Integrand size = 15, antiderivative size = 51 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2}{5} a^3 x^{5/2}+\frac {6}{13} a^2 c x^{13/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2} \]
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Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {276} \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2}{5} a^3 x^{5/2}+\frac {6}{13} a^2 c x^{13/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^{3/2}+3 a^2 c x^{11/2}+3 a c^2 x^{19/2}+c^3 x^{27/2}\right ) \, dx \\ & = \frac {2}{5} a^3 x^{5/2}+\frac {6}{13} a^2 c x^{13/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2 \left (2639 a^3 x^{5/2}+3045 a^2 c x^{13/2}+1885 a c^2 x^{21/2}+455 c^3 x^{29/2}\right )}{13195} \]
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Time = 3.95 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 a^{3} x^{\frac {5}{2}}}{5}+\frac {6 a^{2} c \,x^{\frac {13}{2}}}{13}+\frac {2 a \,c^{2} x^{\frac {21}{2}}}{7}+\frac {2 c^{3} x^{\frac {29}{2}}}{29}\) | \(36\) |
default | \(\frac {2 a^{3} x^{\frac {5}{2}}}{5}+\frac {6 a^{2} c \,x^{\frac {13}{2}}}{13}+\frac {2 a \,c^{2} x^{\frac {21}{2}}}{7}+\frac {2 c^{3} x^{\frac {29}{2}}}{29}\) | \(36\) |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (455 c^{3} x^{12}+1885 a \,c^{2} x^{8}+3045 a^{2} c \,x^{4}+2639 a^{3}\right )}{13195}\) | \(38\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (455 c^{3} x^{12}+1885 a \,c^{2} x^{8}+3045 a^{2} c \,x^{4}+2639 a^{3}\right )}{13195}\) | \(38\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (455 c^{3} x^{12}+1885 a \,c^{2} x^{8}+3045 a^{2} c \,x^{4}+2639 a^{3}\right )}{13195}\) | \(38\) |
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none
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2}{13195} \, {\left (455 \, c^{3} x^{14} + 1885 \, a c^{2} x^{10} + 3045 \, a^{2} c x^{6} + 2639 \, a^{3} x^{2}\right )} \sqrt {x} \]
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Time = 1.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2 a^{3} x^{\frac {5}{2}}}{5} + \frac {6 a^{2} c x^{\frac {13}{2}}}{13} + \frac {2 a c^{2} x^{\frac {21}{2}}}{7} + \frac {2 c^{3} x^{\frac {29}{2}}}{29} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {2}{7} \, a c^{2} x^{\frac {21}{2}} + \frac {6}{13} \, a^{2} c x^{\frac {13}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {2}{7} \, a c^{2} x^{\frac {21}{2}} + \frac {6}{13} \, a^{2} c x^{\frac {13}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{3/2} \left (a+c x^4\right )^3 \, dx=\frac {2\,a^3\,x^{5/2}}{5}+\frac {2\,c^3\,x^{29/2}}{29}+\frac {6\,a^2\,c\,x^{13/2}}{13}+\frac {2\,a\,c^2\,x^{21/2}}{7} \]
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