Optimal. Leaf size=59 \[ \frac {2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}-\frac {4 a \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^3} \]
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Rubi [A]
time = 0.02, 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 \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^3}-\frac {4 a \left (a+b x^3\right )^{5/2}}{15 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^8 \sqrt {a+b x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac {2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}-\frac {4 a \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.85 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (8 a^3-4 a^2 b x^3+3 a b^2 x^6+15 b^3 x^9\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 71, normalized size = 1.20
method | result | size |
gosper | \(\frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (15 b^{2} x^{6}-12 a b \,x^{3}+8 a^{2}\right )}{315 b^{3}}\) | \(36\) |
trager | \(\frac {2 \left (15 b^{3} x^{9}+3 a \,b^{2} x^{6}-4 a^{2} b \,x^{3}+8 a^{3}\right ) \sqrt {b \,x^{3}+a}}{315 b^{3}}\) | \(47\) |
risch | \(\frac {2 \left (15 b^{3} x^{9}+3 a \,b^{2} x^{6}-4 a^{2} b \,x^{3}+8 a^{3}\right ) \sqrt {b \,x^{3}+a}}{315 b^{3}}\) | \(47\) |
default | \(\frac {2 x^{9} \sqrt {b \,x^{3}+a}}{21}+\frac {2 a \,x^{6} \sqrt {b \,x^{3}+a}}{105 b}-\frac {8 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{315 b^{2}}+\frac {16 a^{3} \sqrt {b \,x^{3}+a}}{315 b^{3}}\) | \(71\) |
elliptic | \(\frac {2 x^{9} \sqrt {b \,x^{3}+a}}{21}+\frac {2 a \,x^{6} \sqrt {b \,x^{3}+a}}{105 b}-\frac {8 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{315 b^{2}}+\frac {16 a^{3} \sqrt {b \,x^{3}+a}}{315 b^{3}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 47, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{21 \, b^{3}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{15 \, b^{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{9 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 46, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (15 \, b^{3} x^{9} + 3 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {b x^{3} + a}}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 90, normalized size = 1.53 \begin {gather*} \begin {cases} \frac {16 a^{3} \sqrt {a + b x^{3}}}{315 b^{3}} - \frac {8 a^{2} x^{3} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 a x^{6} \sqrt {a + b x^{3}}}{105 b} + \frac {2 x^{9} \sqrt {a + b x^{3}}}{21} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{9}}{9} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.09, size = 43, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} - 42 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}\right )}}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 70, normalized size = 1.19 \begin {gather*} \frac {2\,x^9\,\sqrt {b\,x^3+a}}{21}+\frac {16\,a^3\,\sqrt {b\,x^3+a}}{315\,b^3}+\frac {2\,a\,x^6\,\sqrt {b\,x^3+a}}{105\,b}-\frac {8\,a^2\,x^3\,\sqrt {b\,x^3+a}}{315\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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