Optimal. Leaf size=133 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^2-b}}{\sqrt {a x^2-b}+\sqrt {c} x^2}\right )}{\sqrt {2} \sqrt [4]{c}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {a x^2-b}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} \sqrt [4]{c}} \]
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Rubi [C] time = 15.78, antiderivative size = 2670, normalized size of antiderivative = 20.08, number of steps used = 18, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1692, 399, 490, 1217, 220, 1707}
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Warning: Unable to verify antiderivative.
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Rule 220
Rule 399
Rule 490
Rule 1217
Rule 1692
Rule 1707
Rubi steps
\begin {align*} \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx &=\int \left (\frac {a-\sqrt {a^2+4 b c}}{\sqrt [4]{-b+a x^2} \left (a-\sqrt {a^2+4 b c}+2 c x^2\right )}+\frac {a+\sqrt {a^2+4 b c}}{\sqrt [4]{-b+a x^2} \left (a+\sqrt {a^2+4 b c}+2 c x^2\right )}\right ) \, dx\\ &=\left (a-\sqrt {a^2+4 b c}\right ) \int \frac {1}{\sqrt [4]{-b+a x^2} \left (a-\sqrt {a^2+4 b c}+2 c x^2\right )} \, dx+\left (a+\sqrt {a^2+4 b c}\right ) \int \frac {1}{\sqrt [4]{-b+a x^2} \left (a+\sqrt {a^2+4 b c}+2 c x^2\right )} \, dx\\ &=\frac {\left (2 \left (a-\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}} \left (2 b c+a \left (a-\sqrt {a^2+4 b c}\right )+2 c x^4\right )} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}+\frac {\left (2 \left (a+\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}} \left (2 b c+a \left (a+\sqrt {a^2+4 b c}\right )+2 c x^4\right )} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}\\ &=-\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}+\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}-\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}+\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} x}\\ &=-\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\sqrt {b} \left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {2} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}+\frac {\left (\sqrt {b} \left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {2} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (\sqrt {b} \left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {2} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}-\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}+\frac {\left (\sqrt {b} \left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {2} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}+\sqrt {2} \sqrt {c} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}\\ &=\frac {\sqrt {b} \left (a^3+4 a b c+\left (a^2+2 b c\right ) \sqrt {a^2+4 b c}\right ) \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b c}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c-a \sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt {-a-\sqrt {a^2+4 b c}} \sqrt [4]{-a^2-2 b c-a \sqrt {a^2+4 b c}} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b c}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c-a \sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c-a \sqrt {a^2+4 b c}} x}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b c}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c+a \sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c+a \sqrt {a^2+4 b c}} x}-\frac {\sqrt {b} \sqrt {-a+\sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a+\sqrt {a^2+4 b c}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c+a \sqrt {a^2+4 b c}} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{c} \sqrt [4]{-a^2-2 b c+a \sqrt {a^2+4 b c}} x}-\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {2} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {c} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {2} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {c} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}-\sqrt {2} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (2 \sqrt {b} \sqrt {c}+\sqrt {2} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) x}+\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}-\frac {\left (a+\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {-a^2-2 b c-a \sqrt {a^2+4 b c}} \left (a^2+4 b c+a \sqrt {a^2+4 b c}\right ) x}+\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}} x}-\frac {\left (a-\sqrt {a^2+4 b c}\right ) \left (\sqrt {2} \sqrt {b} \sqrt {c}-\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b} \sqrt {c}+\sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {c} \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {c} \left (a^2+4 b c-a \sqrt {a^2+4 b c}\right ) \sqrt {-a^2-2 b c+a \sqrt {a^2+4 b c}} x}\\ \end {align*}
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Mathematica [F] time = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.38, size = 133, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^2-b}}{\sqrt {a x^2-b}+\sqrt {c} x^2}\right )}{\sqrt {2} \sqrt [4]{c}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {a x^2-b}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} \sqrt [4]{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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} - 2 \, b}{{\left (c x^{4} + a x^{2} - b\right )} {\left (a x^{2} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,x^{2}-2 b}{\left (a \,x^{2}-b \right )^{\frac {1}{4}} \left (c \,x^{4}+a \,x^{2}-b \right )}\, dx \end {gather*}
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 {a x^{2} - 2 \, b}{{\left (c x^{4} + a x^{2} - b\right )} {\left (a x^{2} - b\right )}^{\frac {1}{4}}}\,{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 = -0.01 \begin {gather*} \int -\frac {2\,b-a\,x^2}{{\left (a\,x^2-b\right )}^{1/4}\,\left (c\,x^4+a\,x^2-b\right )} \,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 {a x^{2} - 2 b}{\sqrt [4]{a x^{2} - b} \left (a x^{2} - b + c x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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