Integrand size = 20, antiderivative size = 61 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=-\frac {a c \sqrt {c x^2}}{b^2}+\frac {c x \sqrt {c x^2}}{2 b}+\frac {a^2 c \sqrt {c x^2} \log (a+b x)}{b^3 x} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=\frac {a^2 c \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a c \sqrt {c x^2}}{b^2}+\frac {c x \sqrt {c x^2}}{2 b} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {x^2}{a+b x} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{x} \\ & = -\frac {a c \sqrt {c x^2}}{b^2}+\frac {c x \sqrt {c x^2}}{2 b}+\frac {a^2 c \sqrt {c x^2} \log (a+b x)}{b^3 x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=c \sqrt {c x^2} \left (\frac {-2 a+b x}{2 b^2}+\frac {a^2 \log (a+b x)}{b^3 x}\right ) \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 b^{3} x^{3}}\) | \(40\) |
risch | \(\frac {c \sqrt {c \,x^{2}}\, \left (\frac {1}{2} b \,x^{2}-a x \right )}{x \,b^{2}}+\frac {a^{2} c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{3} x}\) | \(52\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=\frac {{\left (b^{2} c x^{2} - 2 \, a b c x + 2 \, a^{2} c \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} x} \]
[In]
[Out]
\[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x \left (a + b x\right )}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.52 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{2} c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{3}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3}} + \frac {\sqrt {c x^{2}} c x}{2 \, b} - \frac {\sqrt {c x^{2}} a c}{b^{2}} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=\frac {1}{2} \, c^{\frac {3}{2}} {\left (\frac {2 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3}} - \frac {2 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3}} + \frac {b x^{2} \mathrm {sgn}\left (x\right ) - 2 \, a x \mathrm {sgn}\left (x\right )}{b^{2}}\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x\,\left (a+b\,x\right )} \,d x \]
[In]
[Out]