Optimal. Leaf size=157 \[ \frac {b \log \left (i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{8 \sqrt {2} a^{3/2}}-\frac {i \left (2 i a x^4 \sqrt {a^2 x^4+b}+i x^2 \left (2 a^2 x^4-b\right )\right )}{8 a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}} \]
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Rubi [F] time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx &=\int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.56, size = 157, normalized size = 1.00 \begin {gather*} -\frac {i \left (2 i a x^4 \sqrt {b+a^2 x^4}+i x^2 \left (-b+2 a^2 x^4\right )\right )}{8 a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {b \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.82, size = 236, normalized size = 1.50 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {a} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) + 4 \, {\left (3 \, a^{2} x^{3} - \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a^{2}}, -\frac {\sqrt {2} \sqrt {-a} b \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (3 \, a^{2} x^{3} - \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\, dx\]
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*} \int \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} x^{2}\,{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 x^2\,\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2} \,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 x^{2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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