Integrand size = 18, antiderivative size = 147 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=-\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {5 (7 A b-3 a B)}{4 a^4 \sqrt {x}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 147, 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, 211} \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {5 (7 A b-3 a B)}{4 a^4 \sqrt {x}}-\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}-\frac {\left (-\frac {7 A b}{2}+\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} (a+b x)^2} \, dx}{2 a b} \\ & = \frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {(5 (7 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{8 a^2 b} \\ & = -\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}-\frac {(5 (7 A b-3 a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{8 a^3} \\ & = -\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {5 (7 A b-3 a B)}{4 a^4 \sqrt {x}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {(5 b (7 A b-3 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^4} \\ & = -\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {5 (7 A b-3 a B)}{4 a^4 \sqrt {x}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {(5 b (7 A b-3 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^4} \\ & = -\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {5 (7 A b-3 a B)}{4 a^4 \sqrt {x}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {5 \sqrt {b} (7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=\frac {105 A b^3 x^3+a^2 b x (56 A-75 B x)+5 a b^2 x^2 (35 A-9 B x)-8 a^3 (A+3 B x)}{12 a^4 x^{3/2} (a+b x)^2}+\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 1.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.67
method | result | size |
risch | \(-\frac {2 \left (-9 A b x +3 B a x +A a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {11}{8} b^{2} A -\frac {7}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (13 A b -9 B a \right ) \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {5 \left (7 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{4}}\) | \(98\) |
derivativedivides | \(\frac {2 b \left (\frac {\left (\frac {11}{8} b^{2} A -\frac {7}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (13 A b -9 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {5 \left (7 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 \left (-3 A b +B a \right )}{a^{4} \sqrt {x}}\) | \(101\) |
default | \(\frac {2 b \left (\frac {\left (\frac {11}{8} b^{2} A -\frac {7}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (13 A b -9 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {5 \left (7 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 \left (-3 A b +B a \right )}{a^{4} \sqrt {x}}\) | \(101\) |
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Time = 0.24 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.59 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=\left [-\frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 1703 vs. \(2 (139) = 278\).
Time = 38.49 (sec) , antiderivative size = 1703, normalized size of antiderivative = 11.59 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=-\frac {8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} + 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} - \frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=-\frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} - \frac {2 \, {\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} - \frac {7 \, B a b^{2} x^{\frac {3}{2}} - 11 \, A b^{3} x^{\frac {3}{2}} + 9 \, B a^{2} b \sqrt {x} - 13 \, A a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4}} \]
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Time = 0.55 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx=\frac {\frac {2\,x\,\left (7\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {5\,b^2\,x^3\,\left (7\,A\,b-3\,B\,a\right )}{4\,a^4}+\frac {25\,b\,x^2\,\left (7\,A\,b-3\,B\,a\right )}{12\,a^3}}{a^2\,x^{3/2}+b^2\,x^{7/2}+2\,a\,b\,x^{5/2}}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b-3\,B\,a\right )}{4\,a^{9/2}} \]
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