Optimal. Leaf size=1356 \[ \frac {a^3 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right ) d^5}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}-\frac {a^3 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}\right ) d^5}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}+\frac {a^2 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right ) d^3}{\left (b c^4+a^2 d^4\right ) \sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}+\frac {a^2 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}\right ) d^3}{\left (b c^4+a^2 d^4\right ) \sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}+\frac {a c \sqrt {a x^2+\sqrt {a^2 x^4+b}} d^3}{\left (b c^4+a^2 d^4\right ) \left (c^2 x^2-d^2\right )}+\frac {a^{3/2} \text {RootSum}\left [c^2 \text {$\#$1}^4-4 i a d^2 \text {$\#$1}^3+2 b c^2 \text {$\#$1}^2+4 i a b d^2 \text {$\#$1}+b^2 c^2\& ,\frac {2 i b \log \left (i a x^2+i \sqrt {2} \sqrt {a} \sqrt {a x^2+\sqrt {a^2 x^4+b}} x-\text {$\#$1}+i \sqrt {a^2 x^4+b}\right ) \text {$\#$1} c^2-a d^2 \log \left (i a x^2+i \sqrt {2} \sqrt {a} \sqrt {a x^2+\sqrt {a^2 x^4+b}} x-\text {$\#$1}+i \sqrt {a^2 x^4+b}\right ) \text {$\#$1}^2+a b d^2 \log \left (i a x^2+i \sqrt {2} \sqrt {a} \sqrt {a x^2+\sqrt {a^2 x^4+b}} x-\text {$\#$1}+i \sqrt {a^2 x^4+b}\right )}{-i b \text {$\#$1}^3 c^6-i b^2 \text {$\#$1} c^6+a b^2 d^2 c^4-3 a b d^2 \text {$\#$1}^2 c^4-i a^2 d^4 \text {$\#$1}^3 c^2-i a^2 b d^4 \text {$\#$1} c^2+a^3 b d^6-3 a^3 d^6 \text {$\#$1}^2}\& \right ] d^2}{\sqrt {2}}-\frac {a b c^4 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right ) d}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}+\frac {a b c^4 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}\right ) d}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}+\frac {b c^3 d}{\left (b c^4+a^2 d^4\right ) \left (c^2 x^2-d^2\right ) \sqrt {a x^2+\sqrt {a^2 x^4+b}}}+\frac {i \sqrt {a} x \left (\frac {i a^{3/2} c^2 x^2 d^2}{b c^4+a^2 d^4}+\frac {i \sqrt {a} c^2 \sqrt {a^2 x^4+b} d^2}{b c^4+a^2 d^4}\right )}{\left (c^2 x^2-d^2\right ) \sqrt {a x^2+\sqrt {a^2 x^4+b}}}-\frac {b c^4 x}{\left (b c^4+a^2 d^4\right ) \left (c^2 x^2-d^2\right ) \sqrt {a x^2+\sqrt {a^2 x^4+b}}} \]
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Rubi [F] time = 0.63, 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 \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 {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x)^2 \sqrt {b+a^2 x^4}} \, dx &=\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x)^2 \sqrt {b+a^2 x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c 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 = 8.88, size = 2109, normalized size = 1.56 \begin {gather*} \text {Result too large to show} \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(-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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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\left (c x +d \right )^{2} \sqrt {a^{2} x^{4}+b}}\, dx\]
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 \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (c x + d\right )}^{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 = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{\sqrt {a^2\,x^4+b}\,{\left (d+c\,x\right )}^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 \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} \left (c x + d\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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