Optimal. Leaf size=120 \[ -\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {397} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 397
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 165, normalized size = 1.38 \begin {gather*} \frac {6 a x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right ) \left (6 a F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.26, size = 136, normalized size = 1.13 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}{2 \sqrt {a} \sqrt {a+b x^2}+b x^2}\right )}{4 a^{3/4} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {a+b x^2}}{\sqrt {b}}-\frac {\sqrt {b} x^2}{2 \sqrt [4]{a}}}{x \sqrt [4]{a+b x^2}}\right )}{4 a^{3/4} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 137.83, size = 337, normalized size = 2.81 \begin {gather*} \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {b x^{2} + a} a \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )} \sqrt {-a b \sqrt {-\frac {1}{a^{3} b^{2}}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (b x^{2} + a\right )}^{\frac {1}{4}} a \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) \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*} \int \frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (b \,x^{2}+2 a \right )}\, dx \end {gather*}
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*} \int \frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,x^2+a\right )}^{1/4}\,\left (b\,x^2+2\,a\right )} \,d x \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 {1}{\sqrt [4]{a + b x^{2}} \left (2 a + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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