Integrand size = 15, antiderivative size = 97 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}-\frac {35 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 214} \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=-\frac {35 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {35}{12 a^3 x^{3/2}}-\frac {7}{4 a^2 x^{3/2} (a-b x)}-\frac {1}{2 a x^{3/2} (a-b x)^2} \]
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7 \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx}{4 a} \\ & = -\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}+\frac {35 \int \frac {1}{x^{5/2} (-a+b x)} \, dx}{8 a^2} \\ & = \frac {35}{12 a^3 x^{3/2}}-\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}+\frac {(35 b) \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{8 a^3} \\ & = \frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}+\frac {\left (35 b^2\right ) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 a^4} \\ & = \frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}+\frac {\left (35 b^2\right ) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^4} \\ & = \frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}-\frac {35 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {8 a^3+56 a^2 b x-175 a b^2 x^2+105 b^3 x^3}{12 a^4 x^{3/2} (a-b x)^2}-\frac {35 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {6 b x +\frac {2 a}{3}}{a^{4} x^{\frac {3}{2}}}+\frac {b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{4}-\frac {13 a \sqrt {x}}{4}}{\left (b x -a \right )^{2}}-\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{4}}\) | \(66\) |
derivativedivides | \(-\frac {2 b^{2} \left (\frac {-\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}+\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}\) | \(68\) |
default | \(-\frac {2 b^{2} \left (\frac {-\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}+\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}\) | \(68\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\left [\frac {105 \, {\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) + 2 \, {\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {105 \, {\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (88) = 176\).
Time = 59.60 (sec) , antiderivative size = 799, normalized size of antiderivative = 8.24 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2}{3 a^{3} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{9 b^{3} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\\frac {16 a^{3} \sqrt {\frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} + \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} - \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} + \frac {112 a^{2} b x \sqrt {\frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} - \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} + \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} - \frac {350 a b^{2} x^{2} \sqrt {\frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} + \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} - \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} + \frac {210 b^{3} x^{3} \sqrt {\frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 48 a^{5} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} - 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, b^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} a^{4}} + \frac {2 \, {\left (9 \, b x + a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} + \frac {11 \, b^{3} x^{\frac {3}{2}} - 13 \, a b^{2} \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} a^{4}} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {\frac {2}{3\,a}-\frac {175\,b^2\,x^2}{12\,a^3}+\frac {35\,b^3\,x^3}{4\,a^4}+\frac {14\,b\,x}{3\,a^2}}{a^2\,x^{3/2}+b^2\,x^{7/2}-2\,a\,b\,x^{5/2}}-\frac {35\,b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \]
[In]
[Out]