Optimal. Leaf size=82 \[ \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c} \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1914, 1107, 621, 206} \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c} \sqrt {a x+b x^3+c x^5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 1107
Rule 1914
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a x+b x^3+c x^5}}\\ &=\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{\sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 82, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c} \sqrt {x \left (a+b x^2+c x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 67, normalized size = 0.82 \begin {gather*} \frac {\log \left (\sqrt {x}\right )}{2 \sqrt {c}}-\frac {\log \left (-2 \sqrt {c} \sqrt {a x+b x^3+c x^5}+b \sqrt {x}+2 c x^{5/2}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 135, normalized size = 1.65 \begin {gather*} \left [\frac {\log \left (-\frac {8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {c} \sqrt {x} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right )}{4 \, \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {-c} \sqrt {x}}{2 \, {\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right )}{2 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 60, normalized size = 0.73 \begin {gather*} -\frac {\log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c}} + \frac {\log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right )}{2 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )}{2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}\, \sqrt {x}} \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 {x^{\frac {3}{2}}}{\sqrt {c x^{5} + b x^{3} + a x}}\,{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 {x^{3/2}}{\sqrt {c\,x^5+b\,x^3+a\,x}} \,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 {x^{\frac {3}{2}}}{\sqrt {x \left (a + b x^{2} + c x^{4}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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