Optimal. Leaf size=115 \[ \frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 115, 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, 738, 640, 621, 206} \[ \frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 738
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\operatorname {Subst}\left (\int \frac {2 a+b x}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\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 )}{c}\\ &=\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 107, normalized size = 0.93 \[ \frac {\frac {2 \sqrt {c} \left (a \left (b-2 c x^2\right )+b^2 x^2\right )}{\sqrt {a+b x^2+c x^4}}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 387, normalized size = 3.37 \[ \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x^{2}\right )}}{4 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x^{2}\right )}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 101, normalized size = 0.88 \[ -\frac {\frac {{\left (b^{2} - 2 \, a c\right )} x^{2}}{b^{2} c - 4 \, a c^{2}} + \frac {a b}{b^{2} c - 4 \, a c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} - \frac {\log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 149, normalized size = 1.30 \[ \frac {b^{2} x^{2}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}+\frac {b^{3}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}-\frac {x^{2}}{2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}+\frac {\ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}+\frac {b}{4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}} \]
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 = 4.76, size = 84, normalized size = 0.73 \[ \frac {\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )}{2\,c^{3/2}}+\frac {\frac {a\,b}{2}-x^2\,\left (a\,c-\frac {b^2}{2}\right )}{2\,c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^4+b\,x^2+a}} \]
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 \[ \int \frac {x^{5}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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