Optimal. Leaf size=177 \[ \frac {1}{12} \sqrt [3]{a} b \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3+b x}+\left (a x^3+b x\right )^{2/3}\right )-\frac {1}{6} \sqrt [3]{a} b \log \left (\sqrt [3]{a x^3+b x}-\sqrt [3]{a} x\right )-\frac {\sqrt [3]{a} b \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2 \sqrt [3]{a x^3+b x}+\sqrt [3]{a} x}\right )}{2 \sqrt {3}}+\frac {\sqrt [3]{a x^3+b x} \left (4 a x^4-3 a x^2-3 b\right )}{8 x^3} \]
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Rubi [A] time = 0.41, antiderivative size = 267, normalized size of antiderivative = 1.51, number of steps used = 14, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2052, 2004, 2032, 329, 275, 331, 292, 31, 634, 617, 204, 628, 2014} \begin {gather*} \frac {\sqrt [3]{a} b x^{2/3} \left (a x^2+b\right )^{2/3} \log \left (\frac {a^{2/3} x^{4/3}}{\left (a x^2+b\right )^{2/3}}+\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{a x^2+b}}+1\right )}{12 \left (a x^3+b x\right )^{2/3}}+\frac {1}{2} a x \sqrt [3]{a x^3+b x}-\frac {3 \left (a x^3+b x\right )^{4/3}}{8 x^4}-\frac {\sqrt [3]{a} b x^{2/3} \left (a x^2+b\right )^{2/3} \log \left (1-\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{a x^2+b}}\right )}{6 \left (a x^3+b x\right )^{2/3}}-\frac {\sqrt [3]{a} b x^{2/3} \left (a x^2+b\right )^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x^{2/3}}{\sqrt [3]{a x^2+b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \left (a x^3+b x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 617
Rule 628
Rule 634
Rule 2004
Rule 2014
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{b x+a x^3} \left (b+a x^4\right )}{x^4} \, dx &=\int \left (a \sqrt [3]{b x+a x^3}+\frac {b \sqrt [3]{b x+a x^3}}{x^4}\right ) \, dx\\ &=a \int \sqrt [3]{b x+a x^3} \, dx+b \int \frac {\sqrt [3]{b x+a x^3}}{x^4} \, dx\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}+\frac {1}{3} (a b) \int \frac {x}{\left (b x+a x^3\right )^{2/3}} \, dx\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}+\frac {\left (a b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (b+a x^2\right )^{2/3}} \, dx}{3 \left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}+\frac {\left (a b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (b+a x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}+\frac {\left (a b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (b+a x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}+\frac {\left (a b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-a x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{2 \left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}+\frac {\left (a^{2/3} b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{a} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{6 \left (b x+a x^3\right )^{2/3}}-\frac {\left (a^{2/3} b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [3]{a} x}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{6 \left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}-\frac {\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3} \log \left (1-\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{6 \left (b x+a x^3\right )^{2/3}}+\frac {\left (\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a}+2 a^{2/3} x}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{12 \left (b x+a x^3\right )^{2/3}}-\frac {\left (a^{2/3} b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{4 \left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}-\frac {\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3} \log \left (1-\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{6 \left (b x+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3} \log \left (1+\frac {a^{2/3} x^{4/3}}{\left (b+a x^2\right )^{2/3}}+\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{12 \left (b x+a x^3\right )^{2/3}}+\frac {\left (\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{2 \left (b x+a x^3\right )^{2/3}}\\ &=\frac {1}{2} a x \sqrt [3]{b x+a x^3}-\frac {3 \left (b x+a x^3\right )^{4/3}}{8 x^4}-\frac {\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (b x+a x^3\right )^{2/3}}-\frac {\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3} \log \left (1-\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{6 \left (b x+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} b x^{2/3} \left (b+a x^2\right )^{2/3} \log \left (1+\frac {a^{2/3} x^{4/3}}{\left (b+a x^2\right )^{2/3}}+\frac {\sqrt [3]{a} x^{2/3}}{\sqrt [3]{b+a x^2}}\right )}{12 \left (b x+a x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 69, normalized size = 0.39 \begin {gather*} \frac {3 \sqrt [3]{x \left (a x^2+b\right )} \left (\frac {2 a x^4 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {a x^2}{b}\right )}{\sqrt [3]{\frac {a x^2}{b}+1}}-a x^2-b\right )}{8 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.57, size = 177, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{b x+a x^3} \left (-3 b-3 a x^2+4 a x^4\right )}{8 x^3}-\frac {\sqrt [3]{a} b \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{b x+a x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \sqrt [3]{a} b \log \left (-\sqrt [3]{a} x+\sqrt [3]{b x+a x^3}\right )+\frac {1}{12} \sqrt [3]{a} b \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{b x+a x^3}+\left (b x+a x^3\right )^{2/3}\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(-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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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}+b x \right )^{\frac {1}{3}} \left (a \,x^{4}+b \right )}{x^{4}}\, dx\]
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 (a x^{4} + b\right )} {\left (a x^{3} + b x\right )}^{\frac {1}{3}}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.85, size = 65, normalized size = 0.37 \begin {gather*} \frac {3\,a\,x\,{\left (a\,x^3+b\,x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ -\frac {a\,x^2}{b}\right )}{4\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/3}}-\frac {3\,{\left (a\,x^3+b\,x\right )}^{1/3}\,\left (a\,x^2+b\right )}{8\,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 {\sqrt [3]{x \left (a x^{2} + b\right )} \left (a x^{4} + b\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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