Integrand size = 15, antiderivative size = 70 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=-\frac {5}{3 a^2 x^{3/2}}-\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a-b x)}+\frac {5 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 214} \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\frac {5 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {5 b}{a^3 \sqrt {x}}-\frac {5}{3 a^2 x^{3/2}}+\frac {1}{a x^{3/2} (a-b x)} \]
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {1}{a x^{3/2} (a-b x)}-\frac {5 \int \frac {1}{x^{5/2} (-a+b x)} \, dx}{2 a} \\ & = -\frac {5}{3 a^2 x^{3/2}}+\frac {1}{a x^{3/2} (a-b x)}-\frac {(5 b) \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{2 a^2} \\ & = -\frac {5}{3 a^2 x^{3/2}}-\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a-b x)}-\frac {\left (5 b^2\right ) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{2 a^3} \\ & = -\frac {5}{3 a^2 x^{3/2}}-\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a-b x)}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{a^3} \\ & = -\frac {5}{3 a^2 x^{3/2}}-\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a-b x)}+\frac {5 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\frac {-2 a^2-10 a b x+15 b^2 x^2}{3 a^3 x^{3/2} (a-b x)}+\frac {5 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {2 \left (6 b x +a \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {b^{2} \left (\frac {\sqrt {x}}{b x -a}-\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{3}}\) | \(56\) |
derivativedivides | \(\frac {2 b^{2} \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}-\frac {4 b}{a^{3} \sqrt {x}}\) | \(59\) |
default | \(\frac {2 b^{2} \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}-\frac {4 b}{a^{3} \sqrt {x}}\) | \(59\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.67 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\left [\frac {15 \, {\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) - 2 \, {\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt {x}}{6 \, {\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}, -\frac {15 \, {\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (65) = 130\).
Time = 22.12 (sec) , antiderivative size = 416, normalized size of antiderivative = 5.94 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{2} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{2} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {4 a^{2} \sqrt {\frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {20 a b x \sqrt {\frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {30 b^{2} x^{2} \sqrt {\frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=-\frac {15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}}{3 \, {\left (a^{3} b x^{\frac {5}{2}} - a^{4} x^{\frac {3}{2}}\right )}} - \frac {5 \, b^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=-\frac {5 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{3}} - \frac {b^{2} \sqrt {x}}{{\left (b x - a\right )} a^{3}} - \frac {2 \, {\left (6 \, b x + a\right )}}{3 \, a^{3} x^{\frac {3}{2}}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\frac {5\,b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {\frac {2}{3\,a}-\frac {5\,b^2\,x^2}{a^3}+\frac {10\,b\,x}{3\,a^2}}{a\,x^{3/2}-b\,x^{5/2}} \]
[In]
[Out]