Optimal. Leaf size=70 \[ \frac {\sqrt {a^2 x^4+b^2}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}}\right ) \]
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Rubi [A] time = 0.97, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {6725, 277, 305, 220, 1196, 1209, 1198, 1211, 1699, 208} \begin {gather*} \frac {\sqrt {a^2 x^4+b^2}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 220
Rule 277
Rule 305
Rule 1196
Rule 1198
Rule 1209
Rule 1211
Rule 1699
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx &=\int \left (-\frac {\sqrt {b^2+a^2 x^4}}{x^2}+\frac {2 a \sqrt {b^2+a^2 x^4}}{-b+a x^2}\right ) \, dx\\ &=(2 a) \int \frac {\sqrt {b^2+a^2 x^4}}{-b+a x^2} \, dx-\int \frac {\sqrt {b^2+a^2 x^4}}{x^2} \, dx\\ &=\frac {\sqrt {b^2+a^2 x^4}}{x}-\frac {2 \int \frac {-a^2 b-a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx}{a}-\left (2 a^2\right ) \int \frac {x^2}{\sqrt {b^2+a^2 x^4}} \, dx+\left (4 a b^2\right ) \int \frac {1}{\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx\\ &=\frac {\sqrt {b^2+a^2 x^4}}{x}-2 \left ((2 a b) \int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx\right )-(2 a b) \int \frac {-b-a x^2}{\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx+(4 a b) \int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx\\ &=\frac {\sqrt {b^2+a^2 x^4}}{x}+\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b+2 a b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^2+a^2 x^4}}\right )\\ &=\frac {\sqrt {b^2+a^2 x^4}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.30, size = 147, normalized size = 2.10 \begin {gather*} \frac {\sqrt {\frac {i a}{b}} \left (a^2 x^4+b^2\right )-2 i a b x \sqrt {\frac {a^2 x^4}{b^2}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a}{b}} x\right )\right |-1\right )+4 i a b x \sqrt {\frac {a^2 x^4}{b^2}+1} \Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {\frac {i a}{b}} x\right )\right |-1\right )}{x \sqrt {\frac {i a}{b}} \sqrt {a^2 x^4+b^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 70, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b^2+a^2 x^4}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 159, normalized size = 2.27 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {a b} x \log \left (\frac {a^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {a b} x + b^{2}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{4} + b^{2}}}{2 \, x}, \frac {\sqrt {2} \sqrt {-a b} x \arctan \left (\frac {\sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {-a b}}{2 \, a b x}\right ) + \sqrt {a^{2} x^{4} + b^{2}}}{x}\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 {\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 63, normalized size = 0.90
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {a^{2} x^{4}+b^{2}}\, \sqrt {2}}{x}-\frac {2 a b \arctanh \left (\frac {\sqrt {a^{2} x^{4}+b^{2}}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{\sqrt {a b}}\right ) \sqrt {2}}{2}\) | \(63\) |
risch | \(\frac {\sqrt {a^{2} x^{4}+b^{2}}}{x}+2 b a \left (\frac {\sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticF \left (x \sqrt {\frac {i a}{b}}, i\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}-\frac {2 \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticPi \left (x \sqrt {\frac {i a}{b}}, -i, \frac {\sqrt {-\frac {i a}{b}}}{\sqrt {\frac {i a}{b}}}\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}\right )\) | \(174\) |
default | \(2 a \left (\frac {b \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticF \left (x \sqrt {\frac {i a}{b}}, i\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}+\frac {i b \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticF \left (x \sqrt {\frac {i a}{b}}, i\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}-\frac {i b \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticE \left (x \sqrt {\frac {i a}{b}}, i\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}-\frac {2 b \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticPi \left (x \sqrt {\frac {i a}{b}}, -i, \frac {\sqrt {-\frac {i a}{b}}}{\sqrt {\frac {i a}{b}}}\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}\right )+\frac {\sqrt {a^{2} x^{4}+b^{2}}}{x}-\frac {2 i a b \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \left (\EllipticF \left (x \sqrt {\frac {i a}{b}}, i\right )-\EllipticE \left (x \sqrt {\frac {i a}{b}}, i\right )\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}\) | \(397\) |
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^{2} x^{4} + b^{2}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{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 = -0.01 \begin {gather*} -\int \frac {\sqrt {a^2\,x^4+b^2}\,\left (a\,x^2+b\right )}{x^2\,\left (b-a\,x^2\right )} \,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 {\left (a x^{2} + b\right ) \sqrt {a^{2} x^{4} + b^{2}}}{x^{2} \left (a x^{2} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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