Optimal. Leaf size=297 \[ -\frac {2}{a \sqrt {x}}-\frac {\sqrt [8]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {\sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}} \]
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Rubi [A]
time = 0.19, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {331, 335,
307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} -\frac {\sqrt [8]{c} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac {\sqrt [8]{c} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {2}{a \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 303
Rule 304
Rule 307
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx &=-\frac {2}{a \sqrt {x}}-\frac {c \int \frac {x^{5/2}}{a+c x^4} \, dx}{a}\\ &=-\frac {2}{a \sqrt {x}}-\frac {(2 c) \text {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {2}{a \sqrt {x}}+\frac {\sqrt {c} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{a}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {2}{a \sqrt {x}}+\frac {\sqrt [4]{c} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 a}-\frac {\sqrt [4]{c} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 a}+\frac {\sqrt [4]{c} \text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 a}-\frac {\sqrt [4]{c} \text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 a}\\ &=-\frac {2}{a \sqrt {x}}+\frac {\sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {\sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{4 a}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{4 a}+\frac {\sqrt [8]{c} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} (-a)^{9/8}}\\ &=-\frac {2}{a \sqrt {x}}+\frac {\sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {\sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}\\ &=-\frac {2}{a \sqrt {x}}-\frac {\sqrt [8]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {\sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 276, normalized size = 0.93 \begin {gather*} \frac {-8 \sqrt [8]{a}+\sqrt {2+\sqrt {2}} \sqrt [8]{c} \sqrt {x} \tan ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )} \tan ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2+\sqrt {2}} \sqrt [8]{c} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+\sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 a^{9/8} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.15, size = 38, normalized size = 0.13
method | result | size |
derivativedivides | \(-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 a}-\frac {2}{a \sqrt {x}}\) | \(38\) |
default | \(-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 a}-\frac {2}{a \sqrt {x}}\) | \(38\) |
risch | \(-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 a}-\frac {2}{a \sqrt {x}}\) | \(38\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 446 vs.
\(2 (198) = 396\).
time = 0.38, size = 446, normalized size = 1.50 \begin {gather*} \frac {4 \, \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} - \sqrt {2} \sqrt {\sqrt {2} a^{8} c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} - a^{7} c \left (-\frac {c}{a^{9}}\right )^{\frac {3}{4}} + c^{2} x} a \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} - c}{c}\right ) + 4 \, \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} - \sqrt {2} \sqrt {-\sqrt {2} a^{8} c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} - a^{7} c \left (-\frac {c}{a^{9}}\right )^{\frac {3}{4}} + c^{2} x} a \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} + c}{c}\right ) - \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{8} c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} - a^{7} c \left (-\frac {c}{a^{9}}\right )^{\frac {3}{4}} + c^{2} x\right ) + \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{8} c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} - a^{7} c \left (-\frac {c}{a^{9}}\right )^{\frac {3}{4}} + c^{2} x\right ) + 8 \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a c \sqrt {x} \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} - \sqrt {-a^{7} c \left (-\frac {c}{a^{9}}\right )^{\frac {3}{4}} + c^{2} x} a \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}}}{c}\right ) - 2 \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) + 2 \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (-a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) - 16 \, \sqrt {x}}{8 \, a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 20.13, size = 299, normalized size = 1.01 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {2}{9 c x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: c = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} + \frac {\log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \sqrt [8]{- \frac {a}{c}}} + \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \sqrt [8]{- \frac {a}{c}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 a \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (198) = 396\).
time = 1.11, size = 461, normalized size = 1.55 \begin {gather*} -\frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {2}{a \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 126, normalized size = 0.42 \begin {gather*} -\frac {2}{a\,\sqrt {x}}-\frac {{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{2\,a^{9/8}}-\frac {{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{2\,a^{9/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{a^{9/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{a^{9/8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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