Integrand size = 18, antiderivative size = 41 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {1}{4} a c^2 x^3 \sqrt {c x^2}+\frac {1}{5} b c^2 x^4 \sqrt {c x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {1}{4} a c^2 x^3 \sqrt {c x^2}+\frac {1}{5} b c^2 x^4 \sqrt {c x^2} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int x^3 (a+b x) \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x} \\ & = \frac {1}{4} a c^2 x^3 \sqrt {c x^2}+\frac {1}{5} b c^2 x^4 \sqrt {c x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {1}{20} c x \left (c x^2\right )^{3/2} (5 a+4 b x) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {\left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{20 x}\) | \(21\) |
default | \(\frac {\left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{20 x}\) | \(21\) |
risch | \(\frac {a \,c^{2} x^{3} \sqrt {c \,x^{2}}}{4}+\frac {b \,c^{2} x^{4} \sqrt {c \,x^{2}}}{5}\) | \(34\) |
trager | \(\frac {c^{2} \left (4 b \,x^{4}+5 a \,x^{3}+4 b \,x^{3}+5 a \,x^{2}+4 b \,x^{2}+5 a x +4 b x +5 a +4 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{20 x}\) | \(64\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {1}{20} \, {\left (4 \, b c^{2} x^{4} + 5 \, a c^{2} x^{3}\right )} \sqrt {c x^{2}} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.59 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{4 x} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}}}{5} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.59 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {1}{5} \, \left (c x^{2}\right )^{\frac {5}{2}} b + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a}{4 \, x} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {1}{20} \, {\left (4 \, b c^{2} x^{5} \mathrm {sgn}\left (x\right ) + 5 \, a c^{2} x^{4} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx=\frac {c^{5/2}\,\left (4\,b\,\sqrt {x^{10}}+5\,a\,x^3\,\sqrt {x^2}\right )}{20} \]
[In]
[Out]