};-\frac {2}{3};-\frac {b x^3}{a}\right )+5 x \left (9 a^3 f \, _2F_1\left (-\frac {4}{3},-\frac {1}{2};-\frac {1}{3};-\frac {b x^3}{a}\right )+12 a^2 g x \left (a \sqrt {\frac {b x^3}{a}+1}+b x^3 \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )\right )+8 b^2 d x^4 \left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x^3}{a}+1\right )\right )\right )\right )}{1260 a^3 x^7 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{8}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1376, normalized size = 1.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{8}}\,{d x} \]
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 \[ \int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.60, size = 308, normalized size = 0.43 \[ \frac {\sqrt {a} c \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} f \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} - \frac {a d}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} d}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} g \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} d}{12 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} + \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{12 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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