Optimal. Leaf size=36 \[ \frac {a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{b^2 \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{b^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{5/2}}+\frac {1}{b (a+b x)^{3/2}}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{b^2 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.78 \[ \frac {-2 a-3 b x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 47, normalized size = 1.31 \[ -\frac {{\left (3 \, b x^{2} + 2 \, a\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 24, normalized size = 0.67 \[ -\frac {3 \, b x^{2} + 2 \, a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \]
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 \[ -\frac {3 b \,x^{2}+2 a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 33, normalized size = 0.92 \[ -\frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {2 \, a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 24, normalized size = 0.67 \[ -\frac {3\,b\,x^2+2\,a}{3\,b^2\,{\left (b\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 92, normalized size = 2.56 \[ \begin {cases} - \frac {2 a}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} - \frac {3 b x^{2}}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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