Optimal. Leaf size=43 \[ \frac {2}{5} \tan ^{-1}\left (x \sqrt [4]{a x^6-b x}\right )-\frac {2}{5} \tanh ^{-1}\left (x \sqrt [4]{a x^6-b x}\right ) \]
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Rubi [C] time = 1.48, antiderivative size = 189, normalized size of antiderivative = 4.40, number of steps used = 12, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {1593, 2056, 6728, 466, 465, 511, 510} \begin {gather*} -\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b-\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \left (b-\sqrt {4 a+b^2}\right ) \sqrt [4]{a x^6-b x}}-\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b+\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \left (\sqrt {4 a+b^2}+b\right ) \sqrt [4]{a x^6-b x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 465
Rule 466
Rule 510
Rule 511
Rule 1593
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx &=\int \frac {x^3 \left (-b+2 a x^5\right )}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {x^{11/4} \left (-b+2 a x^5\right )}{\sqrt [4]{-b+a x^5} \left (-1-b x^5+a x^{10}\right )} \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \left (\frac {2 a x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b-\sqrt {4 a+b^2}+2 a x^5\right )}+\frac {2 a x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b+\sqrt {4 a+b^2}+2 a x^5\right )}\right ) \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b-\sqrt {4 a+b^2}+2 a x^5\right )} \, dx}{\sqrt [4]{-b x+a x^6}}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b+\sqrt {4 a+b^2}+2 a x^5\right )} \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{20}} \left (-b-\sqrt {4 a+b^2}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{20}} \left (-b+\sqrt {4 a+b^2}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^4} \left (-b-\sqrt {4 a+b^2}+2 a x^4\right )} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^4} \left (-b+\sqrt {4 a+b^2}+2 a x^4\right )} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b-\sqrt {4 a+b^2}+2 a x^4\right ) \sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+\sqrt {4 a+b^2}+2 a x^4\right ) \sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}\\ &=-\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b-\sqrt {4 a+b^2}},\frac {a x^5}{b}\right )}{15 \left (b-\sqrt {4 a+b^2}\right ) \sqrt [4]{-b x+a x^6}}-\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b+\sqrt {4 a+b^2}},\frac {a x^5}{b}\right )}{15 \left (b+\sqrt {4 a+b^2}\right ) \sqrt [4]{-b x+a x^6}}\\ \end {align*}
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Mathematica [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 19.97, size = 43, normalized size = 1.00 \begin {gather*} \frac {2}{5} \tan ^{-1}\left (x \sqrt [4]{-b x+a x^6}\right )-\frac {2}{5} \tanh ^{-1}\left (x \sqrt [4]{-b x+a x^6}\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 {2 \, a x^{8} - b x^{3}}{{\left (a x^{10} - b x^{5} - 1\right )} {\left (a x^{6} - 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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{8}-b \,x^{3}}{\left (a \,x^{6}-b x \right )^{\frac {1}{4}} \left (a \,x^{10}-b \,x^{5}-1\right )}\, dx\]
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 {2 \, a x^{8} - b x^{3}}{{\left (a x^{10} - b x^{5} - 1\right )} {\left (a x^{6} - 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 {2\,a\,x^8-b\,x^3}{{\left (a\,x^6-b\,x\right )}^{1/4}\,\left (-a\,x^{10}+b\,x^5+1\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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