Integrand size = 16, antiderivative size = 67 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=\frac {2}{3 a \sqrt {x} (a-b x)^{3/2}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a-b x}}-\frac {16 \sqrt {a-b x}}{3 a^3 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=-\frac {16 \sqrt {a-b x}}{3 a^3 \sqrt {x}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a-b x}}+\frac {2}{3 a \sqrt {x} (a-b x)^{3/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3 a \sqrt {x} (a-b x)^{3/2}}+\frac {4 \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx}{3 a} \\ & = \frac {2}{3 a \sqrt {x} (a-b x)^{3/2}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a-b x}}+\frac {8 \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a^2} \\ & = \frac {2}{3 a \sqrt {x} (a-b x)^{3/2}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a-b x}}-\frac {16 \sqrt {a-b x}}{3 a^3 \sqrt {x}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=-\frac {2 \left (3 a^2-12 a b x+8 b^2 x^2\right )}{3 a^3 \sqrt {x} (a-b x)^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {2 \left (8 b^{2} x^{2}-12 a b x +3 a^{2}\right )}{3 \sqrt {x}\, \left (-b x +a \right )^{\frac {3}{2}} a^{3}}\) | \(36\) |
risch | \(-\frac {2 \sqrt {-b x +a}}{a^{3} \sqrt {x}}+\frac {2 b \left (-5 b x +6 a \right ) \sqrt {x}}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{3}}\) | \(43\) |
default | \(-\frac {2}{a \left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}+\frac {4 b \left (\frac {2 \sqrt {x}}{3 a \left (-b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {-b x +a}}\right )}{a}\) | \(57\) |
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none
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=-\frac {2 \, {\left (8 \, b^{2} x^{2} - 12 \, a b x + 3 \, a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{3} b^{2} x^{3} - 2 \, a^{4} b x^{2} + a^{5} x\right )}} \]
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Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 4.69 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=\begin {cases} - \frac {6 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} + \frac {24 a b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {16 b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {6 i a^{2} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} + \frac {24 i a b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {16 i b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=\frac {2 \, {\left (b^{2} - \frac {6 \, {\left (b x - a\right )} b}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3}} - \frac {2 \, \sqrt {-b x + a}}{a^{3} \sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (49) = 98\).
Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.82 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=-\frac {2 \, \sqrt {-b x + a} b^{2}}{\sqrt {{\left (b x - a\right )} b + a b} a^{3} {\left | b \right |}} - \frac {4 \, {\left (3 \, {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt {-b} b^{2} - 12 \, a {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt {-b} b^{3} + 5 \, a^{2} \sqrt {-b} b^{4}\right )}}{3 \, {\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} a^{2} {\left | b \right |}} \]
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Time = 0.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^{3/2} (a-b x)^{5/2}} \, dx=\frac {6\,a^2\,\sqrt {a-b\,x}+16\,b^2\,x^2\,\sqrt {a-b\,x}-24\,a\,b\,x\,\sqrt {a-b\,x}}{\sqrt {x}\,\left (x\,\left (6\,a^4\,b-3\,a^3\,b^2\,x\right )-3\,a^5\right )} \]
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