Optimal. Leaf size=38 \[ \frac {2 a}{3 b^2 \sqrt {a+b x^3}}+\frac {2 \sqrt {a+b x^3}}{3 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, 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 {2 a}{3 b^2 \sqrt {a+b x^3}}+\frac {2 \sqrt {a+b x^3}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,x^3\right )\\ &=\frac {2 a}{3 b^2 \sqrt {a+b x^3}}+\frac {2 \sqrt {a+b x^3}}{3 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.71 \begin {gather*} \frac {2 \left (2 a+b x^3\right )}{3 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 35, normalized size = 0.92
method | result | size |
gosper | \(\frac {\frac {2 b \,x^{3}}{3}+\frac {4 a}{3}}{\sqrt {b \,x^{3}+a}\, b^{2}}\) | \(24\) |
trager | \(\frac {\frac {2 b \,x^{3}}{3}+\frac {4 a}{3}}{\sqrt {b \,x^{3}+a}\, b^{2}}\) | \(24\) |
risch | \(\frac {2 a}{3 b^{2} \sqrt {b \,x^{3}+a}}+\frac {2 \sqrt {b \,x^{3}+a}}{3 b^{2}}\) | \(31\) |
default | \(\frac {2 a}{3 b^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 \sqrt {b \,x^{3}+a}}{3 b^{2}}\) | \(35\) |
elliptic | \(\frac {2 a}{3 b^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 \sqrt {b \,x^{3}+a}}{3 b^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 30, normalized size = 0.79 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a}}{3 \, b^{2}} + \frac {2 \, a}{3 \, \sqrt {b x^{3} + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 35, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 46, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {4 a}{3 b^{2} \sqrt {a + b x^{3}}} + \frac {2 x^{3}}{3 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 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 = 1.56, size = 33, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {b x^{3} + a}}{b} + \frac {a}{\sqrt {b x^{3} + a} b}\right )}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 24, normalized size = 0.63 \begin {gather*} \frac {2\,b\,x^3+4\,a}{3\,b^2\,\sqrt {b\,x^3+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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