Integrand size = 15, antiderivative size = 41 \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {1}{6} a c^2 x^5 \sqrt {c x^2}+\frac {1}{7} b c^2 x^6 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 45} \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {1}{6} a c^2 x^5 \sqrt {c x^2}+\frac {1}{7} b c^2 x^6 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (a+b x) \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^5+b x^6\right ) \, dx}{x} \\ & = \frac {1}{6} a c^2 x^5 \sqrt {c x^2}+\frac {1}{7} b c^2 x^6 \sqrt {c x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {1}{42} x \left (c x^2\right )^{5/2} (7 a+6 b x) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(\frac {x \left (6 b x +7 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{42}\) | \(19\) |
default | \(\frac {x \left (6 b x +7 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{42}\) | \(19\) |
risch | \(\frac {a \,c^{2} x^{5} \sqrt {c \,x^{2}}}{6}+\frac {b \,c^{2} x^{6} \sqrt {c \,x^{2}}}{7}\) | \(34\) |
trager | \(\frac {c^{2} \left (6 b \,x^{6}+7 a \,x^{5}+6 b \,x^{5}+7 a \,x^{4}+6 b \,x^{4}+7 a \,x^{3}+6 b \,x^{3}+7 a \,x^{2}+6 b \,x^{2}+7 a x +6 b x +7 a +6 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{42 x}\) | \(88\) |
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {1}{42} \, {\left (6 \, b c^{2} x^{6} + 7 \, a c^{2} x^{5}\right )} \sqrt {c x^{2}} \]
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Time = 0.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {a x \left (c x^{2}\right )^{\frac {5}{2}}}{6} + \frac {b x^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{7} \]
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Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {1}{6} \, \left (c x^{2}\right )^{\frac {5}{2}} a x + \frac {\left (c x^{2}\right )^{\frac {7}{2}} b}{7 \, c} \]
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Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {1}{42} \, {\left (6 \, b c^{2} x^{7} \mathrm {sgn}\left (x\right ) + 7 \, a c^{2} x^{6} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Timed out. \[ \int \left (c x^2\right )^{5/2} (a+b x) \, dx=\int {\left (c\,x^2\right )}^{5/2}\,\left (a+b\,x\right ) \,d x \]
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