Optimal. Leaf size=91 \[ \frac{1}{2} x \sqrt [4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt{a} \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} \sqrt{\frac{b x^2}{a}+1}} \]
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Rubi [A] time = 0.0195423, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1089, 195, 215} \[ \frac{1}{2} x \sqrt [4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt{a} \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 1089
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{\sqrt [4]{a^2+2 a b x^2+b^2 x^4} \int \sqrt{1+\frac{b x^2}{a}} \, dx}{\sqrt{1+\frac{b x^2}{a}}}\\ &=\frac{1}{2} x \sqrt [4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt [4]{a^2+2 a b x^2+b^2 x^4} \int \frac{1}{\sqrt{1+\frac{b x^2}{a}}} \, dx}{2 \sqrt{1+\frac{b x^2}{a}}}\\ &=\frac{1}{2} x \sqrt [4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt{a} \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} \sqrt{1+\frac{b x^2}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0376042, size = 59, normalized size = 0.65 \[ \frac{1}{2} \sqrt [4]{\left (a+b x^2\right )^2} \left (\frac{a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{\sqrt{b} \sqrt{a+b x^2}}+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 58, normalized size = 0.6 \begin{align*}{\frac{x}{2}\sqrt [4]{ \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \sqrt [4]{ \left ( b{x}^{2}+a \right ) ^{2}}{\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20101, size = 347, normalized size = 3.81 \begin{align*} \left [\frac{a \sqrt{b} \log \left (-2 \, b x^{2} - 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} \sqrt{b} x - a\right ) + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x}{4 \, b}, -\frac{a \sqrt{-b} \arctan \left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} \sqrt{-b} x}{b x^{2} + a}\right ) -{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x}{2 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{a^{2} + 2 a b x^{2} + b^{2} x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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