Integrand size = 18, antiderivative size = 37 \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{7} a c x^6 \sqrt {c x^2}+\frac {1}{8} b c x^7 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{7} a c x^6 \sqrt {c x^2}+\frac {1}{8} b c x^7 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int x^6 (a+b x) \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (a x^6+b x^7\right ) \, dx}{x} \\ & = \frac {1}{7} a c x^6 \sqrt {c x^2}+\frac {1}{8} b c x^7 \sqrt {c x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{56} x^4 \left (c x^2\right )^{3/2} (8 a+7 b x) \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(\frac {x^{4} \left (7 b x +8 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{56}\) | \(21\) |
default | \(\frac {x^{4} \left (7 b x +8 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{56}\) | \(21\) |
risch | \(\frac {a c \,x^{6} \sqrt {c \,x^{2}}}{7}+\frac {b c \,x^{7} \sqrt {c \,x^{2}}}{8}\) | \(30\) |
trager | \(\frac {c \left (7 b \,x^{7}+8 a \,x^{6}+7 b \,x^{6}+8 a \,x^{5}+7 b \,x^{5}+8 a \,x^{4}+7 b \,x^{4}+8 a \,x^{3}+7 b \,x^{3}+8 a \,x^{2}+7 b \,x^{2}+8 a x +7 b x +8 a +7 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{56 x}\) | \(98\) |
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{56} \, {\left (7 \, b c x^{7} + 8 \, a c x^{6}\right )} \sqrt {c x^{2}} \]
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Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {a x^{4} \left (c x^{2}\right )^{\frac {3}{2}}}{7} + \frac {b x^{5} \left (c x^{2}\right )^{\frac {3}{2}}}{8} \]
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none
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {\left (c x^{2}\right )^{\frac {5}{2}} b x^{3}}{8 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a x^{2}}{7 \, c} \]
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Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.59 \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{56} \, {\left (7 \, b x^{8} \mathrm {sgn}\left (x\right ) + 8 \, a x^{7} \mathrm {sgn}\left (x\right )\right )} c^{\frac {3}{2}} \]
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Timed out. \[ \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx=\int x^3\,{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right ) \,d x \]
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