Integrand size = 20, antiderivative size = 56 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 56, 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} \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{c^2 \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(a+b x)^2}{x^2} \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {x \int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx}{c^2 \sqrt {c x^2}} \\ & = -\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.62 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {-a^2 x^4+b^2 x^6+2 a b x^5 \log (x)}{\left (c x^2\right )^{5/2}} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.57
method | result | size |
default | \(\frac {x^{4} \left (2 a b \ln \left (x \right ) x +b^{2} x^{2}-a^{2}\right )}{\left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(32\) |
risch | \(-\frac {a^{2}}{c^{2} \sqrt {c \,x^{2}}}+\frac {b^{2} x^{2}}{c^{2} \sqrt {c \,x^{2}}}+\frac {2 a b x \ln \left (x \right )}{c^{2} \sqrt {c \,x^{2}}}\) | \(51\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt {c x^{2}}}{c^{3} x^{2}} \]
[In]
[Out]
Time = 1.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=- \frac {a^{2} x^{4}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {2 a b x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b^{2} x^{6}}{\left (c x^{2}\right )^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {b^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a^{2} x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} + \frac {2 \, a b \log \left (x\right )}{c^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {b^{2} x}{c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, a b \log \left ({\left | x \right |}\right )}{c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} - \frac {a^{2}}{c^{\frac {5}{2}} x \mathrm {sgn}\left (x\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{5/2}} \,d x \]
[In]
[Out]