Optimal. Leaf size=96 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}} \]
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Rubi [F] time = 2.09, 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 {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 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 {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx &=\int \left (\frac {2 c}{a \left (-b+a x^6\right )^{3/4}}+\frac {x^2}{\left (-b+a x^6\right )^{3/4}}+\frac {2 b c+3 a b x^2+4 c^2 x^4}{a \left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {2 b c+3 a b x^2+4 c^2 x^4}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx}{a}+\frac {(2 c) \int \frac {1}{\left (-b+a x^6\right )^{3/4}} \, dx}{a}+\int \frac {x^2}{\left (-b+a x^6\right )^{3/4}} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\left (-b+a x^2\right )^{3/4}} \, dx,x,x^3\right )+\frac {\int \left (-\frac {2 b c}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}-\frac {4 c^2 x^4}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}+\frac {3 a b x^2}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )}\right ) \, dx}{a}+\frac {\left (2 c \left (1-\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {a x^6}{b}\right )^{3/4}} \, dx}{a \left (-b+a x^6\right )^{3/4}}\\ &=\frac {2 c x \left (1-\frac {a x^6}{b}\right )^{3/4} \, _2F_1\left (\frac {1}{6},\frac {3}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+(3 b) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx-\frac {(2 b c) \int \frac {1}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}-\frac {\left (4 c^2\right ) \int \frac {x^4}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}+\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a x^3}\\ &=\frac {\sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 a \sqrt [4]{b} x^3}+\frac {2 c x \left (1-\frac {a x^6}{b}\right )^{3/4} \, _2F_1\left (\frac {1}{6},\frac {3}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+(3 b) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx-\frac {(2 b c) \int \frac {1}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}-\frac {\left (4 c^2\right ) \int \frac {x^4}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}\\ \end {align*}
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Mathematica [F] time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 15.72, size = 96, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}} \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 {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} - 2 \, c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a \,x^{6}+2 b \right )}{\left (a \,x^{6}-b \right )^{\frac {3}{4}} \left (a \,x^{6}-2 c \,x^{4}-b \right )}\, dx\]
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*} \int \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} - 2 \, c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{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 {x^2\,\left (a\,x^6+2\,b\right )}{{\left (a\,x^6-b\right )}^{3/4}\,\left (-a\,x^6+2\,c\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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