Optimal. Leaf size=36 \[ \frac {\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac {a \sqrt {a+b x^2}}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5, 266, 43} \[ \frac {\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac {a \sqrt {a+b x^2}}{b^2} \]
Antiderivative was successfully verified.
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Rule 5
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {x^3}{\sqrt {a+b x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {a+b x^2}}{b^2}+\frac {\left (a+b x^2\right )^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.75 \[ \frac {\left (b x^2-2 a\right ) \sqrt {a+b x^2}}{3 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 23, normalized size = 0.64 \[ \frac {\sqrt {b x^{2} + a} {\left (b x^{2} - 2 \, a\right )}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 30, normalized size = 0.83 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {\sqrt {b x^{2} + a} a}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.69 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-b \,x^{2}+2 a \right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 33, normalized size = 0.92 \[ \frac {\sqrt {b x^{2} + a} x^{2}}{3 \, b} - \frac {2 \, \sqrt {b x^{2} + a} a}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.60, size = 24, normalized size = 0.67 \[ -\frac {\sqrt {b\,x^2+a}\,\left (2\,a-b\,x^2\right )}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 44, normalized size = 1.22 \[ \begin {cases} - \frac {2 a \sqrt {a + b x^{2}}}{3 b^{2}} + \frac {x^{2} \sqrt {a + b x^{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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