Optimal. Leaf size=413 \[ -\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^5-b x^3}-2^{2/3} x \sqrt [4]{a x^5-b x^3}}{2^{2/3} x \sqrt [4]{a x^5-b x^3}+x^2 \left (-\sqrt [4]{a x^5-b x^3}\right )-\sqrt {2} x+2 \sqrt [6]{2}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^5-b x^3}-2^{2/3} x \sqrt [4]{a x^5-b x^3}}{2^{2/3} x \sqrt [4]{a x^5-b x^3}+x^2 \left (-\sqrt [4]{a x^5-b x^3}\right )+\sqrt {2} x-2 \sqrt [6]{2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-\sqrt {2} x^3 \sqrt [4]{a x^5-b x^3}-2\ 2^{5/6} x \sqrt [4]{a x^5-b x^3}+4 \sqrt [6]{2} x^2 \sqrt [4]{a x^5-b x^3}}{2\ 2^{2/3} x^3 \sqrt {a x^5-b x^3}+x^4 \left (-\sqrt {a x^5-b x^3}\right )-2 \sqrt [3]{2} x^2 \sqrt {a x^5-b x^3}-x^2+2\ 2^{2/3} x-2 \sqrt [3]{2}}\right ) \]
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Rubi [F] time = 2.71, 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^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{17/4} \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (1-b x^7+a x^9\right )} \, dx}{\sqrt [4]{-b x^3+a x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{20} \left (-7 b+9 a x^8\right )}{\sqrt [4]{-b+a x^8} \left (1-b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{-b+a x^8} \left (-1+b x^{28}-a x^{36}\right )}+\frac {9 a x^{28}}{\sqrt [4]{-b+a x^8} \left (1-b x^{28}+a x^{36}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}\\ &=\frac {\left (36 a x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{28}}{\sqrt [4]{-b+a x^8} \left (1-b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}+\frac {\left (28 b x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{-b+a x^8} \left (-1+b x^{28}-a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}\\ \end {align*}
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Mathematica [F] time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [F] time = 78.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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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 (9 \, a x^{2} - 7 \, b\right )} x^{5}}{{\left (a x^{9} - b x^{7} + 1\right )} {\left (a x^{5} - b x^{3}\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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (9 a \,x^{2}-7 b \right )}{\left (a \,x^{5}-b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{9}-b \,x^{7}+1\right )}\, dx\]
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 (9 \, a x^{2} - 7 \, b\right )} x^{5}}{{\left (a x^{9} - b x^{7} + 1\right )} {\left (a x^{5} - b x^{3}\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 = -0.00 \begin {gather*} -\int \frac {x^5\,\left (7\,b-9\,a\,x^2\right )}{{\left (a\,x^5-b\,x^3\right )}^{1/4}\,\left (a\,x^9-b\,x^7+1\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 {x^{5} \left (9 a x^{2} - 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{2} - b\right )} \left (a x^{9} - b x^{7} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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