Integrand size = 15, antiderivative size = 84 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\frac {15}{4 a^3 \sqrt {x}}-\frac {1}{2 a \sqrt {x} (a-b x)^2}-\frac {5}{4 a^2 \sqrt {x} (a-b x)}-\frac {15 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 84, 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} \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=-\frac {15 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {15}{4 a^3 \sqrt {x}}-\frac {5}{4 a^2 \sqrt {x} (a-b x)}-\frac {1}{2 a \sqrt {x} (a-b x)^2} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a \sqrt {x} (a-b x)^2}-\frac {5 \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx}{4 a} \\ & = -\frac {1}{2 a \sqrt {x} (a-b x)^2}-\frac {5}{4 a^2 \sqrt {x} (a-b x)}+\frac {15 \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{8 a^2} \\ & = \frac {15}{4 a^3 \sqrt {x}}-\frac {1}{2 a \sqrt {x} (a-b x)^2}-\frac {5}{4 a^2 \sqrt {x} (a-b x)}+\frac {(15 b) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 a^3} \\ & = \frac {15}{4 a^3 \sqrt {x}}-\frac {1}{2 a \sqrt {x} (a-b x)^2}-\frac {5}{4 a^2 \sqrt {x} (a-b x)}+\frac {(15 b) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^3} \\ & = \frac {15}{4 a^3 \sqrt {x}}-\frac {1}{2 a \sqrt {x} (a-b x)^2}-\frac {5}{4 a^2 \sqrt {x} (a-b x)}-\frac {15 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\frac {8 a^2-25 a b x+15 b^2 x^2}{4 a^3 \sqrt {x} (a-b x)^2}-\frac {15 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(-\frac {2 b \left (\frac {-\frac {7 b \,x^{\frac {3}{2}}}{8}+\frac {9 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {15 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}+\frac {2}{a^{3} \sqrt {x}}\) | \(57\) |
default | \(-\frac {2 b \left (\frac {-\frac {7 b \,x^{\frac {3}{2}}}{8}+\frac {9 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {15 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}+\frac {2}{a^{3} \sqrt {x}}\) | \(57\) |
risch | \(\frac {2}{a^{3} \sqrt {x}}+\frac {b \left (\frac {\frac {7 b \,x^{\frac {3}{2}}}{4}-\frac {9 a \sqrt {x}}{4}}{\left (b x -a \right )^{2}}-\frac {15 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{3}}\) | \(58\) |
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Time = 0.24 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.54 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\left [\frac {15 \, {\left (b^{2} x^{3} - 2 \, a b x^{2} + a^{2} x\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) + 2 \, {\left (15 \, b^{2} x^{2} - 25 \, a b x + 8 \, a^{2}\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{2} x^{3} - 2 \, a^{4} b x^{2} + a^{5} x\right )}}, \frac {15 \, {\left (b^{2} x^{3} - 2 \, a b x^{2} + a^{2} x\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (15 \, b^{2} x^{2} - 25 \, a b x + 8 \, a^{2}\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{2} x^{3} - 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (75) = 150\).
Time = 24.37 (sec) , antiderivative size = 716, normalized size of antiderivative = 8.52 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2}{a^{3} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\\frac {15 a^{2} \sqrt {x} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {15 a^{2} \sqrt {x} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {16 a^{2} \sqrt {\frac {a}{b}}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {30 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {30 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {50 a b x \sqrt {\frac {a}{b}}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} 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 )}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} 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 )}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {30 b^{2} x^{2} \sqrt {\frac {a}{b}}}{8 a^{5} \sqrt {x} \sqrt {\frac {a}{b}} - 16 a^{4} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}} + 8 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\frac {15 \, b^{2} x^{2} - 25 \, a b x + 8 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{\frac {5}{2}} - 2 \, a^{4} b x^{\frac {3}{2}} + a^{5} \sqrt {x}\right )}} + \frac {15 \, b \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\frac {15 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} a^{3}} + \frac {2}{a^{3} \sqrt {x}} + \frac {7 \, b^{2} x^{\frac {3}{2}} - 9 \, a b \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} a^{3}} \]
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Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{3/2} (-a+b x)^3} \, dx=\frac {\frac {2}{a}+\frac {15\,b^2\,x^2}{4\,a^3}-\frac {25\,b\,x}{4\,a^2}}{a^2\,\sqrt {x}+b^2\,x^{5/2}-2\,a\,b\,x^{3/2}}-\frac {15\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{7/2}} \]
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