Optimal. Leaf size=71 \[ \frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}-c^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^6}}\right )-c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^6}}\right ) \]
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Rubi [F]
time = 0.92, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (-2 b+a x^6\right ) \left (b+a x^6\right )^{3/4}}{x^4 \left (b-c x^4+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-2 b+a x^6\right ) \left (b+a x^6\right )^{3/4}}{x^4 \left (b-c x^4+a x^6\right )} \, dx &=\int \left (-\frac {2 \left (b+a x^6\right )^{3/4}}{x^4}+\frac {\left (2 c-3 a x^2\right ) \left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\left (b+a x^6\right )^{3/4}}{x^4} \, dx\right )+\int \frac {\left (2 c-3 a x^2\right ) \left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx\\ &=-\left (\frac {2}{3} \text {Subst}\left (\int \frac {\left (b+a x^2\right )^{3/4}}{x^2} \, dx,x,x^3\right )\right )+\int \left (\frac {2 c \left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6}+\frac {3 a x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6}\right ) \, dx\\ &=\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}-a \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^2}} \, dx,x,x^3\right )+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx\\ &=\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx-\frac {\left (a \sqrt [4]{1+\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {a x^2}{b}}} \, dx,x,x^3\right )}{\sqrt [4]{b+a x^6}}\\ &=-\frac {2 a x^3}{\sqrt [4]{b+a x^6}}+\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx+\frac {\left (a \sqrt [4]{1+\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,x^3\right )}{\sqrt [4]{b+a x^6}}\\ &=-\frac {2 a x^3}{\sqrt [4]{b+a x^6}}+\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}+\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {a x^6}{b}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a} x^3}{\sqrt {b}}\right )\right |2\right )}{\sqrt [4]{b+a x^6}}+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx\\ \end {align*}
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Mathematica [A]
time = 2.25, size = 71, normalized size = 1.00 \begin {gather*} \frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}-c^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^6}}\right )-c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{6}-2 b \right ) \left (a \,x^{6}+b \right )^{\frac {3}{4}}}{x^{4} \left (a \,x^{6}-c \,x^{4}+b \right )}\, dx\]
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 [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 {{\left (a\,x^6+b\right )}^{3/4}\,\left (2\,b-a\,x^6\right )}{x^4\,\left (a\,x^6-c\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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