Integrand size = 18, antiderivative size = 113 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=\frac {2 a^2 (A b-a B) \sqrt {x}}{b^4}-\frac {2 a (A b-a B) x^{3/2}}{3 b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {2 a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 113, 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 = {81, 52, 65, 211} \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=-\frac {2 a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}}+\frac {2 a^2 \sqrt {x} (A b-a B)}{b^4}-\frac {2 a x^{3/2} (A b-a B)}{3 b^3}+\frac {2 x^{5/2} (A b-a B)}{5 b^2}+\frac {2 B x^{7/2}}{7 b} \]
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Rule 52
Rule 65
Rule 81
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {2 B x^{7/2}}{7 b}+\frac {\left (2 \left (\frac {7 A b}{2}-\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{a+b x} \, dx}{7 b} \\ & = \frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(a (A b-a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{b^2} \\ & = -\frac {2 a (A b-a B) x^{3/2}}{3 b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{b^3} \\ & = \frac {2 a^2 (A b-a B) \sqrt {x}}{b^4}-\frac {2 a (A b-a B) x^{3/2}}{3 b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {\left (a^3 (A b-a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^4} \\ & = \frac {2 a^2 (A b-a B) \sqrt {x}}{b^4}-\frac {2 a (A b-a B) x^{3/2}}{3 b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {\left (2 a^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^4} \\ & = \frac {2 a^2 (A b-a B) \sqrt {x}}{b^4}-\frac {2 a (A b-a B) x^{3/2}}{3 b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=\frac {2 \sqrt {x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )}{105 b^4}+\frac {2 a^{5/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \]
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Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {2 \left (15 b^{3} B \,x^{3}+21 A \,b^{3} x^{2}-21 B a \,b^{2} x^{2}-35 a \,b^{2} A x +35 a^{2} b B x +105 a^{2} b A -105 a^{3} B \right ) \sqrt {x}}{105 b^{4}}-\frac {2 a^{3} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(100\) |
derivativedivides | \(\frac {\frac {2 b^{3} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {2 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {2 A a \,b^{2} x^{\frac {3}{2}}}{3}+\frac {2 B \,a^{2} b \,x^{\frac {3}{2}}}{3}+2 a^{2} b A \sqrt {x}-2 a^{3} B \sqrt {x}}{b^{4}}-\frac {2 a^{3} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(106\) |
default | \(\frac {\frac {2 b^{3} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {2 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {2 A a \,b^{2} x^{\frac {3}{2}}}{3}+\frac {2 B \,a^{2} b \,x^{\frac {3}{2}}}{3}+2 a^{2} b A \sqrt {x}-2 a^{3} B \sqrt {x}}{b^{4}}-\frac {2 a^{3} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(106\) |
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Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.03 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=\left [-\frac {105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}}{105 \, b^{4}}, \frac {2 \, {\left (105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}\right )}}{105 \, b^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (109) = 218\).
Time = 3.37 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.60 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{b} & \text {for}\: a = 0 \\- \frac {A a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{4} \sqrt {- \frac {a}{b}}} + \frac {A a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{4} \sqrt {- \frac {a}{b}}} + \frac {2 A a^{2} \sqrt {x}}{b^{3}} - \frac {2 A a x^{\frac {3}{2}}}{3 b^{2}} + \frac {2 A x^{\frac {5}{2}}}{5 b} + \frac {B a^{4} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{5} \sqrt {- \frac {a}{b}}} - \frac {B a^{4} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{5} \sqrt {- \frac {a}{b}}} - \frac {2 B a^{3} \sqrt {x}}{b^{4}} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3 b^{3}} - \frac {2 B a x^{\frac {5}{2}}}{5 b^{2}} + \frac {2 B x^{\frac {7}{2}}}{7 b} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=\frac {2 \, {\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (15 \, B b^{3} x^{\frac {7}{2}} - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{\frac {5}{2}} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{\frac {3}{2}} - 105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {x}\right )}}{105 \, b^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=\frac {2 \, {\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (15 \, B b^{6} x^{\frac {7}{2}} - 21 \, B a b^{5} x^{\frac {5}{2}} + 21 \, A b^{6} x^{\frac {5}{2}} + 35 \, B a^{2} b^{4} x^{\frac {3}{2}} - 35 \, A a b^{5} x^{\frac {3}{2}} - 105 \, B a^{3} b^{3} \sqrt {x} + 105 \, A a^{2} b^{4} \sqrt {x}\right )}}{105 \, b^{7}} \]
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Time = 0.43 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.11 \[ \int \frac {x^{5/2} (A+B x)}{a+b x} \, dx=x^{5/2}\,\left (\frac {2\,A}{5\,b}-\frac {2\,B\,a}{5\,b^2}\right )+\frac {2\,B\,x^{7/2}}{7\,b}+\frac {a^2\,\sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{b^2}+\frac {2\,a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,\sqrt {x}\,\left (A\,b-B\,a\right )}{B\,a^4-A\,a^3\,b}\right )\,\left (A\,b-B\,a\right )}{b^{9/2}}-\frac {a\,x^{3/2}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{3\,b} \]
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