Optimal. Leaf size=398 \[ \frac {\sqrt {a} \log \left (i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{\sqrt {2} c}-\frac {\sqrt {a} \text {RootSum}\left [\text {$\#$1}^4 c+4 i \text {$\#$1}^3 a d+2 \text {$\#$1}^2 b c-4 i \text {$\#$1} a b d+b^2 c\& ,\frac {\text {$\#$1}^2 (-a) d \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )+2 i \text {$\#$1} b c \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )+a b d \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{i \text {$\#$1}^3 c-3 \text {$\#$1}^2 a d+i \text {$\#$1} b c+a b d}\& \right ]}{\sqrt {2} c} \]
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Rubi [F] time = 0.44, 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 {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx &=\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}-\sqrt {-c} x} \, dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}+\sqrt {-c} x} \, dx}{2 \sqrt {d}}\\ \end {align*}
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Mathematica [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 398, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a} \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 )}{\sqrt {2} c}+\frac {\sqrt {a} \text {RootSum}\left [b^2 c+4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2-4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {a b d \log \left (-i a x^2-i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )-2 i b c \log \left (-i a x^2-i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}-a d \log \left (-i a x^2-i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{a b d-i b c \text {$\#$1}-3 a d \text {$\#$1}^2-i c \text {$\#$1}^3}\&\right ]}{\sqrt {2} c} \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 {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d}\,{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 \[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{c \,x^{2}+d}\, dx\]
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 {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d}\,{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}}{c\,x^2+d} \,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}}}{c x^{2} + d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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