Optimal. Leaf size=103 \[ \frac {2}{5} a^3 x^{5/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{13} a \left (b^2+a c\right ) x^{13/2}+\frac {2}{17} b \left (b^2+6 a c\right ) x^{17/2}+\frac {2}{7} c \left (b^2+a c\right ) x^{21/2}+\frac {6}{25} b c^2 x^{25/2}+\frac {2}{29} c^3 x^{29/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1122}
\begin {gather*} \frac {2}{5} a^3 x^{5/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {2}{7} c x^{21/2} \left (a c+b^2\right )+\frac {2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac {6}{13} a x^{13/2} \left (a c+b^2\right )+\frac {6}{25} b c^2 x^{25/2}+\frac {2}{29} c^3 x^{29/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1122
Rubi steps
\begin {align*} \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 b x^{7/2}+3 a \left (b^2+a c\right ) x^{11/2}+b \left (b^2+6 a c\right ) x^{15/2}+3 c \left (b^2+a c\right ) x^{19/2}+3 b c^2 x^{23/2}+c^3 x^{27/2}\right ) \, dx\\ &=\frac {2}{5} a^3 x^{5/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{13} a \left (b^2+a c\right ) x^{13/2}+\frac {2}{17} b \left (b^2+6 a c\right ) x^{17/2}+\frac {2}{7} c \left (b^2+a c\right ) x^{21/2}+\frac {6}{25} b c^2 x^{25/2}+\frac {2}{29} c^3 x^{29/2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 111, normalized size = 1.08 \begin {gather*} \frac {2 \left (672945 a^3 x^{5/2}+1121575 a^2 b x^{9/2}+776475 a b^2 x^{13/2}+776475 a^2 c x^{13/2}+197925 b^3 x^{17/2}+1187550 a b c x^{17/2}+480675 b^2 c x^{21/2}+480675 a c^2 x^{21/2}+403767 b c^2 x^{25/2}+116025 c^3 x^{29/2}\right )}{3364725} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 111, normalized size = 1.08
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 x^{8} c^{2} a +480675 x^{8} b^{2} c +1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 x^{4} a^{2} c +776475 a \,b^{2} x^{4}+1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 x^{8} c^{2} a +480675 x^{8} b^{2} c +1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 x^{4} a^{2} c +776475 a \,b^{2} x^{4}+1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 x^{8} c^{2} a +480675 x^{8} b^{2} c +1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 x^{4} a^{2} c +776475 a \,b^{2} x^{4}+1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
derivativedivides | \(\frac {2 c^{3} x^{\frac {29}{2}}}{29}+\frac {6 b \,c^{2} x^{\frac {25}{2}}}{25}+\frac {2 \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{2} b \,x^{\frac {9}{2}}}{3}+\frac {2 a^{3} x^{\frac {5}{2}}}{5}\) | \(111\) |
default | \(\frac {2 c^{3} x^{\frac {29}{2}}}{29}+\frac {6 b \,c^{2} x^{\frac {25}{2}}}{25}+\frac {2 \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{2} b \,x^{\frac {9}{2}}}{3}+\frac {2 a^{3} x^{\frac {5}{2}}}{5}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 81, normalized size = 0.79 \begin {gather*} \frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {6}{25} \, b c^{2} x^{\frac {25}{2}} + \frac {2}{7} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {6}{13} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {13}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 86, normalized size = 0.83 \begin {gather*} \frac {2}{3364725} \, {\left (116025 \, c^{3} x^{14} + 403767 \, b c^{2} x^{12} + 480675 \, {\left (b^{2} c + a c^{2}\right )} x^{10} + 197925 \, {\left (b^{3} + 6 \, a b c\right )} x^{8} + 1121575 \, a^{2} b x^{4} + 776475 \, {\left (a b^{2} + a^{2} c\right )} x^{6} + 672945 \, a^{3} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.65, size = 129, normalized size = 1.25 \begin {gather*} \frac {2 a^{3} x^{\frac {5}{2}}}{5} + \frac {2 a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 a b^{2} x^{\frac {13}{2}}}{13} + \frac {12 a b c x^{\frac {17}{2}}}{17} + \frac {2 a c^{2} x^{\frac {21}{2}}}{7} + \frac {2 b^{3} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c x^{\frac {21}{2}}}{7} + \frac {6 b c^{2} x^{\frac {25}{2}}}{25} + \frac {2 c^{3} x^{\frac {29}{2}}}{29} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.85, size = 87, normalized size = 0.84 \begin {gather*} \frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {6}{25} \, b c^{2} x^{\frac {25}{2}} + \frac {2}{7} \, b^{2} c x^{\frac {21}{2}} + \frac {2}{7} \, a c^{2} x^{\frac {21}{2}} + \frac {2}{17} \, b^{3} x^{\frac {17}{2}} + \frac {12}{17} \, a b c x^{\frac {17}{2}} + \frac {6}{13} \, a b^{2} x^{\frac {13}{2}} + \frac {6}{13} \, a^{2} c x^{\frac {13}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 76, normalized size = 0.74 \begin {gather*} x^{17/2}\,\left (\frac {2\,b^3}{17}+\frac {12\,a\,c\,b}{17}\right )+\frac {2\,a^3\,x^{5/2}}{5}+\frac {2\,c^3\,x^{29/2}}{29}+\frac {2\,a^2\,b\,x^{9/2}}{3}+\frac {6\,b\,c^2\,x^{25/2}}{25}+\frac {6\,a\,x^{13/2}\,\left (b^2+a\,c\right )}{13}+\frac {2\,c\,x^{21/2}\,\left (b^2+a\,c\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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