Optimal. Leaf size=231 \[ -\frac {\sqrt {-\left (\sqrt {d} \left (\sqrt {c}+\sqrt {d}\right )\right )} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {-\sqrt {c} \sqrt {d}-d}}{\sqrt {d} (x-1) \left (x^2+x+1\right )}\right )}{3 c^{3/2} d}+\frac {\sqrt {\sqrt {d} \left (\sqrt {c}-\sqrt {d}\right )} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {\sqrt {c} \sqrt {d}-d}}{\sqrt {d} (x-1) \left (x^2+x+1\right )}\right )}{3 c^{3/2} d}+\frac {a \sqrt {x^4-x} x}{3 c}-\frac {a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )}{3 c} \]
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Rubi [A] time = 0.85, antiderivative size = 268, normalized size of antiderivative = 1.16, number of steps used = 18, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2056, 6715, 1693, 195, 217, 206, 1175, 402, 377, 205, 208} \begin {gather*} \frac {\sqrt {x^4-x} \sqrt {\sqrt {c}-\sqrt {d}} (a d+b c) \tan ^{-1}\left (\frac {x^{3/2} \sqrt {\sqrt {c}-\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3-1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3-1} \sqrt {x}}+\frac {\sqrt {x^4-x} \sqrt {\sqrt {c}+\sqrt {d}} (a d+b c) \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {\sqrt {c}+\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3-1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3-1} \sqrt {x}}+\frac {a \sqrt {x^4-x} x}{3 c}-\frac {a \sqrt {x^4-x} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 c \sqrt {x^3-1} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 205
Rule 206
Rule 208
Rule 217
Rule 377
Rule 402
Rule 1175
Rule 1693
Rule 2056
Rule 6715
Rubi steps
\begin {align*} \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx &=\frac {\sqrt {-x+x^4} \int \frac {\sqrt {x} \sqrt {-1+x^3} \left (b+a x^6\right )}{-d+c x^6} \, dx}{\sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2} \left (b+a x^4\right )}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {a \sqrt {-1+x^2}}{c}+\frac {(b c+a d) \sqrt {-1+x^2}}{c \left (-d+c x^4\right )}\right ) \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 a \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (2 (b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\left (a \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left ((b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {c} \sqrt {d}-c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}-\frac {\left ((b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {c} \sqrt {d}+c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\left (a \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {c} \sqrt {d}-c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {c} \sqrt {d}+c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {a x \sqrt {-x+x^4}}{3 c}-\frac {a \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (-c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {a x \sqrt {-x+x^4}}{3 c}+\frac {\sqrt {\sqrt {c}-\sqrt {d}} (b c+a d) \sqrt {-x+x^4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {c}-\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {-1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {-1+x^3}}-\frac {a \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}+\frac {\sqrt {\sqrt {c}+\sqrt {d}} (b c+a d) \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c}+\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {-1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] time = 1.60, size = 487, normalized size = 2.11 \begin {gather*} \frac {\sqrt {x} \sqrt {x^3-1} \left (-\frac {(a d+b c) \left (\sqrt [4]{d} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )+\sqrt {\sqrt {c}+\sqrt {d}} \tanh ^{-1}\left (\frac {-\sqrt [4]{d} x^{3/2}+i \sqrt [4]{c}}{\sqrt {x^3-1} \sqrt {\sqrt {c}+\sqrt {d}}}\right )\right )}{d^{3/4}}-\frac {(a d+b c) \left (\sqrt [4]{d} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )-\sqrt {\sqrt {c}+\sqrt {d}} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x^{3/2}+i \sqrt [4]{c}}{\sqrt {x^3-1} \sqrt {\sqrt {c}+\sqrt {d}}}\right )\right )}{d^{3/4}}+\frac {(a d+b c) \left (\sqrt {\sqrt {c}-\sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{c}-\sqrt [4]{d} x^{3/2}}{\sqrt {x^3-1} \sqrt {\sqrt {c}-\sqrt {d}}}\right )+\sqrt [4]{d} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )\right )}{d^{3/4}}+\frac {(a d+b c) \left (\sqrt [4]{d} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )-\sqrt {\sqrt {c}-\sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{c}+\sqrt [4]{d} x^{3/2}}{\sqrt {x^3-1} \sqrt {\sqrt {c}-\sqrt {d}}}\right )\right )}{d^{3/4}}+\frac {2 a \sqrt {c} \left (x^3-1\right ) \left (\sin ^{-1}\left (x^{3/2}\right )+\sqrt {1-x^3} x^{3/2}\right )}{\sqrt {-\left (x^3-1\right )^2}}\right )}{6 c^{3/2} \sqrt {x \left (x^3-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.74, size = 261, normalized size = 1.13 \begin {gather*} \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\sqrt {-\left (\left (\sqrt {c}+\sqrt {d}\right ) \sqrt {d}\right )} (b c+a d) \tan ^{-1}\left (\frac {\sqrt {-\sqrt {c} \sqrt {d}-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^{3/2} d}+\frac {\sqrt {\left (\sqrt {c}-\sqrt {d}\right ) \sqrt {d}} (b c+a d) \tan ^{-1}\left (\frac {\sqrt {\sqrt {c} \sqrt {d}-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^{3/2} d}+\frac {a \log \left (-x^2+\sqrt {-x+x^4}\right )}{6 c}-\frac {a \log \left (c x^2+c \sqrt {-x+x^4}\right )}{6 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.59, size = 681, normalized size = 2.95
method | result | size |
default | \(\frac {a \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{c}-\frac {\left (a d +b c \right ) \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{6}-d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} c \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} c -3 \underline {\hspace {1.25 ex}}\alpha ^{5} c -i \sqrt {3}\, d +3 d}{6 d}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{d}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (c -d \right ) \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 c}\) | \(681\) |
elliptic | \(\frac {a x \sqrt {x^{4}-x}}{3 c}-\frac {a \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{c \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{6}-d \right )}{\sum }\frac {\left (a d +b c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} c \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} c -3 \underline {\hspace {1.25 ex}}\alpha ^{5} c -i \sqrt {3}\, d +3 d}{6 d}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{d}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (c -d \right ) \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 c}\) | \(681\) |
risch | \(\frac {a \,x^{2} \left (x^{3}-1\right )}{3 c \sqrt {x \left (x^{3}-1\right )}}-\frac {\frac {2 a \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{6}-d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a d -\underline {\hspace {1.25 ex}}\alpha ^{3} b c +a d +b c \right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} c \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} c -3 \underline {\hspace {1.25 ex}}\alpha ^{5} c -i \sqrt {3}\, d +3 d}{6 d}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{d}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (c -d \right ) \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3}}{2 c}\) | \(697\) |
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 \begin {gather*} \int \frac {{\left (a x^{6} + b\right )} \sqrt {x^{4} - x}}{c x^{6} - d}\,{d x} \end {gather*}
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 \begin {gather*} \int -\frac {\sqrt {x^4-x}\,\left (a\,x^6+b\right )}{d-c\,x^6} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{6} + b\right )}{c x^{6} - d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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