Integrand size = 18, antiderivative size = 136 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=-\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {2 a^{7/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
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Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {81, 52, 65, 211} \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=\frac {2 a^{7/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}-\frac {2 a^3 \sqrt {x} (A b-a B)}{b^5}+\frac {2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac {2 a x^{5/2} (A b-a B)}{5 b^3}+\frac {2 x^{7/2} (A b-a B)}{7 b^2}+\frac {2 B x^{9/2}}{9 b} \]
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Rule 52
Rule 65
Rule 81
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {2 B x^{9/2}}{9 b}+\frac {\left (2 \left (\frac {9 A b}{2}-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{a+b x} \, dx}{9 b} \\ & = \frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {(a (A b-a B)) \int \frac {x^{5/2}}{a+b x} \, dx}{b^2} \\ & = -\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {x^{3/2}}{a+b x} \, dx}{b^3} \\ & = \frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {\left (a^3 (A b-a B)\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{b^4} \\ & = -\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^4 (A b-a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^5} \\ & = -\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (2 a^4 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^5} \\ & = -\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=\frac {2 \sqrt {x} \left (315 a^4 B-105 a^3 b (3 A+B x)+21 a^2 b^2 x (5 A+3 B x)-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )}{315 b^5}-\frac {2 a^{7/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
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Time = 1.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {2 \left (-35 B \,x^{4} b^{4}-45 A \,x^{3} b^{4}+45 B \,x^{3} a \,b^{3}+63 A \,x^{2} a \,b^{3}-63 B \,x^{2} a^{2} b^{2}-105 A x \,a^{2} b^{2}+105 B x \,a^{3} b +315 A \,a^{3} b -315 B \,a^{4}\right ) \sqrt {x}}{315 b^{5}}+\frac {2 a^{4} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{5} \sqrt {a b}}\) | \(124\) |
derivativedivides | \(-\frac {2 \left (-\frac {B \,x^{\frac {9}{2}} b^{4}}{9}-\frac {A \,b^{4} x^{\frac {7}{2}}}{7}+\frac {B a \,b^{3} x^{\frac {7}{2}}}{7}+\frac {A a \,b^{3} x^{\frac {5}{2}}}{5}-\frac {B \,a^{2} b^{2} x^{\frac {5}{2}}}{5}-\frac {A \,a^{2} b^{2} x^{\frac {3}{2}}}{3}+\frac {B \,a^{3} b \,x^{\frac {3}{2}}}{3}+A \,a^{3} b \sqrt {x}-B \,a^{4} \sqrt {x}\right )}{b^{5}}+\frac {2 a^{4} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{5} \sqrt {a b}}\) | \(130\) |
default | \(-\frac {2 \left (-\frac {B \,x^{\frac {9}{2}} b^{4}}{9}-\frac {A \,b^{4} x^{\frac {7}{2}}}{7}+\frac {B a \,b^{3} x^{\frac {7}{2}}}{7}+\frac {A a \,b^{3} x^{\frac {5}{2}}}{5}-\frac {B \,a^{2} b^{2} x^{\frac {5}{2}}}{5}-\frac {A \,a^{2} b^{2} x^{\frac {3}{2}}}{3}+\frac {B \,a^{3} b \,x^{\frac {3}{2}}}{3}+A \,a^{3} b \sqrt {x}-B \,a^{4} \sqrt {x}\right )}{b^{5}}+\frac {2 a^{4} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{5} \sqrt {a b}}\) | \(130\) |
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Time = 0.24 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.03 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=\left [-\frac {315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{315 \, b^{5}}, -\frac {2 \, {\left (315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt {x}\right )}}{315 \, b^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (131) = 262\).
Time = 10.89 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.41 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {11}{2}}}{11}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{b} & \text {for}\: a = 0 \\\frac {A a^{4} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{5} \sqrt {- \frac {a}{b}}} - \frac {A a^{4} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{5} \sqrt {- \frac {a}{b}}} - \frac {2 A a^{3} \sqrt {x}}{b^{4}} + \frac {2 A a^{2} x^{\frac {3}{2}}}{3 b^{3}} - \frac {2 A a x^{\frac {5}{2}}}{5 b^{2}} + \frac {2 A x^{\frac {7}{2}}}{7 b} - \frac {B a^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{6} \sqrt {- \frac {a}{b}}} + \frac {B a^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{6} \sqrt {- \frac {a}{b}}} + \frac {2 B a^{4} \sqrt {x}}{b^{5}} - \frac {2 B a^{3} x^{\frac {3}{2}}}{3 b^{4}} + \frac {2 B a^{2} x^{\frac {5}{2}}}{5 b^{3}} - \frac {2 B a x^{\frac {7}{2}}}{7 b^{2}} + \frac {2 B x^{\frac {9}{2}}}{9 b} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.94 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=-\frac {2 \, {\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (35 \, B b^{4} x^{\frac {9}{2}} - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{\frac {7}{2}} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{\frac {5}{2}} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{\frac {3}{2}} + 315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {x}\right )}}{315 \, b^{5}} \]
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Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.02 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=-\frac {2 \, {\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (35 \, B b^{8} x^{\frac {9}{2}} - 45 \, B a b^{7} x^{\frac {7}{2}} + 45 \, A b^{8} x^{\frac {7}{2}} + 63 \, B a^{2} b^{6} x^{\frac {5}{2}} - 63 \, A a b^{7} x^{\frac {5}{2}} - 105 \, B a^{3} b^{5} x^{\frac {3}{2}} + 105 \, A a^{2} b^{6} x^{\frac {3}{2}} + 315 \, B a^{4} b^{4} \sqrt {x} - 315 \, A a^{3} b^{5} \sqrt {x}\right )}}{315 \, b^{9}} \]
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Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.11 \[ \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx=x^{7/2}\,\left (\frac {2\,A}{7\,b}-\frac {2\,B\,a}{7\,b^2}\right )+\frac {2\,B\,x^{9/2}}{9\,b}+\frac {a^2\,x^{3/2}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{3\,b^2}-\frac {a^3\,\sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{b^3}-\frac {2\,a^{7/2}\,\mathrm {atan}\left (\frac {a^{7/2}\,\sqrt {b}\,\sqrt {x}\,\left (A\,b-B\,a\right )}{B\,a^5-A\,a^4\,b}\right )\,\left (A\,b-B\,a\right )}{b^{11/2}}-\frac {a\,x^{5/2}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{5\,b} \]
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