Optimal. Leaf size=32 \[ \frac {a}{b^2 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, 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 = {272, 45}
\begin {gather*} \frac {a}{b^2 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{b^2 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.75 \begin {gather*} \frac {2 a+b x^2}{b^2 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 33, normalized size = 1.03
method | result | size |
gosper | \(\frac {b \,x^{2}+2 a}{\sqrt {b \,x^{2}+a}\, b^{2}}\) | \(23\) |
trager | \(\frac {b \,x^{2}+2 a}{\sqrt {b \,x^{2}+a}\, b^{2}}\) | \(23\) |
risch | \(\frac {a}{b^{2} \sqrt {b \,x^{2}+a}}+\frac {\sqrt {b \,x^{2}+a}}{b^{2}}\) | \(29\) |
default | \(\frac {x^{2}}{\sqrt {b \,x^{2}+a}\, b}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 32, normalized size = 1.00 \begin {gather*} \frac {x^{2}}{\sqrt {b x^{2} + a} b} + \frac {2 \, a}{\sqrt {b x^{2} + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.14, size = 34, normalized size = 1.06 \begin {gather*} \frac {{\left (b x^{2} + 2 \, a\right )} \sqrt {b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 41, normalized size = 1.28 \begin {gather*} \begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 32, normalized size = 1.00 \begin {gather*} \frac {\frac {\sqrt {b x^{2} + a}}{b} + \frac {a}{\sqrt {b x^{2} + a} b}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.70, size = 22, normalized size = 0.69 \begin {gather*} \frac {b\,x^2+2\,a}{b^2\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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