Optimal. Leaf size=73 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}-\frac {3 a x \sqrt {a+b x^2}}{8 b^2}+\frac {x^3 \sqrt {a+b x^2}}{4 b} \]
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Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {5, 321, 217, 206} \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}-\frac {3 a x \sqrt {a+b x^2}}{8 b^2}+\frac {x^3 \sqrt {a+b x^2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 5
Rule 206
Rule 217
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {x^4}{\sqrt {a+b x^2}} \, dx\\ &=\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {(3 a) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 b}\\ &=-\frac {3 a x \sqrt {a+b x^2}}{8 b^2}+\frac {x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^2}\\ &=-\frac {3 a x \sqrt {a+b x^2}}{8 b^2}+\frac {x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^2}\\ &=-\frac {3 a x \sqrt {a+b x^2}}{8 b^2}+\frac {x^3 \sqrt {a+b x^2}}{4 b}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.85 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\sqrt {b} x \sqrt {a+b x^2} \left (2 b x^2-3 a\right )}{8 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 124, normalized size = 1.70 \[ \left [\frac {3 \, a^{2} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, b^{2} x^{3} - 3 \, a b x\right )} \sqrt {b x^{2} + a}}{16 \, b^{3}}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b^{2} x^{3} - 3 \, a b x\right )} \sqrt {b x^{2} + a}}{8 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 54, normalized size = 0.74 \[ \frac {1}{8} \, \sqrt {b x^{2} + a} x {\left (\frac {2 \, x^{2}}{b} - \frac {3 \, a}{b^{2}}\right )} - \frac {3 \, a^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.81 \[ \frac {\sqrt {b \,x^{2}+a}\, x^{3}}{4 b}+\frac {3 a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}-\frac {3 \sqrt {b \,x^{2}+a}\, a x}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 51, normalized size = 0.70 \[ \frac {\sqrt {b x^{2} + a} x^{3}}{4 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a x}{8 \, b^{2}} + \frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} \]
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 \[ \int \frac {x^4}{\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.61, size = 95, normalized size = 1.30 \[ - \frac {3 a^{\frac {3}{2}} x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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