Optimal. Leaf size=51 \[ \frac {2}{3} a^3 x^{3/2}+\frac {6}{11} a^2 c x^{11/2}+\frac {6}{19} a c^2 x^{19/2}+\frac {2}{27} c^3 x^{27/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {276}
\begin {gather*} \frac {2}{3} a^3 x^{3/2}+\frac {6}{11} a^2 c x^{11/2}+\frac {6}{19} a c^2 x^{19/2}+\frac {2}{27} c^3 x^{27/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+c x^4\right )^3 \, dx &=\int \left (a^3 \sqrt {x}+3 a^2 c x^{9/2}+3 a c^2 x^{17/2}+c^3 x^{25/2}\right ) \, dx\\ &=\frac {2}{3} a^3 x^{3/2}+\frac {6}{11} a^2 c x^{11/2}+\frac {6}{19} a c^2 x^{19/2}+\frac {2}{27} c^3 x^{27/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 x^{3/2} \left (1881 a^3+1539 a^2 c x^4+891 a c^2 x^8+209 c^3 x^{12}\right )}{5643} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 36, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} c \,x^{\frac {11}{2}}}{11}+\frac {6 a \,c^{2} x^{\frac {19}{2}}}{19}+\frac {2 c^{3} x^{\frac {27}{2}}}{27}\) | \(36\) |
default | \(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} c \,x^{\frac {11}{2}}}{11}+\frac {6 a \,c^{2} x^{\frac {19}{2}}}{19}+\frac {2 c^{3} x^{\frac {27}{2}}}{27}\) | \(36\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (209 c^{3} x^{12}+891 a \,c^{2} x^{8}+1539 a^{2} c \,x^{4}+1881 a^{3}\right )}{5643}\) | \(38\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (209 c^{3} x^{12}+891 a \,c^{2} x^{8}+1539 a^{2} c \,x^{4}+1881 a^{3}\right )}{5643}\) | \(38\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (209 c^{3} x^{12}+891 a \,c^{2} x^{8}+1539 a^{2} c \,x^{4}+1881 a^{3}\right )}{5643}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{19} \, a c^{2} x^{\frac {19}{2}} + \frac {6}{11} \, a^{2} c x^{\frac {11}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 38, normalized size = 0.75 \begin {gather*} \frac {2}{5643} \, {\left (209 \, c^{3} x^{13} + 891 \, a c^{2} x^{9} + 1539 \, a^{2} c x^{5} + 1881 \, a^{3} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.83, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 a^{3} x^{\frac {3}{2}}}{3} + \frac {6 a^{2} c x^{\frac {11}{2}}}{11} + \frac {6 a c^{2} x^{\frac {19}{2}}}{19} + \frac {2 c^{3} x^{\frac {27}{2}}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{19} \, a c^{2} x^{\frac {19}{2}} + \frac {6}{11} \, a^{2} c x^{\frac {11}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,c^3\,x^{27/2}}{27}+\frac {6\,a^2\,c\,x^{11/2}}{11}+\frac {6\,a\,c^2\,x^{19/2}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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