Optimal. Leaf size=44 \[ \frac {\sqrt {x^2+1} \sqrt {a x^4}}{2 x}-\frac {\sqrt {a x^4} \sinh ^{-1}(x)}{2 x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {15, 321, 215} \[ \frac {\sqrt {x^2+1} \sqrt {a x^4}}{2 x}-\frac {\sqrt {a x^4} \sinh ^{-1}(x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 215
Rule 321
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^4}}{\sqrt {1+x^2}} \, dx &=\frac {\sqrt {a x^4} \int \frac {x^2}{\sqrt {1+x^2}} \, dx}{x^2}\\ &=\frac {\sqrt {a x^4} \sqrt {1+x^2}}{2 x}-\frac {\sqrt {a x^4} \int \frac {1}{\sqrt {1+x^2}} \, dx}{2 x^2}\\ &=\frac {\sqrt {a x^4} \sqrt {1+x^2}}{2 x}-\frac {\sqrt {a x^4} \sinh ^{-1}(x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.73 \[ \frac {\sqrt {a x^4} \left (x \sqrt {x^2+1}-\sinh ^{-1}(x)\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 42, normalized size = 0.95 \[ \frac {\sqrt {a x^{4}} \sqrt {x^{2} + 1} x + \sqrt {a x^{4}} \log \left (-x + \sqrt {x^{2} + 1}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 27, normalized size = 0.61 \[ \frac {1}{2} \, {\left (\sqrt {x^{2} + 1} x + \log \left (-x + \sqrt {x^{2} + 1}\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.61 \[ \frac {\sqrt {a \,x^{4}}\, \left (\sqrt {x^{2}+1}\, x -\arcsinh \relax (x )\right )}{2 x^{2}} \]
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 \[ \int \frac {\sqrt {a x^{4}}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a\,x^4}}{\sqrt {x^2+1}} \,d x \]
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 \[ \int \frac {\sqrt {a x^{4}}}{\sqrt {x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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