Optimal. Leaf size=376 \[ -\sqrt {2} a^{3/2} \text {RootSum}\left [\text {$\#$1}^8 c-16 \text {$\#$1}^6 a^2 d+4 \text {$\#$1}^6 b c+32 \text {$\#$1}^4 a^2 b d+6 \text {$\#$1}^4 b^2 c-16 \text {$\#$1}^2 a^2 b^2 d+4 \text {$\#$1}^2 b^3 c+b^4 c\& ,\frac {\text {$\#$1}^4 \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )-2 \text {$\#$1}^2 b \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )+b^2 \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{\text {$\#$1}^6 c-12 \text {$\#$1}^4 a^2 d+3 \text {$\#$1}^4 b c+16 \text {$\#$1}^2 a^2 b d+3 \text {$\#$1}^2 b^2 c-4 a^2 b^2 d+b^3 c}\& \right ] \]
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Rubi [F] time = 4.73, 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}}}{\sqrt {b+a^2 x^4} \left (d+c x^4\right )} \, 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}}}{\sqrt {b+a^2 x^4} \left (d+c x^4\right )} \, dx &=\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt {d}-\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}} \, dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt {d}+\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}} \, dx}{2 \sqrt {d}}\\ &=\frac {\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}-\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}+\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}}\right ) \, dx}{2 \sqrt {d}}+\frac {\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}-\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}+\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}}\right ) \, dx}{2 \sqrt {d}}\\ &=\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}-\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}+\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}-\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}+\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}\\ \end {align*}
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Mathematica [F] time = 0.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4} \left (d+c x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 12.09, size = 376, normalized size = 1.00 \begin {gather*} -\sqrt {2} a^{3/2} \text {RootSum}\left [b^4 c+4 b^3 c \text {$\#$1}^2-16 a^2 b^2 d \text {$\#$1}^2+6 b^2 c \text {$\#$1}^4+32 a^2 b d \text {$\#$1}^4+4 b c \text {$\#$1}^6-16 a^2 d \text {$\#$1}^6+c \text {$\#$1}^8\&,\frac {b^2 \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )-2 b \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^3 c-4 a^2 b^2 d+3 b^2 c \text {$\#$1}^2+16 a^2 b d \text {$\#$1}^2+3 b c \text {$\#$1}^4-12 a^2 d \text {$\#$1}^4+c \text {$\#$1}^6}\&\right ] \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}}}{\sqrt {a^{2} x^{4} + b} {\left (c x^{4} + d\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 \[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}\, \left (c \,x^{4}+d \right )}\, dx\]
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}}}{\sqrt {a^{2} x^{4} + b} {\left (c x^{4} + d\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 (c\,x^4+d\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}}}{\sqrt {a^{2} x^{4} + b} \left (c x^{4} + d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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