Integrand size = 17, antiderivative size = 55 \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{2} a^2 x \sqrt {c x^2}+\frac {2}{3} a b x^2 \sqrt {c x^2}+\frac {1}{4} b^2 x^3 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45} \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{2} a^2 x \sqrt {c x^2}+\frac {2}{3} a b x^2 \sqrt {c x^2}+\frac {1}{4} b^2 x^3 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int x (a+b x)^2 \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x} \\ & = \frac {1}{2} a^2 x \sqrt {c x^2}+\frac {2}{3} a b x^2 \sqrt {c x^2}+\frac {1}{4} b^2 x^3 \sqrt {c x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.60 \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{12} x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]
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Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(\frac {x \left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) \sqrt {c \,x^{2}}}{12}\) | \(30\) |
default | \(\frac {x \left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) \sqrt {c \,x^{2}}}{12}\) | \(30\) |
risch | \(\frac {a^{2} x \sqrt {c \,x^{2}}}{2}+\frac {2 a b \,x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b^{2} x^{3} \sqrt {c \,x^{2}}}{4}\) | \(44\) |
trager | \(\frac {\left (3 b^{2} x^{3}+8 a b \,x^{2}+3 b^{2} x^{2}+6 a^{2} x +8 a b x +3 b^{2} x +6 a^{2}+8 a b +3 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\) | \(71\) |
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none
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{12} \, {\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt {c x^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {a^{2} x \sqrt {c x^{2}}}{2} + \frac {2 a b x^{2} \sqrt {c x^{2}}}{3} + \frac {b^{2} x^{3} \sqrt {c x^{2}}}{4} \]
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none
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{2} \, \sqrt {c x^{2}} a^{2} x + \frac {\left (c x^{2}\right )^{\frac {3}{2}} b^{2} x}{4 \, c} + \frac {2 \, \left (c x^{2}\right )^{\frac {3}{2}} a b}{3 \, c} \]
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none
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.64 \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\frac {1}{12} \, {\left (3 \, b^{2} x^{4} \mathrm {sgn}\left (x\right ) + 8 \, a b x^{3} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} x^{2} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Timed out. \[ \int \sqrt {c x^2} (a+b x)^2 \, dx=\int \sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2 \,d x \]
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