Optimal. Leaf size=321 \[ -\frac {27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}+\frac {27\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{640 b^3 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]
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Rubi [A]
time = 0.21, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {285, 327, 247,
242, 225} \begin {gather*} \frac {27\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{640 b^3 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}+\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 242
Rule 247
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^4 \sqrt [6]{a+b x^2} \, dx &=\frac {3}{16} x^5 \sqrt [6]{a+b x^2}+\frac {1}{16} a \int \frac {x^4}{\left (a+b x^2\right )^{5/6}} \, dx\\ &=\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}-\frac {\left (9 a^2\right ) \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx}{160 b}\\ &=-\frac {27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}+\frac {\left (27 a^3\right ) \int \frac {1}{\left (a+b x^2\right )^{5/6}} \, dx}{640 b^2}\\ &=-\frac {27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}+\frac {\left (27 a^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-b x^2\right )^{2/3}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{640 b^2 \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=-\frac {27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}-\frac {\left (81 a^3 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{1280 b^3 x \sqrt [3]{\frac {a}{a+b x^2}}}\\ &=-\frac {27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac {3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac {3}{16} x^5 \sqrt [6]{a+b x^2}+\frac {27\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{640 b^3 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.82, size = 93, normalized size = 0.29 \begin {gather*} \frac {3 x \sqrt [6]{a+b x^2} \left (\sqrt [6]{1+\frac {b x^2}{a}} \left (-9 a^2+a b x^2+10 b^2 x^4\right )+9 a^2 \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )\right )}{160 b^2 \sqrt [6]{1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{4} \left (b \,x^{2}+a \right )^{\frac {1}{6}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 29, normalized size = 0.09 \begin {gather*} \frac {\sqrt [6]{a} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} \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*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int x^4\,{\left (b\,x^2+a\right )}^{1/6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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