Integrand size = 29, antiderivative size = 107 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 79, 53, 65, 211} \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}-\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{a b x^{3/2} (a+b x)} \]
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Rule 27
Rule 53
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx \\ & = \frac {A b-a B}{a b x^{3/2} (a+b x)}-\frac {\left (-\frac {5 A b}{2}+\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{a b} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}-\frac {(5 A b-3 a B) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a^2} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {(b (5 A b-3 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^3} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {(b (5 A b-3 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^3} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {15 A b^2 x^2+a b x (10 A-9 B x)-2 a^2 (A+3 B x)}{3 a^3 x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {2 \left (-6 A b x +3 a B x +a A \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{\sqrt {b a}}\right )}{a^{3}}\) | \(77\) |
derivativedivides | \(\frac {2 b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{2 \sqrt {b a}}\right )}{a^{3}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}\) | \(81\) |
default | \(\frac {2 b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{2 \sqrt {b a}}\right )}{a^{3}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}\) | \(81\) |
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Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.45 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\left [-\frac {3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt {x}}{6 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (95) = 190\).
Time = 30.24 (sec) , antiderivative size = 882, normalized size of antiderivative = 8.24 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {4 A a^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {20 A a b x \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {30 A b^{2} x^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {9 B a^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {9 B a^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {12 B a^{2} x \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {9 B a b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {9 B a b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {18 B a b x^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x}{3 \, {\left (a^{3} b x^{\frac {5}{2}} + a^{4} x^{\frac {3}{2}}\right )}} - \frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {B a b \sqrt {x} - A b^{2} \sqrt {x}}{{\left (b x + a\right )} a^{3}} - \frac {2 \, {\left (3 \, B a x - 6 \, A b x + A a\right )}}{3 \, a^{3} x^{\frac {3}{2}}} \]
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Time = 9.89 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {\frac {2\,x\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {b\,x^2\,\left (5\,A\,b-3\,B\,a\right )}{a^3}}{a\,x^{3/2}+b\,x^{5/2}}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (5\,A\,b-3\,B\,a\right )}{a^{7/2}} \]
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