Integrand size = 20, antiderivative size = 67 \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a d^4 x (d x)^{-4+m}}{c^2 (4-m) \sqrt {c x^2}}-\frac {b d^3 x (d x)^{-3+m}}{c^2 (3-m) \sqrt {c x^2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 16, 45} \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt {c x^2}}-\frac {b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 16
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(d x)^m (a+b x)}{x^5} \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {\left (d^5 x\right ) \int (d x)^{-5+m} (a+b x) \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {\left (d^5 x\right ) \int \left (a (d x)^{-5+m}+\frac {b (d x)^{-4+m}}{d}\right ) \, dx}{c^2 \sqrt {c x^2}} \\ & = -\frac {a d^4 x (d x)^{-4+m}}{c^2 (4-m) \sqrt {c x^2}}-\frac {b d^3 x (d x)^{-3+m}}{c^2 (3-m) \sqrt {c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.57 \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {x (d x)^m (a (-3+m)+b (-4+m) x)}{(-4+m) (-3+m) \left (c x^2\right )^{5/2}} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {x \left (b m x +a m -4 b x -3 a \right ) \left (d x \right )^{m}}{\left (-3+m \right ) \left (-4+m \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(40\) |
risch | \(\frac {\left (b m x +a m -4 b x -3 a \right ) \left (d x \right )^{m}}{c^{2} x^{3} \sqrt {c \,x^{2}}\, \left (-3+m \right ) \left (-4+m \right )}\) | \(45\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c x^{2}} {\left (a m + {\left (b m - 4 \, b\right )} x - 3 \, a\right )} \left (d x\right )^{m}}{{\left (c^{3} m^{2} - 7 \, c^{3} m + 12 \, c^{3}\right )} x^{5}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (58) = 116\).
Time = 3.68 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.66 \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\begin {cases} d^{3} \left (- \frac {a x^{4}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}}\right ) & \text {for}\: m = 3 \\d^{4} \left (\frac {a x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b x^{6}}{\left (c x^{2}\right )^{\frac {5}{2}}}\right ) & \text {for}\: m = 4 \\\frac {a m x \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {5}{2}} - 7 m \left (c x^{2}\right )^{\frac {5}{2}} + 12 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {3 a x \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {5}{2}} - 7 m \left (c x^{2}\right )^{\frac {5}{2}} + 12 \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b m x^{2} \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {5}{2}} - 7 m \left (c x^{2}\right )^{\frac {5}{2}} + 12 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {4 b x^{2} \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {5}{2}} - 7 m \left (c x^{2}\right )^{\frac {5}{2}} + 12 \left (c x^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58 \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {b d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 3\right )} x^{3}} + \frac {a d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 4\right )} x^{4}} \]
[In]
[Out]
\[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x + a\right )} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=-\frac {{\left (d\,x\right )}^m\,\left (3\,a-a\,m+4\,b\,x-b\,m\,x\right )}{c^2\,x^3\,\sqrt {c\,x^2}\,\left (m^2-7\,m+12\right )} \]
[In]
[Out]