Optimal. Leaf size=108 \[ \frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}-\frac {b \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 c} \]
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Rubi [A] time = 0.08, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1114, 640, 612, 621, 206} \[ \frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}-\frac {b \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1114
Rubi steps
\begin {align*} \int x^3 \sqrt {a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sqrt {a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {b \operatorname {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {b \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 c}+\frac {\left (b \left (b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 c^2}\\ &=-\frac {b \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 c}+\frac {\left (b \left (b^2-4 a c\right )\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 )}{16 c^2}\\ &=-\frac {b \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 c}+\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 101, normalized size = 0.94 \[ \frac {2 \sqrt {c} \sqrt {a+b x^2+c x^4} \left (8 c \left (a+c x^4\right )-3 b^2+2 b c x^2\right )+3 b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{96 c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 237, normalized size = 2.19 \[ \left [-\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (8 \, c^{3} x^{4} + 2 \, b c^{2} x^{2} - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, c^{3}}, -\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \, {\left (8 \, c^{3} x^{4} + 2 \, b c^{2} x^{2} - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 98, normalized size = 0.91 \[ \frac {1}{48} \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac {{\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{32 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 139, normalized size = 1.29 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{2}}{8 c}-\frac {a b \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}+\frac {b^{3} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {5}{2}}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2}}{16 c^{2}}+\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 87, normalized size = 0.81 \[ \frac {\left (8\,c\,\left (c\,x^4+a\right )-3\,b^2+2\,b\,c\,x^2\right )\,\sqrt {c\,x^4+b\,x^2+a}}{48\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^4+b\,x^2+a}+\frac {2\,c\,x^2+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{32\,c^{5/2}} \]
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 \[ \int x^{3} \sqrt {a + b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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