Integrand size = 15, antiderivative size = 97 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}-\frac {35 a^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]
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Time = 0.03 (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 = {43, 52, 65, 214} \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=-\frac {35 a^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}+\frac {35 a \sqrt {x}}{4 b^4}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {35 x^{3/2}}{12 b^3} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 \int \frac {x^{5/2}}{(-a+b x)^2} \, dx}{4 b} \\ & = -\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {35 \int \frac {x^{3/2}}{-a+b x} \, dx}{8 b^2} \\ & = \frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {(35 a) \int \frac {\sqrt {x}}{-a+b x} \, dx}{8 b^3} \\ & = \frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {\left (35 a^2\right ) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 b^4} \\ & = \frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^4} \\ & = \frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}-\frac {35 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=\frac {\sqrt {x} \left (105 a^3-175 a^2 b x+56 a b^2 x^2+8 b^3 x^3\right )}{12 b^4 (a-b x)^2}-\frac {35 a^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]
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Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {2 \left (b x +9 a \right ) \sqrt {x}}{3 b^{4}}+\frac {a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{4}+\frac {11 a \sqrt {x}}{4}}{\left (b x -a \right )^{2}}-\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{4}}\) | \(67\) |
derivativedivides | \(\frac {\frac {2 b \,x^{\frac {3}{2}}}{3}+6 a \sqrt {x}}{b^{4}}-\frac {2 a^{2} \left (\frac {\frac {13 b \,x^{\frac {3}{2}}}{8}-\frac {11 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(69\) |
default | \(\frac {\frac {2 b \,x^{\frac {3}{2}}}{3}+6 a \sqrt {x}}{b^{4}}-\frac {2 a^{2} \left (\frac {\frac {13 b \,x^{\frac {3}{2}}}{8}-\frac {11 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(69\) |
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Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.34 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=\left [\frac {105 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, {\left (8 \, b^{3} x^{3} + 56 \, a b^{2} x^{2} - 175 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {105 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{3} x^{3} + 56 \, a b^{2} x^{2} - 175 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (87) = 174\).
Time = 64.02 (sec) , antiderivative size = 695, normalized size of antiderivative = 7.16 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {9}{2}}}{9 a^{3}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b^{3}} & \text {for}\: a = 0 \\\frac {105 a^{4} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} - \frac {105 a^{4} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} + \frac {210 a^{3} b \sqrt {x} \sqrt {\frac {a}{b}}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} - \frac {210 a^{3} b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} + \frac {210 a^{3} b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} - \frac {350 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} + \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} - \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} + \frac {112 a b^{3} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} + \frac {16 b^{4} x^{\frac {7}{2}} \sqrt {\frac {a}{b}}}{24 a^{2} b^{5} \sqrt {\frac {a}{b}} - 48 a b^{6} x \sqrt {\frac {a}{b}} + 24 b^{7} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=-\frac {13 \, a^{2} b x^{\frac {3}{2}} - 11 \, a^{3} \sqrt {x}}{4 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} + \frac {2 \, {\left (b x^{\frac {3}{2}} + 9 \, a \sqrt {x}\right )}}{3 \, b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=\frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} b^{4}} - \frac {13 \, a^{2} b x^{\frac {3}{2}} - 11 \, a^{3} \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} b^{4}} + \frac {2 \, {\left (b^{6} x^{\frac {3}{2}} + 9 \, a b^{5} \sqrt {x}\right )}}{3 \, b^{9}} \]
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Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {x^{7/2}}{(-a+b x)^3} \, dx=\frac {\frac {11\,a^3\,\sqrt {x}}{4}-\frac {13\,a^2\,b\,x^{3/2}}{4}}{a^2\,b^4-2\,a\,b^5\,x+b^6\,x^2}+\frac {2\,x^{3/2}}{3\,b^3}+\frac {6\,a\,\sqrt {x}}{b^4}+\frac {a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,35{}\mathrm {i}}{4\,b^{9/2}} \]
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