Optimal. Leaf size=110 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}+\frac {4 \left (a x^4+b x\right )^{3/4} \left (11 a x^3-3 b\right )}{63 b x^6} \]
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Rubi [A] time = 0.27, antiderivative size = 167, normalized size of antiderivative = 1.52, number of steps used = 12, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2052, 2011, 329, 275, 240, 212, 206, 203, 2016, 2014} \begin {gather*} -\frac {4 \left (a x^4+b x\right )^{3/4}}{21 x^6}+\frac {44 a \left (a x^4+b x\right )^{3/4}}{63 b x^3}+\frac {2 \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} \sqrt [4]{a x^4+b x}}+\frac {2 \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} \sqrt [4]{a x^4+b x}} \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 {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx &=\int \left (\frac {1}{\sqrt [4]{b x+a x^4}}+\frac {b}{x^6 \sqrt [4]{b x+a x^4}}-\frac {a}{x^3 \sqrt [4]{b x+a x^4}}\right ) \, dx\\ &=-\left (a \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx\right )+b \int \frac {1}{x^6 \sqrt [4]{b x+a x^4}} \, dx+\int \frac {1}{\sqrt [4]{b x+a x^4}} \, dx\\ &=-\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {4 a \left (b x+a x^4\right )^{3/4}}{9 b x^3}-\frac {1}{7} (4 a) \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx+\frac {\left (\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 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (4 \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 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (4 \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 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (4 \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 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (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 (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 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {2 \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]{a} \sqrt [4]{b x+a x^4}}+\frac {2 \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]{a} \sqrt [4]{b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 83, normalized size = 0.75 \begin {gather*} \frac {4 \left (11 a^2 x^6+21 b x^6 \sqrt [4]{\frac {a x^3}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {a x^3}{b}\right )+8 a b x^3-3 b^2\right )}{63 b x^5 \sqrt [4]{x \left (a x^3+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 110, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}+\frac {4 \left (a x^4+b x\right )^{3/4} \left (11 a x^3-3 b\right )}{63 b 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 = 0.28, size = 212, normalized size = 1.93 \begin {gather*} -\frac {\sqrt {2} \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 )}{3 \, \left (-a\right )^{\frac {1}{4}}} - \frac {\sqrt {2} \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 )}{3 \, \left (-a\right )^{\frac {1}{4}}} + \frac {\sqrt {2} \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 )}{6 \, \left (-a\right )^{\frac {1}{4}}} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{6 \, a} - \frac {4 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}} b^{6} - 14 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a b^{6}\right )}}{63 \, b^{7}} \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 \begin {gather*} \int \frac {x^{6}-a \,x^{3}+b}{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 {x^{6} - a x^{3} + b}{{\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 = 1.20, size = 77, normalized size = 0.70 \begin {gather*} \frac {4\,x\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^3}{b}\right )}{3\,{\left (a\,x^4+b\,x\right )}^{1/4}}-\frac {4\,{\left (a\,x^4+b\,x\right )}^{3/4}}{21\,x^6}+\frac {44\,a\,{\left (a\,x^4+b\,x\right )}^{3/4}}{63\,b\,x^3} \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 {- a x^{3} + b + x^{6}}{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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