Integrand size = 20, antiderivative size = 64 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 64, 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} \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int x (a+b x)^2 \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x} \\ & = \frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.56 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\frac {1}{12} c^2 x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(\frac {\left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{3}}\) | \(32\) |
default | \(\frac {\left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{3}}\) | \(32\) |
risch | \(\frac {a^{2} c^{2} x \sqrt {c \,x^{2}}}{2}+\frac {2 a b \,c^{2} x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b^{2} c^{2} x^{3} \sqrt {c \,x^{2}}}{4}\) | \(53\) |
trager | \(\frac {c^{2} \left (3 b^{2} x^{3}+8 a b \,x^{2}+3 b^{2} x^{2}+6 a^{2} x +8 a b x +3 b^{2} x +6 a^{2}+8 a b +3 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\) | \(74\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\frac {1}{12} \, {\left (3 \, b^{2} c^{2} x^{3} + 8 \, a b c^{2} x^{2} + 6 \, a^{2} c^{2} x\right )} \sqrt {c x^{2}} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.77 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{2 x^{3}} + \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{4 x} \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\frac {1}{12} \, {\left (3 \, b^{2} c^{2} x^{4} \mathrm {sgn}\left (x\right ) + 8 \, a b c^{2} x^{3} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} c^{2} x^{2} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx=\int \frac {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \]
[In]
[Out]