Integrand size = 18, antiderivative size = 66 \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {1}{7} a^2 c^2 x^6 \sqrt {c x^2}+\frac {1}{4} a b c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c^2 x^8 \sqrt {c x^2} \]
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Time = 0.02 (sec) , antiderivative size = 66, 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} \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {1}{7} a^2 c^2 x^6 \sqrt {c x^2}+\frac {1}{4} a b c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c^2 x^8 \sqrt {c x^2} \]
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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^6 (a+b x)^2 \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 x^6+2 a b x^7+b^2 x^8\right ) \, dx}{x} \\ & = \frac {1}{7} a^2 c^2 x^6 \sqrt {c x^2}+\frac {1}{4} a b c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c^2 x^8 \sqrt {c x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.56 \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {1}{252} \left (c x^2\right )^{5/2} \left (36 a^2 x^2+63 a b x^3+28 b^2 x^4\right ) \]
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Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.48
method | result | size |
gosper | \(\frac {x^{2} \left (28 b^{2} x^{2}+63 a b x +36 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{252}\) | \(32\) |
default | \(\frac {x^{2} \left (28 b^{2} x^{2}+63 a b x +36 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{252}\) | \(32\) |
risch | \(\frac {a^{2} c^{2} x^{6} \sqrt {c \,x^{2}}}{7}+\frac {a b \,c^{2} x^{7} \sqrt {c \,x^{2}}}{4}+\frac {b^{2} c^{2} x^{8} \sqrt {c \,x^{2}}}{9}\) | \(55\) |
trager | \(\frac {c^{2} \left (28 b^{2} x^{8}+63 a b \,x^{7}+28 b^{2} x^{7}+36 a^{2} x^{6}+63 a b \,x^{6}+28 b^{2} x^{6}+36 a^{2} x^{5}+63 a b \,x^{5}+28 b^{2} x^{5}+36 a^{2} x^{4}+63 a b \,x^{4}+28 b^{2} x^{4}+36 a^{2} x^{3}+63 a b \,x^{3}+28 b^{2} x^{3}+36 a^{2} x^{2}+63 a b \,x^{2}+28 b^{2} x^{2}+36 a^{2} x +63 a b x +28 b^{2} x +36 a^{2}+63 a b +28 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{252 x}\) | \(189\) |
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Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.64 \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {1}{252} \, {\left (28 \, b^{2} c^{2} x^{8} + 63 \, a b c^{2} x^{7} + 36 \, a^{2} c^{2} x^{6}\right )} \sqrt {c x^{2}} \]
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Time = 0.36 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {a^{2} x^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{7} + \frac {a b x^{3} \left (c x^{2}\right )^{\frac {5}{2}}}{4} + \frac {b^{2} x^{4} \left (c x^{2}\right )^{\frac {5}{2}}}{9} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {\left (c x^{2}\right )^{\frac {7}{2}} b^{2} x^{2}}{9 \, c} + \frac {\left (c x^{2}\right )^{\frac {7}{2}} a b x}{4 \, c} + \frac {\left (c x^{2}\right )^{\frac {7}{2}} a^{2}}{7 \, c} \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {1}{252} \, {\left (28 \, b^{2} c^{2} x^{9} \mathrm {sgn}\left (x\right ) + 63 \, a b c^{2} x^{8} \mathrm {sgn}\left (x\right ) + 36 \, a^{2} c^{2} x^{7} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Timed out. \[ \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\int x\,{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2 \,d x \]
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