Optimal. Leaf size=151 \[ \frac {2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {a^2 x^4+b}+3 \sqrt {a} b x^2}{8 \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}-\frac {11 b \log \left (\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{8 \sqrt {2} \sqrt {a}} \]
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Rubi [F] time = 1.24, 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 \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\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 \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx &=\int \left (-\frac {b \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}}+\frac {a^2 x^4 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}}\right ) \, dx\\ &=a^2 \int \frac {x^4 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx-b \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\\ &=a^2 \int \frac {x^4 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx-b \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}}\right )\\ &=-\frac {b \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}}\right )}{\sqrt {2} \sqrt {a}}+a^2 \int \frac {x^4 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.84, size = 151, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt {a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {b+a^2 x^4}}{8 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.40, size = 236, normalized size = 1.56 \begin {gather*} \left [\frac {11 \, \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 (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a}, \frac {11 \, \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 (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, 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 {{\left (a^{2} x^{4} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}}\,{d x} \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 \[\int \frac {\left (a^{2} x^{4}-b \right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}}\, dx\]
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 \frac {{\left (a^{2} x^{4} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}}\,{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 {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\left (b-a^2\,x^4\right )}{\sqrt {a^2\,x^4+b}} \,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 {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \left (a^{2} x^{4} - b\right )}{\sqrt {a^{2} x^{4} + b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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