Optimal. Leaf size=58 \[ \frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3, 2018, 640, 620, 206} \begin {gather*} \frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3
Rule 206
Rule 620
Rule 640
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=\frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c}\\ &=\frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 1.26 \begin {gather*} \frac {x \left (\sqrt {c} x \left (b+c x^2\right )-b \sqrt {b+c x^2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b+c x^2}}\right )\right )}{2 c^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 68, normalized size = 1.17 \begin {gather*} \frac {b \log \left (-2 c^{3/2} \sqrt {b x^2+c x^4}+b c+2 c^2 x^2\right )}{4 c^{3/2}}+\frac {\sqrt {b x^2+c x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 114, normalized size = 1.97 \begin {gather*} \left [\frac {b \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} c}{4 \, c^{2}}, \frac {b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} c}{2 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 59, normalized size = 1.02 \begin {gather*} \frac {b \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} \sqrt {c} - b \right |}\right )}{4 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2}}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 1.10 \begin {gather*} \frac {\sqrt {c \,x^{2}+b}\, \left (-b c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+\sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} x \right ) x}{2 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 52, normalized size = 0.90 \begin {gather*} -\frac {b \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{4 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2}}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.61, size = 53, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}}{2\,c}-\frac {b\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{4\,c^{3/2}} \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^{3}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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