Integrand size = 20, antiderivative size = 57 \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{4} a^2 x^3 \sqrt {c x^2}+\frac {2}{5} a b x^4 \sqrt {c x^2}+\frac {1}{6} b^2 x^5 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{4} a^2 x^3 \sqrt {c x^2}+\frac {2}{5} a b x^4 \sqrt {c x^2}+\frac {1}{6} b^2 x^5 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int x^3 (a+b x)^2 \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (a^2 x^3+2 a b x^4+b^2 x^5\right ) \, dx}{x} \\ & = \frac {1}{4} a^2 x^3 \sqrt {c x^2}+\frac {2}{5} a b x^4 \sqrt {c x^2}+\frac {1}{6} b^2 x^5 \sqrt {c x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{60} x^3 \sqrt {c x^2} \left (15 a^2+24 a b x+10 b^2 x^2\right ) \]
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Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(\frac {x^{3} \left (10 b^{2} x^{2}+24 a b x +15 a^{2}\right ) \sqrt {c \,x^{2}}}{60}\) | \(32\) |
default | \(\frac {x^{3} \left (10 b^{2} x^{2}+24 a b x +15 a^{2}\right ) \sqrt {c \,x^{2}}}{60}\) | \(32\) |
risch | \(\frac {a^{2} x^{3} \sqrt {c \,x^{2}}}{4}+\frac {2 a b \,x^{4} \sqrt {c \,x^{2}}}{5}+\frac {b^{2} x^{5} \sqrt {c \,x^{2}}}{6}\) | \(46\) |
trager | \(\frac {\left (10 b^{2} x^{5}+24 a b \,x^{4}+10 b^{2} x^{4}+15 a^{2} x^{3}+24 a b \,x^{3}+10 b^{2} x^{3}+15 a^{2} x^{2}+24 a b \,x^{2}+10 b^{2} x^{2}+15 a^{2} x +24 a b x +10 b^{2} x +15 a^{2}+24 a b +10 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{60 x}\) | \(117\) |
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none
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.58 \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{60} \, {\left (10 \, b^{2} x^{5} + 24 \, a b x^{4} + 15 \, a^{2} x^{3}\right )} \sqrt {c x^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {a^{2} x^{3} \sqrt {c x^{2}}}{4} + \frac {2 a b x^{4} \sqrt {c x^{2}}}{5} + \frac {b^{2} x^{5} \sqrt {c x^{2}}}{6} \]
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none
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {\left (c x^{2}\right )^{\frac {3}{2}} b^{2} x^{3}}{6 \, c} + \frac {2 \, \left (c x^{2}\right )^{\frac {3}{2}} a b x^{2}}{5 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a^{2} x}{4 \, c} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{60} \, {\left (10 \, b^{2} x^{6} \mathrm {sgn}\left (x\right ) + 24 \, a b x^{5} \mathrm {sgn}\left (x\right ) + 15 \, a^{2} x^{4} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Timed out. \[ \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx=\int x^2\,\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2 \,d x \]
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