Optimal. Leaf size=45 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b x^2}}{x^3}\right )-2 \tanh ^{-1}\left (\frac {x^3}{\sqrt [4]{a x^3+b x^2}}\right ) \]
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Rubi [F] time = 1.98, 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 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {x^{3/2} (10 b+9 a x)}{\sqrt [4]{b+a x} \left (-b-a x+x^{10}\right )} \, dx}{\sqrt [4]{b x^2+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (10 b+9 a x^2\right )}{\sqrt [4]{b+a x^2} \left (-b-a x^2+x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \left (-\frac {10 b x^4}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )}-\frac {9 a x^6}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}}\\ &=-\frac {\left (18 a \sqrt {x} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}}-\frac {\left (20 b \sqrt {x} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}}\\ \end {align*}
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Mathematica [F] time = 1.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.72, size = 45, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{b x^2+a x^3}}{x^3}\right )-2 \tanh ^{-1}\left (\frac {x^3}{\sqrt [4]{b x^2+a x^3}}\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 + 10 \, b\right )} x^{2}}{{\left (x^{10} - a x - b\right )} {\left (a x^{3} + b x^{2}\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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (9 a x +10 b \right )}{\left (a \,x^{3}+b \,x^{2}\right )^{\frac {1}{4}} \left (x^{10}-a x -b \right )}\, dx\]
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 (9 \, a x + 10 \, b\right )} x^{2}}{{\left (x^{10} - a x - b\right )} {\left (a x^{3} + b x^{2}\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^2\,\left (10\,b+9\,a\,x\right )}{{\left (a\,x^3+b\,x^2\right )}^{1/4}\,\left (-x^{10}+a\,x+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 {x^{2} \left (9 a x + 10 b\right )}{\sqrt [4]{x^{2} \left (a x + b\right )} \left (- a x - b + x^{10}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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