Integrand size = 20, antiderivative size = 25 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-\frac {c \sqrt {c x^2}}{b x (a+b x)} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-\frac {c \sqrt {c x^2}}{b x (a+b x)} \]
[In]
[Out]
Rule 15
Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{(a+b x)^2} \, dx}{x} \\ & = -\frac {c \sqrt {c x^2}}{b x (a+b x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-\frac {\left (c x^2\right )^{3/2}}{b x^3 (a+b x)} \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}}}{\left (b x +a \right ) b \,x^{3}}\) | \(23\) |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}}}{\left (b x +a \right ) b \,x^{3}}\) | \(23\) |
risch | \(-\frac {c \sqrt {c \,x^{2}}}{b x \left (b x +a \right )}\) | \(24\) |
trager | \(\frac {c \left (-1+x \right ) \sqrt {c \,x^{2}}}{\left (b x +a \right ) \left (a +b \right ) x}\) | \(28\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-\frac {\sqrt {c x^{2}} c}{b^{2} x^{2} + a b x} \]
[In]
[Out]
Time = 1.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=\begin {cases} - \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{a b x^{3} + b^{2} x^{4}} & \text {for}\: b \neq 0 \\\frac {\left (c x^{2}\right )^{\frac {3}{2}}}{a^{2} x^{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-\frac {c^{\frac {3}{2}}}{b^{2} x + a b} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-c^{\frac {3}{2}} {\left (\frac {\mathrm {sgn}\left (x\right )}{{\left (b x + a\right )} b} - \frac {\mathrm {sgn}\left (x\right )}{a b}\right )} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx=-\frac {c^{3/2}\,\sqrt {x^2}}{b^2\,x^2+a\,b\,x} \]
[In]
[Out]