Integrand size = 18, antiderivative size = 131 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {b^{3/2} (7 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 53, 65, 211} \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {b^{3/2} (7 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {A b-a B}{a b x^{5/2} (a+b x)} \]
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Rule 53
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {\left (-\frac {7 A b}{2}+\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} (a+b x)} \, dx}{a b} \\ & = -\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {(7 A b-5 a B) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{2 a^2} \\ & = -\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}+\frac {(b (7 A b-5 a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a^3} \\ & = -\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {\left (b^2 (7 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^4} \\ & = -\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {\left (b^2 (7 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^4} \\ & = -\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\frac {-105 A b^3 x^3-2 a^3 (3 A+5 B x)+5 a b^2 x^2 (-14 A+15 B x)+2 a^2 b x (7 A+25 B x)}{15 a^4 x^{5/2} (a+b x)}+\frac {b^{3/2} (-7 A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \]
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Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {2 b^{2} \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 b \left (3 A b -2 B a \right )}{a^{4} \sqrt {x}}\) | \(101\) |
default | \(-\frac {2 b^{2} \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 b \left (3 A b -2 B a \right )}{a^{4} \sqrt {x}}\) | \(101\) |
risch | \(-\frac {2 \left (45 A \,b^{2} x^{2}-30 B a b \,x^{2}-10 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 a^{4} x^{\frac {5}{2}}}-\frac {b^{2} \left (\frac {2 \left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{4}}\) | \(103\) |
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Time = 0.23 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\left [-\frac {15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{30 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}, -\frac {15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{15 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (119) = 238\).
Time = 42.15 (sec) , antiderivative size = 1017, normalized size of antiderivative = 7.76 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {12 A a^{3} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {28 A a^{2} b x \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 A a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 A a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {140 A a b^{2} x^{2} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 A b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 A b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {210 A b^{3} x^{3} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {20 B a^{3} x \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {75 B a^{2} b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {75 B a^{2} b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {100 B a^{2} b x^{2} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {75 B a b^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {75 B a b^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {150 B a b^{2} x^{3} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x}{15 \, {\left (a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {B a b^{2} \sqrt {x} - A b^{3} \sqrt {x}}{{\left (b x + a\right )} a^{4}} + \frac {2 \, {\left (30 \, B a b x^{2} - 45 \, A b^{2} x^{2} - 5 \, B a^{2} x + 10 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{4} x^{\frac {5}{2}}} \]
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Time = 0.44 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {\frac {2\,A}{5\,a}-\frac {2\,x\,\left (7\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {b^2\,x^3\,\left (7\,A\,b-5\,B\,a\right )}{a^4}+\frac {2\,b\,x^2\,\left (7\,A\,b-5\,B\,a\right )}{3\,a^3}}{a\,x^{5/2}+b\,x^{7/2}}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b-5\,B\,a\right )}{a^{9/2}} \]
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