Optimal. Leaf size=289 \[ \frac {\sqrt {2-\sqrt {3}} \left (\frac {b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}-\frac {a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.16, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1113, 325, 236, 219} \[ \frac {\sqrt {2-\sqrt {3}} \left (\frac {b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}-\frac {a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 236
Rule 325
Rule 1113
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (1+\frac {b x^2}{a}\right )^{2/3} \int \frac {1}{x^2 \left (1+\frac {b x^2}{a}\right )^{2/3}} \, dx}{\sqrt [3]{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (1+\frac {b x^2}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{2/3}} \, dx}{3 a \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{2 x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {2-\sqrt {3}} \left (1+\frac {b x^2}{a}\right )^{2/3} \left (1-\sqrt [3]{1+\frac {b x^2}{a}}\right ) \sqrt {\frac {1+\sqrt [3]{1+\frac {b x^2}{a}}+\left (1+\frac {b x^2}{a}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}{1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt {-\frac {1-\sqrt [3]{1+\frac {b x^2}{a}}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.18 \[ -\frac {\left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac {1}{2},\frac {2}{3};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \sqrt [3]{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}}{b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {1}{3}} x^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{3}} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt [3]{\left (a + b x^{2}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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