Optimal. Leaf size=53 \[ -\frac {\tanh ^{-1}\left (\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^3+b x^5+c x^7}}\right )}{2 \sqrt {a}} \]
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Rubi [A] time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1997, 1913, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^3+b x^5+c x^7}}\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 1913
Rule 1997
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x^2+c x^4\right )}} \, dx &=\int \frac {\sqrt {x}}{\sqrt {a x^3+b x^5+c x^7}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x^{3/2} \left (2 a+b x^2\right )}{\sqrt {a x^3+b x^5+c x^7}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^3+b x^5+c x^7}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 85, normalized size = 1.60 \begin {gather*} -\frac {x^{3/2} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a} \sqrt {x^3 \left (a+b x^2+c x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 54, normalized size = 1.02 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {c} x^{7/2}-\sqrt {a x^3+b x^5+c x^7}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 145, normalized size = 2.74 \begin {gather*} \left [\frac {\log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{4} + 8 \, a^{2} x^{2} - 4 \, \sqrt {c x^{7} + b x^{5} + a x^{3}} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{6}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{7} + b x^{5} + a x^{3}} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{6} + a b x^{4} + a^{2} x^{2}\right )}}\right )}{2 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 56, normalized size = 1.06 \begin {gather*} \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 1.40 \begin {gather*} -\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{\frac {3}{2}} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{2 \sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x^{3}}\, \sqrt {a}} \end {gather*}
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 {x}}{\sqrt {{\left (c x^{4} + b x^{2} + a\right )} x^{3}}}\,{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 = -0.02 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {x^3\,\left (c\,x^4+b\,x^2+a\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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