Integrand size = 18, antiderivative size = 73 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 53, 65, 214} \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}} \]
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Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{a x \sqrt {a+b x}}+\frac {\left (-\frac {3 A b}{2}+a B\right ) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{a} \\ & = -\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}-\frac {(3 A b-2 a B) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a^2} \\ & = -\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}-\frac {(3 A b-2 a B) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^2 b} \\ & = -\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {-a A-3 A b x+2 a B x}{a^2 x \sqrt {a+b x}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.54 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 \left (A b -B a \right )}{\sqrt {b x +a}}-\frac {A \sqrt {b x +a}}{x}+\frac {\left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{2}}\) | \(61\) |
derivativedivides | \(-\frac {2 \left (A b -B a \right )}{a^{2} \sqrt {b x +a}}+\frac {-\frac {A \sqrt {b x +a}}{x}+\frac {\left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{2}}\) | \(67\) |
default | \(-\frac {2 \left (A b -B a \right )}{a^{2} \sqrt {b x +a}}+\frac {-\frac {A \sqrt {b x +a}}{x}+\frac {\left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{2}}\) | \(67\) |
risch | \(-\frac {A \sqrt {b x +a}}{a^{2} x}-\frac {-\frac {2 \left (-2 A b +2 B a \right )}{\sqrt {b x +a}}-\frac {2 \left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a^{2}}\) | \(68\) |
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Time = 0.24 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.89 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b x + a}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, \frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b x + a}}{a^{3} b x^{2} + a^{4} x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (65) = 130\).
Time = 20.01 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.07 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=A \left (- \frac {1}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 \sqrt {b}}{a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x}{a}}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{2} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.47 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=-\frac {1}{2} \, b {\left (\frac {2 \, {\left (2 \, B a^{2} - 2 \, A a b - {\left (2 \, B a - 3 \, A b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} a^{2} b - \sqrt {b x + a} a^{3} b} - \frac {{\left (2 \, B a - 3 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}} b}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, {\left (b x + a\right )} B a - 2 \, B a^{2} - 3 \, {\left (b x + a\right )} A b + 2 \, A a b}{{\left ({\left (b x + a\right )}^{\frac {3}{2}} - \sqrt {b x + a} a\right )} a^{2}} \]
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Time = 0.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (3\,A\,b-2\,B\,a\right )}{a^{5/2}}-\frac {\frac {2\,\left (A\,b-B\,a\right )}{a}-\frac {\left (3\,A\,b-2\,B\,a\right )\,\left (a+b\,x\right )}{a^2}}{a\,\sqrt {a+b\,x}-{\left (a+b\,x\right )}^{3/2}} \]
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