Optimal. Leaf size=300 \[ -\frac {\sqrt [6]{a+b x^2}}{3 a x^3}+\frac {8 b \sqrt [6]{a+b x^2}}{9 a^2 x}+\frac {16 \sqrt {2-\sqrt {3}} b \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a^2 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]
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Rubi [A]
time = 0.17, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {331, 247, 242,
225} \begin {gather*} \frac {16 \sqrt {2-\sqrt {3}} b \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 )}{9 \sqrt [4]{3} a^2 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 {8 b \sqrt [6]{a+b x^2}}{9 a^2 x}-\frac {\sqrt [6]{a+b x^2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 242
Rule 247
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^{5/6}} \, dx &=-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}-\frac {(8 b) \int \frac {1}{x^2 \left (a+b x^2\right )^{5/6}} \, dx}{9 a}\\ &=-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}+\frac {8 b \sqrt [6]{a+b x^2}}{9 a^2 x}+\frac {\left (16 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{5/6}} \, dx}{27 a^2}\\ &=-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}+\frac {8 b \sqrt [6]{a+b x^2}}{9 a^2 x}+\frac {\left (16 b^2\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 )}{27 a^2 \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}+\frac {8 b \sqrt [6]{a+b x^2}}{9 a^2 x}-\frac {\left (8 b \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 )}{9 a^2 x \sqrt [3]{\frac {a}{a+b x^2}}}\\ &=-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}+\frac {8 b \sqrt [6]{a+b x^2}}{9 a^2 x}+\frac {16 \sqrt {2-\sqrt {3}} b \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 )}{9 \sqrt [4]{3} a^2 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 = 10.01, size = 51, normalized size = 0.17 \begin {gather*} -\frac {\left (1+\frac {b x^2}{a}\right )^{5/6} \, _2F_1\left (-\frac {3}{2},\frac {5}{6};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \left (a+b x^2\right )^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{6}}}\, dx\]
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 = 0.60, size = 32, normalized size = 0.11 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{6} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{6}} x^{3}} \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 \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{5/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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