Optimal. Leaf size=1480 \[ \frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right ) a}{b^2 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} a+\left (1-\sqrt {3}\right ) b \sqrt [3]{c}\right ) \sqrt [3]{d} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right ) a}{\sqrt [4]{3} b^3 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt [3]{d} \sqrt {d x^3+c} a}{b^2 \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [6]{c} \sqrt {b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt {d^{2/3} a^2+b \sqrt [3]{c} \sqrt [3]{d} a+b^2 c^{2/3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {3}} \sqrt {d^{2/3} a^2+b \sqrt [3]{c} \sqrt [3]{d} a+b^2 c^{2/3}} \sqrt {1-\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}}}{\sqrt [4]{3} \sqrt {b} \sqrt [6]{c} \sqrt {b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7}}\right )}{b^{5/2} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^3 \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \Pi \left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{b^2 \left (-d^{2/3} a^2+2 b \sqrt [3]{c} \sqrt [3]{d} a+2 b^2 c^{2/3}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}+\frac {2 \sqrt {d x^3+c}}{3 b} \]
[Out]
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Rubi [A] time = 2.81, antiderivative size = 1482, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2147, 261, 1878, 218, 1877, 2136, 2142, 2113, 537, 571, 93, 208} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 208
Rule 218
Rule 261
Rule 537
Rule 571
Rule 1877
Rule 1878
Rule 2113
Rule 2136
Rule 2142
Rule 2147
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx &=\frac {(a d) \int \frac {a-b x}{\sqrt {c+d x^3}} \, dx}{b^3}+\frac {d \int \frac {x^2}{\sqrt {c+d x^3}} \, dx}{b}-\left (-c+\frac {a^3 d}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x^3}} \, dx\\ &=\frac {2 \sqrt {c+d x^3}}{3 b}-\frac {\left (a d^{2/3}\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{b^2}+\frac {\left (a \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{b^3}+\frac {\left (b \left (c-\frac {a^3 d}{b^3}\right )\right ) \int \frac {1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{(a+b x) \sqrt {c+d x^3}} \, dx}{b+\sqrt {3} b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}}-\frac {\left (\sqrt [3]{d} \left (c-\frac {a^3 d}{b^3}\right )\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}}\\ &=\frac {2 \sqrt {c+d x^3}}{3 b}-\frac {2 a \sqrt [3]{d} \sqrt {c+d x^3}}{b^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} a \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^3 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (c-\frac {a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (c-\frac {a^3 d}{b^3}\right ) \left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {\frac {1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}+\left (\left (1+\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}\right )}{\left (b+\sqrt {3} b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) \sqrt {\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt {c+d x^3}}\\ &=\frac {2 \sqrt {c+d x^3}}{3 b}-\frac {2 a \sqrt [3]{d} \sqrt {c+d x^3}}{b^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} a \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^3 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (c-\frac {a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (c-\frac {a^3 d}{b^3}\right ) \left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {\frac {1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (1+\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}\right )}{\sqrt {\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt {c+d x^3}}+\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (b-\sqrt {3} b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) \left (c-\frac {a^3 d}{b^3}\right ) \left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {\frac {1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (1+\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}\right )}{\left (b+\sqrt {3} b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) \sqrt {\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt {c+d x^3}}\\ &=\frac {2 \sqrt {c+d x^3}}{3 b}-\frac {2 a \sqrt [3]{d} \sqrt {c+d x^3}}{b^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} a \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^3 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (c-\frac {a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3} \left (1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \Pi \left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};-\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \left (2 b^2 c^{2/3}+2 a b \sqrt [3]{c} \sqrt [3]{d}-a^2 d^{2/3}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\left (2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (c-\frac {a^3 d}{b^3}\right ) \left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {\frac {1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (1+\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}\right )}{\sqrt {\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt {c+d x^3}}\\ &=\frac {2 \sqrt {c+d x^3}}{3 b}-\frac {2 a \sqrt [3]{d} \sqrt {c+d x^3}}{b^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} a \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^3 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (c-\frac {a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3} \left (1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \Pi \left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};-\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \left (2 b^2 c^{2/3}+2 a b \sqrt [3]{c} \sqrt [3]{d}-a^2 d^{2/3}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (c-\frac {a^3 d}{b^3}\right ) \left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {\frac {1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{-\left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2+\left (\left (1+\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2+\left (7-4 \sqrt {3}\right ) \left (\left (1+\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2\right ) x^2} \, dx,x,\frac {\sqrt {1-\frac {\left (-1+\sqrt {3}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}}{\sqrt {7-4 \sqrt {3}+\frac {\left (-1+\sqrt {3}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}}\right )}{\sqrt {\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt {3}+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt {c+d x^3}}\\ &=\frac {2 \sqrt {c+d x^3}}{3 b}-\frac {2 a \sqrt [3]{d} \sqrt {c+d x^3}}{b^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [6]{c} \sqrt {b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt {b^2 c^{2/3}+a b \sqrt [3]{c} \sqrt [3]{d}+a^2 d^{2/3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3} \left (1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {3}} \sqrt {b^2 c^{2/3}+a b \sqrt [3]{c} \sqrt [3]{d}+a^2 d^{2/3}} \sqrt {1-\frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}}}{\sqrt [4]{3} \sqrt {b} \sqrt [6]{c} \sqrt {b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt {7-4 \sqrt {3}+\frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}}}\right )}{b^{5/2} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} a \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^3 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (c-\frac {a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3} \left (1-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \Pi \left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};-\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{b^2 \left (2 b^2 c^{2/3}+2 a b \sqrt [3]{c} \sqrt [3]{d}-a^2 d^{2/3}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [C] time = 2.15, size = 820, normalized size = 0.55 \[ \frac {2 \left (\frac {\sqrt [3]{-1} \sqrt {3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c} d \sqrt {\frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}} \sqrt {\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1} \Pi \left (\frac {i \sqrt {3} b \sqrt [3]{c}}{\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}\right )|\sqrt [3]{-1}\right ) a^3}{b^2 \left (\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}\right )}-\frac {3^{3/4} d^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {\frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}} \sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{d} x}{\sqrt [3]{c}}} F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}\right )|\sqrt [3]{-1}\right ) a^2}{b^2 \sqrt {\frac {(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}}+\frac {3^{3/4} \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{-1} \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {-\frac {2 i \sqrt [3]{d} x}{\sqrt [3]{c}}+\sqrt {3}+i} \sqrt {\frac {i \left (\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{3 i+\sqrt {3}}} \left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{d} x}{\sqrt [3]{c}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )+F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{d} x}{\sqrt [3]{c}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right ) a}{b \sqrt {\frac {(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}}+d x^3+c-\frac {3 i b c^{4/3} \sqrt {\frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}} \sqrt {\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1} \Pi \left (\frac {i \sqrt {3} b \sqrt [3]{c}}{\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}}\right )}{3 b \sqrt {d x^3+c}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 22.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c}}{b x + a}, x\right ) \]
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 \[ \int \frac {\sqrt {d x^{3} + c}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 1126, normalized size = 0.76 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {d x^{3} + c}}{b x + a}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^3+c}}{a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d x^{3}}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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