Integrand size = 18, antiderivative size = 136 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=\frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}+\frac {5 b (7 A b-4 a B)}{4 a^4 \sqrt {a+b x}}-\frac {5 b (7 A b-4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 44, 53, 65, 214} \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=-\frac {5 b (7 A b-4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {5 b (7 A b-4 a B)}{4 a^4 \sqrt {a+b x}}+\frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {\left (-\frac {7 A b}{2}+2 a B\right ) \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx}{2 a} \\ & = -\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}+\frac {(5 b (7 A b-4 a B)) \int \frac {1}{x (a+b x)^{5/2}} \, dx}{8 a^2} \\ & = \frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}+\frac {(5 b (7 A b-4 a B)) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{8 a^3} \\ & = \frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}+\frac {5 b (7 A b-4 a B)}{4 a^4 \sqrt {a+b x}}+\frac {(5 b (7 A b-4 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx}{8 a^4} \\ & = \frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}+\frac {5 b (7 A b-4 a B)}{4 a^4 \sqrt {a+b x}}+\frac {(5 (7 A b-4 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{4 a^4} \\ & = \frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}+\frac {5 b (7 A b-4 a B)}{4 a^4 \sqrt {a+b x}}-\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=\frac {105 A b^3 x^3+a^2 b x (21 A-80 B x)+20 a b^2 x^2 (7 A-3 B x)-6 a^3 (A+2 B x)}{12 a^4 x^2 (a+b x)^{3/2}}+\frac {5 b (-7 A b+4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.57 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-11 A b x +4 B a x +2 A a \right )}{4 a^{4} x^{2}}+\frac {b \left (-\frac {2 \left (-24 A b +16 B a \right )}{\sqrt {b x +a}}+\frac {16 a \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (35 A b -20 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{8 a^{4}}\) | \(101\) |
pseudoelliptic | \(\frac {-\frac {35 x^{2} \left (b x +a \right )^{\frac {3}{2}} b \left (A b -\frac {4 B a}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{4}+\frac {35 x^{2} \left (-\frac {3 B x}{7}+A \right ) b^{2} a^{\frac {3}{2}}}{3}+\frac {7 b x \left (-\frac {80 B x}{21}+A \right ) a^{\frac {5}{2}}}{4}+\frac {\left (-2 B x -A \right ) a^{\frac {7}{2}}}{2}+\frac {35 A \sqrt {a}\, b^{3} x^{3}}{4}}{a^{\frac {9}{2}} \left (b x +a \right )^{\frac {3}{2}} x^{2}}\) | \(104\) |
derivativedivides | \(2 b \left (-\frac {\frac {\left (-\frac {11 A b}{8}+\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {13}{8} a b A -\frac {1}{2} a^{2} B \right ) \sqrt {b x +a}}{b^{2} x^{2}}+\frac {5 \left (7 A b -4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}-\frac {-3 A b +2 B a}{a^{4} \sqrt {b x +a}}-\frac {-A b +B a}{3 a^{3} \left (b x +a \right )^{\frac {3}{2}}}\right )\) | \(123\) |
default | \(2 b \left (-\frac {\frac {\left (-\frac {11 A b}{8}+\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {13}{8} a b A -\frac {1}{2} a^{2} B \right ) \sqrt {b x +a}}{b^{2} x^{2}}+\frac {5 \left (7 A b -4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}-\frac {-3 A b +2 B a}{a^{4} \sqrt {b x +a}}-\frac {-A b +B a}{3 a^{3} \left (b x +a \right )^{\frac {3}{2}}}\right )\) | \(123\) |
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Time = 0.25 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.90 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (6 \, A a^{4} + 15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{24 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, -\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (6 \, A a^{4} + 15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{12 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 1287 vs. \(2 (129) = 258\).
Time = 59.68 (sec) , antiderivative size = 1287, normalized size of antiderivative = 9.46 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=-\frac {1}{24} \, b^{2} {\left (\frac {2 \, {\left (8 \, B a^{4} - 8 \, A a^{3} b + 15 \, {\left (4 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{3} - 25 \, {\left (4 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{2} + 8 \, {\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {7}{2}} a^{4} b - 2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} b + {\left (b x + a\right )}^{\frac {3}{2}} a^{6} b} + \frac {15 \, {\left (4 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=-\frac {5 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{4}} - \frac {2 \, {\left (6 \, {\left (b x + a\right )} B a b + B a^{2} b - 9 \, {\left (b x + a\right )} A b^{2} - A a b^{2}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x + a} B a^{2} b - 11 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{2} + 13 \, \sqrt {b x + a} A a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
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Time = 0.50 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx=\frac {\frac {2\,\left (A\,b^2-B\,a\,b\right )}{3\,a}+\frac {2\,\left (7\,A\,b^2-4\,B\,a\,b\right )\,\left (a+b\,x\right )}{3\,a^2}-\frac {25\,\left (7\,A\,b^2-4\,B\,a\,b\right )\,{\left (a+b\,x\right )}^2}{12\,a^3}+\frac {5\,\left (7\,A\,b^2-4\,B\,a\,b\right )\,{\left (a+b\,x\right )}^3}{4\,a^4}}{{\left (a+b\,x\right )}^{7/2}-2\,a\,{\left (a+b\,x\right )}^{5/2}+a^2\,{\left (a+b\,x\right )}^{3/2}}-\frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-4\,B\,a\right )}{4\,a^{9/2}} \]
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