Optimal. Leaf size=177 \[ \frac {\sqrt [3]{b x+a x^3} \left (-3 b-3 a x^2+4 a x^4\right )}{8 x^3}-\frac {\sqrt [3]{a} b \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{b x+a x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \sqrt [3]{a} b \log \left (-\sqrt [3]{a} x+\sqrt [3]{b x+a x^3}\right )+\frac {1}{12} \sqrt [3]{a} b \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{b x+a x^3}+\left (b x+a x^3\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.02, number of steps
used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2077, 2029,
2057, 335, 281, 337, 2039} \begin {gather*} -\frac {\sqrt [3]{a} b x^{2/3} \left (a x^2+b\right )^{2/3} \text {ArcTan}\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}}+\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 (\sqrt [3]{a} x^{2/3}-\sqrt [3]{a x^2+b}\right )}{4 \left (a x^3+b x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 281
Rule 335
Rule 337
Rule 2029
Rule 2039
Rule 2057
Rule 2077
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 2.61, size = 239, normalized size = 1.35 \begin {gather*} -\frac {\sqrt [3]{x \left (b+a x^2\right )} \left (9 b \sqrt [3]{b+a x^2}+9 a x^2 \sqrt [3]{b+a x^2}-12 a x^4 \sqrt [3]{b+a x^2}+4 \sqrt {3} \sqrt [3]{a} b x^{8/3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{a} x^{2/3}}{\sqrt [3]{a} x^{2/3}+2 \sqrt [3]{b+a x^2}}\right )+4 \sqrt [3]{a} b x^{8/3} \log \left (-\sqrt [3]{a} x^{2/3}+\sqrt [3]{b+a x^2}\right )-2 \sqrt [3]{a} b x^{8/3} \log \left (a^{2/3} x^{4/3}+\sqrt [3]{a} x^{2/3} \sqrt [3]{b+a x^2}+\left (b+a x^2\right )^{2/3}\right )\right )}{24 x^3 \sqrt [3]{b+a x^2}} \end {gather*}
Antiderivative was successfully verified.
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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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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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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Giac [A]
time = 6.28, size = 129, normalized size = 0.73 \begin {gather*} \frac {12 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} a b x^{2} + 4 \, \sqrt {3} a^{\frac {1}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2 \, a^{\frac {1}{3}} b^{2} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {2}{3}} + {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b^{2} \log \left ({\left | {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) - 9 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {4}{3}} b}{24 \, b} \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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