Optimal. Leaf size=609 \[ \frac {3 x \left (\frac {b x^2}{a}+1\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )}+\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {3^{3/4} a \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} \left (\frac {b x^2}{a}+1\right )^{4/3} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} \left (\frac {b x^2}{a}+1\right )^{4/3} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}} \]
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Rubi [A] time = 0.37, antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1089, 199, 235, 304, 219, 1879} \[ \frac {3 x \left (\frac {b x^2}{a}+1\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )}+\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {3^{3/4} a \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} \left (\frac {b x^2}{a}+1\right )^{4/3} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} \left (\frac {b x^2}{a}+1\right )^{4/3} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 219
Rule 235
Rule 304
Rule 1089
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx &=\frac {\left (1+\frac {b x^2}{a}\right )^{4/3} \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{4/3}} \, dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac {\left (1+\frac {b x^2}{a}\right )^{4/3} \int \frac {1}{\sqrt [3]{1+\frac {b x^2}{a}}} \, dx}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac {\left (3 a \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {\left (3 a \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac {\left (3 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} a \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{2 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {3 x \left (1+\frac {b x^2}{a}\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a \left (1+\frac {b x^2}{a}\right )^{4/3} \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}} E\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 )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \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}}}+\frac {3^{3/4} a \left (1+\frac {b x^2}{a}\right )^{4/3} \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 {2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \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 = 0.02, size = 64, normalized size = 0.11 \[ -\frac {x \left (a+b x^2\right ) \left (\sqrt [3]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )-3\right )}{2 a \left (\left (a+b x^2\right )^2\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{2/3}} \,d x \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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