Optimal. Leaf size=80 \[ -\frac {2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac {2 a \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, 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 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^4}-\frac {2 a \left (a+b x^3\right )^{7/2}}{7 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{11} \sqrt {a+b x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^3 \sqrt {a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac {2 a \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-16 a^4+8 a^3 b x^3-6 a^2 b^2 x^6+5 a b^3 x^9+35 b^4 x^{12}\right )}{945 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 91, normalized size = 1.14
method | result | size |
gosper | \(-\frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (-35 b^{3} x^{9}+30 a \,b^{2} x^{6}-24 a^{2} b \,x^{3}+16 a^{3}\right )}{945 b^{4}}\) | \(47\) |
trager | \(-\frac {2 \left (-35 b^{4} x^{12}-5 a \,b^{3} x^{9}+6 a^{2} b^{2} x^{6}-8 a^{3} b \,x^{3}+16 a^{4}\right ) \sqrt {b \,x^{3}+a}}{945 b^{4}}\) | \(58\) |
risch | \(-\frac {2 \left (-35 b^{4} x^{12}-5 a \,b^{3} x^{9}+6 a^{2} b^{2} x^{6}-8 a^{3} b \,x^{3}+16 a^{4}\right ) \sqrt {b \,x^{3}+a}}{945 b^{4}}\) | \(58\) |
default | \(\frac {2 x^{12} \sqrt {b \,x^{3}+a}}{27}+\frac {2 a \,x^{9} \sqrt {b \,x^{3}+a}}{189 b}-\frac {4 a^{2} x^{6} \sqrt {b \,x^{3}+a}}{315 b^{2}}+\frac {16 a^{3} x^{3} \sqrt {b \,x^{3}+a}}{945 b^{3}}-\frac {32 a^{4} \sqrt {b \,x^{3}+a}}{945 b^{4}}\) | \(91\) |
elliptic | \(\frac {2 x^{12} \sqrt {b \,x^{3}+a}}{27}+\frac {2 a \,x^{9} \sqrt {b \,x^{3}+a}}{189 b}-\frac {4 a^{2} x^{6} \sqrt {b \,x^{3}+a}}{315 b^{2}}+\frac {16 a^{3} x^{3} \sqrt {b \,x^{3}+a}}{945 b^{3}}-\frac {32 a^{4} \sqrt {b \,x^{3}+a}}{945 b^{4}}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}}}{27 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a}{7 \, b^{4}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}}{9 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 57, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{12} + 5 \, a b^{3} x^{9} - 6 \, a^{2} b^{2} x^{6} + 8 \, a^{3} b x^{3} - 16 \, a^{4}\right )} \sqrt {b x^{3} + a}}{945 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 114, normalized size = 1.42 \begin {gather*} \begin {cases} - \frac {32 a^{4} \sqrt {a + b x^{3}}}{945 b^{4}} + \frac {16 a^{3} x^{3} \sqrt {a + b x^{3}}}{945 b^{3}} - \frac {4 a^{2} x^{6} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 a x^{9} \sqrt {a + b x^{3}}}{189 b} + \frac {2 x^{12} \sqrt {a + b x^{3}}}{27} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{12}}{12} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 57, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} - 135 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a + 189 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2} - 105 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}\right )}}{945 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 90, normalized size = 1.12 \begin {gather*} \frac {2\,x^{12}\,\sqrt {b\,x^3+a}}{27}-\frac {32\,a^4\,\sqrt {b\,x^3+a}}{945\,b^4}+\frac {2\,a\,x^9\,\sqrt {b\,x^3+a}}{189\,b}+\frac {16\,a^3\,x^3\,\sqrt {b\,x^3+a}}{945\,b^3}-\frac {4\,a^2\,x^6\,\sqrt {b\,x^3+a}}{315\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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