Optimal. Leaf size=80 \[ -\frac {2 a^3 \sqrt {a+b x^3}}{3 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac {2 a \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^4} \]
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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 \sqrt {a+b x^3}}{3 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^4}-\frac {2 a \left (a+b x^3\right )^{5/2}}{5 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt {a+b x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3}{b^3 \sqrt {a+b x}}+\frac {3 a^2 \sqrt {a+b x}}{b^3}-\frac {3 a (a+b x)^{3/2}}{b^3}+\frac {(a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a^3 \sqrt {a+b x^3}}{3 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac {2 a \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-16 a^3+8 a^2 b x^3-6 a b^2 x^6+5 b^3 x^9\right )}{105 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 74, normalized size = 0.92
method | result | size |
gosper | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-5 b^{3} x^{9}+6 a \,b^{2} x^{6}-8 a^{2} b \,x^{3}+16 a^{3}\right )}{105 b^{4}}\) | \(47\) |
trager | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-5 b^{3} x^{9}+6 a \,b^{2} x^{6}-8 a^{2} b \,x^{3}+16 a^{3}\right )}{105 b^{4}}\) | \(47\) |
risch | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-5 b^{3} x^{9}+6 a \,b^{2} x^{6}-8 a^{2} b \,x^{3}+16 a^{3}\right )}{105 b^{4}}\) | \(47\) |
default | \(\frac {2 x^{9} \sqrt {b \,x^{3}+a}}{21 b}-\frac {4 a \,x^{6} \sqrt {b \,x^{3}+a}}{35 b^{2}}+\frac {16 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{105 b^{3}}-\frac {32 a^{3} \sqrt {b \,x^{3}+a}}{105 b^{4}}\) | \(74\) |
elliptic | \(\frac {2 x^{9} \sqrt {b \,x^{3}+a}}{21 b}-\frac {4 a \,x^{6} \sqrt {b \,x^{3}+a}}{35 b^{2}}+\frac {16 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{105 b^{3}}-\frac {32 a^{3} \sqrt {b \,x^{3}+a}}{105 b^{4}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 64, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{21 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{5 \, b^{4}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{3 \, b^{4}} - \frac {2 \, \sqrt {b x^{3} + a} a^{3}}{3 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 46, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} x^{9} - 6 \, a b^{2} x^{6} + 8 \, a^{2} b x^{3} - 16 \, a^{3}\right )} \sqrt {b x^{3} + a}}{105 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 94, normalized size = 1.18 \begin {gather*} \begin {cases} - \frac {32 a^{3} \sqrt {a + b x^{3}}}{105 b^{4}} + \frac {16 a^{2} x^{3} \sqrt {a + b x^{3}}}{105 b^{3}} - \frac {4 a x^{6} \sqrt {a + b x^{3}}}{35 b^{2}} + \frac {2 x^{9} \sqrt {a + b x^{3}}}{21 b} & \text {for}\: b \neq 0 \\\frac {x^{12}}{12 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 61, normalized size = 0.76 \begin {gather*} -\frac {2 \, \sqrt {b x^{3} + a} a^{3}}{3 \, b^{4}} + \frac {2 \, {\left (5 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 73, normalized size = 0.91 \begin {gather*} \frac {2\,x^9\,\sqrt {b\,x^3+a}}{21\,b}-\frac {32\,a^3\,\sqrt {b\,x^3+a}}{105\,b^4}-\frac {4\,a\,x^6\,\sqrt {b\,x^3+a}}{35\,b^2}+\frac {16\,a^2\,x^3\,\sqrt {b\,x^3+a}}{105\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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