Integrand size = 31, antiderivative size = 238 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (A b-a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {784, 79, 53, 65, 211} \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (a+b x) (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 53
Rule 65
Rule 79
Rule 211
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {A+B x}{x^{9/2} \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )} \, dx}{7 a b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )} \, dx}{7 a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{7 a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 b^2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{7 a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (4 b^2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{7 a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.53 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) \left (\sqrt {a} \left (-105 A b^3 x^3+35 a b^2 x^2 (A+3 B x)-7 a^2 b x (3 A+5 B x)+3 a^3 (5 A+7 B x)\right )-105 b^{5/2} (A b-a B) x^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{105 a^{9/2} x^{7/2} \sqrt {(a+b x)^2}} \]
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Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {2 \left (-105 A \,b^{3} x^{3}+105 B a \,b^{2} x^{3}+35 A a \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-21 A \,a^{2} b x +21 a^{3} B x +15 A \,a^{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{105 a^{4} x^{\frac {7}{2}} \left (b x +a \right )}+\frac {2 \left (A b -B a \right ) b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{4} \sqrt {b a}\, \left (b x +a \right )}\) | \(135\) |
default | \(\frac {2 \left (b x +a \right ) \left (105 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{4}-105 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{3}+105 A \,x^{3} \sqrt {b a}\, b^{3}-105 B \,x^{3} \sqrt {b a}\, a \,b^{2}-35 A \,x^{2} \sqrt {b a}\, a \,b^{2}+35 B \,x^{2} \sqrt {b a}\, a^{2} b +21 A x \sqrt {b a}\, a^{2} b -21 B x \sqrt {b a}\, a^{3}-15 A \,a^{3} \sqrt {b a}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, a^{4} \sqrt {b a}\, x^{\frac {7}{2}}}\) | \(165\) |
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Time = 0.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, a^{4} x^{4}}, \frac {2 \, {\left (105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}\right )}}{105 \, a^{4} x^{4}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (159) = 318\).
Time = 0.34 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {35 \, {\left ({\left (5 \, B a b^{5} - 7 \, A b^{6}\right )} x^{2} + 3 \, {\left (7 \, B a^{2} b^{4} - 9 \, A a b^{5}\right )} x\right )} \sqrt {x} - \frac {70 \, {\left ({\left (5 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{2} - 3 \, {\left (7 \, B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x\right )}}{\sqrt {x}} - \frac {70 \, {\left (3 \, {\left (5 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{2} - {\left (7 \, B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {14 \, {\left (5 \, {\left (5 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} + {\left (7 \, B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )}}{x^{\frac {5}{2}}} + \frac {2 \, {\left (7 \, {\left (5 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{6} - 9 \, A a^{5} b\right )} x\right )}}{x^{\frac {7}{2}}} + \frac {6 \, {\left (7 \, A a^{5} b x^{2} + 5 \, A a^{6} x\right )}}{x^{\frac {9}{2}}}}{105 \, {\left (a^{6} b x + a^{7}\right )}} - \frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {{\left (5 \, B a b^{4} - 7 \, A b^{5}\right )} x^{\frac {3}{2}} + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} \sqrt {x}}{3 \, a^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 \, {\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (105 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 105 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) - 21 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A+B\,x}{x^{9/2}\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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