Integrand size = 18, antiderivative size = 60 \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {1}{5} a^2 c x^4 \sqrt {c x^2}+\frac {1}{3} a b c x^5 \sqrt {c x^2}+\frac {1}{7} b^2 c x^6 \sqrt {c x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 60, 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} \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {1}{5} a^2 c x^4 \sqrt {c x^2}+\frac {1}{3} a b c x^5 \sqrt {c x^2}+\frac {1}{7} b^2 c x^6 \sqrt {c x^2} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int x^4 (a+b x)^2 \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (a^2 x^4+2 a b x^5+b^2 x^6\right ) \, dx}{x} \\ & = \frac {1}{5} a^2 c x^4 \sqrt {c x^2}+\frac {1}{3} a b c x^5 \sqrt {c x^2}+\frac {1}{7} b^2 c x^6 \sqrt {c x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.62 \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {1}{105} \left (c x^2\right )^{3/2} \left (21 a^2 x^2+35 a b x^3+15 b^2 x^4\right ) \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(\frac {x^{2} \left (15 b^{2} x^{2}+35 a b x +21 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{105}\) | \(32\) |
default | \(\frac {x^{2} \left (15 b^{2} x^{2}+35 a b x +21 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{105}\) | \(32\) |
risch | \(\frac {a^{2} c \,x^{4} \sqrt {c \,x^{2}}}{5}+\frac {a b c \,x^{5} \sqrt {c \,x^{2}}}{3}+\frac {b^{2} c \,x^{6} \sqrt {c \,x^{2}}}{7}\) | \(49\) |
trager | \(\frac {c \left (15 b^{2} x^{6}+35 a b \,x^{5}+15 b^{2} x^{5}+21 a^{2} x^{4}+35 a b \,x^{4}+15 b^{2} x^{4}+21 a^{2} x^{3}+35 a b \,x^{3}+15 b^{2} x^{3}+21 a^{2} x^{2}+35 a b \,x^{2}+15 b^{2} x^{2}+21 a^{2} x +35 a b x +15 b^{2} x +21 a^{2}+35 a b +15 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{105 x}\) | \(141\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.60 \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {1}{105} \, {\left (15 \, b^{2} c x^{6} + 35 \, a b c x^{5} + 21 \, a^{2} c x^{4}\right )} \sqrt {c x^{2}} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {a^{2} x^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{5} + \frac {a b x^{3} \left (c x^{2}\right )^{\frac {3}{2}}}{3} + \frac {b^{2} x^{4} \left (c x^{2}\right )^{\frac {3}{2}}}{7} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {\left (c x^{2}\right )^{\frac {5}{2}} b^{2} x^{2}}{7 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a b x}{3 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a^{2}}{5 \, c} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.58 \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {1}{105} \, {\left (15 \, b^{2} x^{7} \mathrm {sgn}\left (x\right ) + 35 \, a b x^{6} \mathrm {sgn}\left (x\right ) + 21 \, a^{2} x^{5} \mathrm {sgn}\left (x\right )\right )} c^{\frac {3}{2}} \]
[In]
[Out]
Timed out. \[ \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\int x\,{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2 \,d x \]
[In]
[Out]