Optimal. Leaf size=59 \[ -\frac {2 a^2}{3 b^3 \sqrt {a+b x^3}}-\frac {4 a \sqrt {a+b x^3}}{3 b^3}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, 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^2}{3 b^3 \sqrt {a+b x^3}}-\frac {4 a \sqrt {a+b x^3}}{3 b^3}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{3/2}}-\frac {2 a}{b^2 \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a^2}{3 b^3 \sqrt {a+b x^3}}-\frac {4 a \sqrt {a+b x^3}}{3 b^3}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.64 \begin {gather*} \frac {2 \left (-8 a^2-4 a b x^3+b^2 x^6\right )}{9 b^3 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 55, normalized size = 0.93
method | result | size |
gosper | \(-\frac {2 \left (-b^{2} x^{6}+4 a b \,x^{3}+8 a^{2}\right )}{9 \sqrt {b \,x^{3}+a}\, b^{3}}\) | \(36\) |
trager | \(-\frac {2 \left (-b^{2} x^{6}+4 a b \,x^{3}+8 a^{2}\right )}{9 \sqrt {b \,x^{3}+a}\, b^{3}}\) | \(36\) |
risch | \(-\frac {2 \left (-b \,x^{3}+5 a \right ) \sqrt {b \,x^{3}+a}}{9 b^{3}}-\frac {2 a^{2}}{3 b^{3} \sqrt {b \,x^{3}+a}}\) | \(43\) |
default | \(-\frac {2 a^{2}}{3 b^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 x^{3} \sqrt {b \,x^{3}+a}}{9 b^{2}}-\frac {10 a \sqrt {b \,x^{3}+a}}{9 b^{3}}\) | \(55\) |
elliptic | \(-\frac {2 a^{2}}{3 b^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 x^{3} \sqrt {b \,x^{3}+a}}{9 b^{2}}-\frac {10 a \sqrt {b \,x^{3}+a}}{9 b^{3}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b^{3}} - \frac {4 \, \sqrt {b x^{3} + a} a}{3 \, b^{3}} - \frac {2 \, a^{2}}{3 \, \sqrt {b x^{3} + a} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 46, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b^{2} x^{6} - 4 \, a b x^{3} - 8 \, a^{2}\right )} \sqrt {b x^{3} + a}}{9 \, {\left (b^{4} x^{3} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 70, normalized size = 1.19 \begin {gather*} \begin {cases} - \frac {16 a^{2}}{9 b^{3} \sqrt {a + b x^{3}}} - \frac {8 a x^{3}}{9 b^{2} \sqrt {a + b x^{3}}} + \frac {2 x^{6}}{9 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{9}}{9 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 = 1.31, size = 52, normalized size = 0.88 \begin {gather*} -\frac {2 \, a^{2}}{3 \, \sqrt {b x^{3} + a} b^{3}} + \frac {2 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x^{3} + a} a b^{6}\right )}}{9 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 41, normalized size = 0.69 \begin {gather*} -\frac {12\,a\,\left (b\,x^3+a\right )-2\,{\left (b\,x^3+a\right )}^2+6\,a^2}{9\,b^3\,\sqrt {b\,x^3+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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