Optimal. Leaf size=351 \[ \frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\sqrt {2} \sqrt {a}}-\frac {i \sqrt {2} \text {RootSum}\left [\text {$\#$1}^4 \sqrt {a}-4 i \text {$\#$1}^3 b+2 \text {$\#$1}^2 a^{3/2} b+4 i \text {$\#$1} a b^2+a^{5/2} b^2\& ,\frac {a b^2 \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )-\text {$\#$1}^2 b \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\text {$\#$1}^3 \sqrt {a}-3 i \text {$\#$1}^2 b+\text {$\#$1} a^{3/2} b+i a b^2}\& \right ]}{\sqrt {a}} \]
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Rubi [F] time = 2.62, 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^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \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^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^4}} \, dx &=\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}}+\frac {2 b \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^4}}\right ) \, dx\\ &=(2 b) \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^4}} \, dx+\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\\ &=(2 b) \int \left (-\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {b} \left (\sqrt {b}-a x\right ) \sqrt {b+a^2 x^4}}-\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {b} \left (\sqrt {b}+a x\right ) \sqrt {b+a^2 x^4}}\right ) \, dx+\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 {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}}\right )}{\sqrt {2} \sqrt {a}}-\sqrt {b} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt {b}-a x\right ) \sqrt {b+a^2 x^4}} \, dx-\sqrt {b} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt {b}+a x\right ) \sqrt {b+a^2 x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [B] time = 36.02, size = 1255, normalized size = 3.58 \begin {gather*} -\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}}+2 \sqrt {2} a^{3/2} b^4 \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right )}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text {$\#$1}^2-16 a b^3 \text {$\#$1}^2+3 a^2 b \text {$\#$1}^4+12 b^2 \text {$\#$1}^4+a \text {$\#$1}^6}\&\right ]-4 \sqrt {2} \sqrt {a} b^3 \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^2}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text {$\#$1}^2-16 a b^3 \text {$\#$1}^2+3 a^2 b \text {$\#$1}^4+12 b^2 \text {$\#$1}^4+a \text {$\#$1}^6}\&\right ]+\frac {2 \sqrt {2} b^2 \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^4}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text {$\#$1}^2-16 a b^3 \text {$\#$1}^2+3 a^2 b \text {$\#$1}^4+12 b^2 \text {$\#$1}^4+a \text {$\#$1}^6}\&\right ]}{\sqrt {a}}+\frac {i b \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a^3 b^3 \log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right )-a^2 b^2 \log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right ) \text {$\#$1}^2+a b \log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right ) \text {$\#$1}^6}{a^4 b^3 \text {$\#$1}+4 a^2 b^4 \text {$\#$1}+3 a^3 b^2 \text {$\#$1}^3-16 a b^3 \text {$\#$1}^3+3 a^2 b \text {$\#$1}^5+12 b^2 \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^{2} + b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (a^{2} x^{2} - b\right )}}\,{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 \[\int \frac {\left (a^{2} x^{2}+b \right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\left (a^{2} x^{2}-b \right ) \sqrt {a^{2} x^{4}+b}}\, dx\]
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^{2} + b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (a^{2} x^{2} - b\right )}}\,{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 = -0.00 \begin {gather*} \int -\frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\left (a^2\,x^2+b\right )}{\left (b-a^2\,x^2\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^{2} + b\right )}{\left (a^{2} x^{2} - 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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