Optimal. Leaf size=58 \[ 2^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{a x^5-b x}}{\sqrt [4]{2}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{a x^5-b x}}{\sqrt [4]{2}}\right ) \]
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Rubi [F] time = 2.64, 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^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+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^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^{11/4} \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b+a x^4} \left (-2-b x^5+a x^9\right )} \, dx}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (-5 b+9 a x^{16}\right )}{\sqrt [4]{-b+a x^{16}} \left (-2-b x^{20}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {5 b x^{14}}{\sqrt [4]{-b+a x^{16}} \left (2+b x^{20}-a x^{36}\right )}+\frac {9 a x^{30}}{\sqrt [4]{-b+a x^{16}} \left (-2-b x^{20}+a x^{36}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (36 a \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{30}}{\sqrt [4]{-b+a x^{16}} \left (-2-b x^{20}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (20 b \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (2+b x^{20}-a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ \end {align*}
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Mathematica [F] time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 15.12, size = 58, normalized size = 1.00 \begin {gather*} 2^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{a x^5-b x}}{\sqrt [4]{2}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{a x^5-b x}}{\sqrt [4]{2}}\right ) \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 (9 \, a x^{4} - 5 \, b\right )} x^{3}}{{\left (a x^{9} - b x^{5} - 2\right )} {\left (a x^{5} - b x\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 = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (9 a \,x^{4}-5 b \right )}{\left (a \,x^{5}-b x \right )^{\frac {1}{4}} \left (a \,x^{9}-b \,x^{5}-2\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 {{\left (9 \, a x^{4} - 5 \, b\right )} x^{3}}{{\left (a x^{9} - b x^{5} - 2\right )} {\left (a x^{5} - b x\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 = -0.02 \begin {gather*} \int \frac {x^3\,\left (5\,b-9\,a\,x^4\right )}{{\left (a\,x^5-b\,x\right )}^{1/4}\,\left (-a\,x^9+b\,x^5+2\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^{3} \left (9 a x^{4} - 5 b\right )}{\sqrt [4]{x \left (a x^{4} - b\right )} \left (a x^{9} - b x^{5} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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