Integrand size = 18, antiderivative size = 130 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=-\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {a^{3/2} (5 A b-7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 130, 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, 52, 65, 211} \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=\frac {a^{3/2} (5 A b-7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}}-\frac {a \sqrt {x} (5 A b-7 a B)}{b^4}+\frac {x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac {x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac {x^{7/2} (A b-a B)}{a b (a+b x)} \]
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Rule 52
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{7/2}}{a b (a+b x)}-\frac {\left (\frac {5 A b}{2}-\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{a+b x} \, dx}{a b} \\ & = -\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {(5 A b-7 a B) \int \frac {x^{3/2}}{a+b x} \, dx}{2 b^2} \\ & = \frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}-\frac {(a (5 A b-7 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b^3} \\ & = -\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {\left (a^2 (5 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^4} \\ & = -\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {\left (a^2 (5 A b-7 a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^4} \\ & = -\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=\frac {\sqrt {x} \left (105 a^3 B+2 b^3 x^2 (5 A+3 B x)-2 a b^2 x (25 A+7 B x)+a^2 (-75 A b+70 b B x)\right )}{15 b^4 (a+b x)}-\frac {a^{3/2} (-5 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \]
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Time = 0.49 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {2 \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +10 B a b x +30 a b A -45 a^{2} B \right ) \sqrt {x}}{15 b^{4}}+\frac {a^{2} \left (\frac {2 \left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{4}}\) | \(99\) |
derivativedivides | \(-\frac {2 \left (-\frac {b^{2} B \,x^{\frac {5}{2}}}{5}-\frac {A \,b^{2} x^{\frac {3}{2}}}{3}+\frac {2 B a b \,x^{\frac {3}{2}}}{3}+2 a b A \sqrt {x}-3 a^{2} B \sqrt {x}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{4}}\) | \(107\) |
default | \(-\frac {2 \left (-\frac {b^{2} B \,x^{\frac {5}{2}}}{5}-\frac {A \,b^{2} x^{\frac {3}{2}}}{3}+\frac {2 B a b \,x^{\frac {3}{2}}}{3}+2 a b A \sqrt {x}-3 a^{2} B \sqrt {x}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{4}}\) | \(107\) |
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Time = 0.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.23 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {x}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {x}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (119) = 238\).
Time = 13.89 (sec) , antiderivative size = 877, normalized size of antiderivative = 6.75 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}\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^{2}} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{b^{2}} & \text {for}\: a = 0 \\\frac {75 A a^{3} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {75 A a^{3} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {150 A a^{2} b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {75 A a^{2} b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {75 A a^{2} b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {100 A a b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {20 A b^{4} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {105 B a^{4} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {105 B a^{4} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {210 B a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {105 B a^{3} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {105 B a^{3} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {140 B a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} - \frac {28 B a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} + \frac {12 B b^{4} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{30 a b^{5} \sqrt {- \frac {a}{b}} + 30 b^{6} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=\frac {{\left (B a^{3} - A a^{2} b\right )} \sqrt {x}}{b^{5} x + a b^{4}} - \frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (3 \, B b^{2} x^{\frac {5}{2}} - 5 \, {\left (2 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} \sqrt {x}\right )}}{15 \, b^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.94 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=-\frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {B a^{3} \sqrt {x} - A a^{2} b \sqrt {x}}{{\left (b x + a\right )} b^{4}} + \frac {2 \, {\left (3 \, B b^{8} x^{\frac {5}{2}} - 10 \, B a b^{7} x^{\frac {3}{2}} + 5 \, A b^{8} x^{\frac {3}{2}} + 45 \, B a^{2} b^{6} \sqrt {x} - 30 \, A a b^{7} \sqrt {x}\right )}}{15 \, b^{10}} \]
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Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx=x^{3/2}\,\left (\frac {2\,A}{3\,b^2}-\frac {4\,B\,a}{3\,b^3}\right )-\sqrt {x}\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b}+\frac {2\,B\,a^2}{b^4}\right )+\frac {2\,B\,x^{5/2}}{5\,b^2}+\frac {\sqrt {x}\,\left (B\,a^3-A\,a^2\,b\right )}{x\,b^5+a\,b^4}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,\sqrt {x}\,\left (5\,A\,b-7\,B\,a\right )}{7\,B\,a^3-5\,A\,a^2\,b}\right )\,\left (5\,A\,b-7\,B\,a\right )}{b^{9/2}} \]
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