Optimal. Leaf size=114 \[ \frac {4}{3} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )+\frac {4}{3} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )+\frac {4 \left (3 b-17 a x^3\right ) \left (a x^4-b x\right )^{3/4}}{63 x^6} \]
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Rubi [A] time = 0.40, antiderivative size = 177, normalized size of antiderivative = 1.55, number of steps used = 12, number of rules used = 10, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {2052, 2011, 329, 275, 240, 212, 206, 203, 2016, 2014} \begin {gather*} \frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {4 b \left (a x^4-b x\right )^{3/4}}{21 x^6}-\frac {68 a \left (a x^4-b x\right )^{3/4}}{63 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 275
Rule 329
Rule 2011
Rule 2014
Rule 2016
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx &=\int \left (\frac {2 a^2}{\sqrt [4]{-b x+a x^4}}+\frac {b^2}{x^6 \sqrt [4]{-b x+a x^4}}-\frac {3 a b}{x^3 \sqrt [4]{-b x+a x^4}}\right ) \, dx\\ &=\left (2 a^2\right ) \int \frac {1}{\sqrt [4]{-b x+a x^4}} \, dx-(3 a b) \int \frac {1}{x^3 \sqrt [4]{-b x+a x^4}} \, dx+b^2 \int \frac {1}{x^6 \sqrt [4]{-b x+a x^4}} \, dx\\ &=\frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {4 a \left (-b x+a x^4\right )^{3/4}}{3 x^3}+\frac {1}{7} (4 a b) \int \frac {1}{x^3 \sqrt [4]{-b x+a x^4}} \, dx+\frac {\left (2 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-b+a x^3}} \, dx}{\sqrt [4]{-b x+a x^4}}\\ &=\frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^4}}\\ &=\frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (4 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}+\frac {\left (4 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 71, normalized size = 0.62 \begin {gather*} \frac {4 \left (a x^4-b x\right )^{3/4} \left (3 \left (b-a x^3\right )-\frac {14 a x^3 \, _2F_1\left (-\frac {3}{4},-\frac {3}{4};\frac {1}{4};\frac {a x^3}{b}\right )}{\left (1-\frac {a x^3}{b}\right )^{3/4}}\right )}{63 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 114, normalized size = 1.00 \begin {gather*} \frac {4}{3} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )+\frac {4}{3} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )+\frac {4 \left (3 b-17 a x^3\right ) \left (a x^4-b x\right )^{3/4}}{63 x^6} \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 [B] time = 0.30, size = 209, normalized size = 1.83 \begin {gather*} \frac {2}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {2}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) - \frac {4}{21} \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}} - \frac {8}{9} \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{3}-b \right ) \left (2 a \,x^{3}-b \right )}{x^{6} \left (a \,x^{4}-b x \right )^{\frac {1}{4}}}\, 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 (2 \, a x^{3} - b\right )} {\left (a x^{3} - b\right )}}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.88, size = 72, normalized size = 0.63 \begin {gather*} -\frac {4\,\left (3\,b^2+17\,a^2\,x^6-20\,a\,b\,x^3-42\,a^2\,x^6\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {a\,x^3}{b}\right )\right )}{63\,x^5\,{\left (a\,x^4-b\,x\right )}^{1/4}} \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 {\left (a x^{3} - b\right ) \left (2 a x^{3} - b\right )}{x^{6} \sqrt [4]{x \left (a x^{3} - b\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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