Optimal. Leaf size=227 \[ \frac {\sqrt {-\left (\sqrt {d} \left (\sqrt {c}+\sqrt {d}\right )\right )} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-\sqrt {c} \sqrt {d}-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^{3/2} d}-\frac {\sqrt {\sqrt {d} \left (\sqrt {c}-\sqrt {d}\right )} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {\sqrt {c} \sqrt {d}-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^{3/2} d}+\frac {a \sqrt {x^4+x} x}{3 c}+\frac {a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )}{3 c} \]
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Rubi [A] time = 0.74, antiderivative size = 250, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2056, 6715, 1693, 195, 215, 1175, 402, 377, 208, 205} \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} \sinh ^{-1}\left (x^{3/2}\right )}{3 c \sqrt {x^3+1} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 205
Rule 208
Rule 215
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 {a \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\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} \sinh ^{-1}\left (x^{3/2}\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 {a \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}-\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 {\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 = 0.79, size = 441, normalized size = 1.94 \begin {gather*} \frac {\sqrt {x^4+x} \left (-\left (x^3+1\right ) (a d+b c) \left (\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )+\sqrt {\sqrt {c}-\sqrt {d}} \tan ^{-1}\left (\frac {-\sqrt [4]{d} x^{3/2}+i \sqrt [4]{c}}{\sqrt {x^3+1} \sqrt {\sqrt {c}-\sqrt {d}}}\right )\right )-\left (x^3+1\right ) (a d+b c) \left (\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )-\sqrt {\sqrt {c}-\sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{d} x^{3/2}+i \sqrt [4]{c}}{\sqrt {x^3+1} \sqrt {\sqrt {c}-\sqrt {d}}}\right )\right )+\left (x^3+1\right ) (a d+b c) \left (\sqrt {\sqrt {c}+\sqrt {d}} \tanh ^{-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} \sinh ^{-1}\left (x^{3/2}\right )\right )+\left (x^3+1\right ) (a d+b c) \left (\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )-\sqrt {\sqrt {c}+\sqrt {d}} \tanh ^{-1}\left (\frac {\sqrt [4]{c}+\sqrt [4]{d} x^{3/2}}{\sqrt {x^3+1} \sqrt {\sqrt {c}+\sqrt {d}}}\right )\right )+2 a \sqrt {c} d^{3/4} \left (x^{3/2} \left (x^3+1\right )^{3/2}+\left (x^3+1\right ) \sinh ^{-1}\left (x^{3/2}\right )\right )\right )}{6 c^{3/2} d^{3/4} \sqrt {x} \left (x^3+1\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.75, size = 227, normalized size = 1.00 \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 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 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 = 0.58, size = 686, normalized size = 3.02
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}\) | \(686\) |
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}\) | \(686\) |
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}\) | \(706\) |
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 {\left (a\,x^6+b\right )\,\sqrt {x^4+x}}{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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