Integrand size = 18, antiderivative size = 108 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=\frac {(3 A b-5 a B) \sqrt {x}}{b^3}-\frac {(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{5/2}}{a b (a+b x)}-\frac {\sqrt {a} (3 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 52, 65, 211} \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=-\frac {\sqrt {a} (3 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}}+\frac {\sqrt {x} (3 A b-5 a B)}{b^3}-\frac {x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac {x^{5/2} (A b-a B)}{a b (a+b x)} \]
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Rule 52
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{5/2}}{a b (a+b x)}-\frac {\left (\frac {3 A b}{2}-\frac {5 a B}{2}\right ) \int \frac {x^{3/2}}{a+b x} \, dx}{a b} \\ & = -\frac {(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{5/2}}{a b (a+b x)}+\frac {(3 A b-5 a B) \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b^2} \\ & = \frac {(3 A b-5 a B) \sqrt {x}}{b^3}-\frac {(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{5/2}}{a b (a+b x)}-\frac {(a (3 A b-5 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^3} \\ & = \frac {(3 A b-5 a B) \sqrt {x}}{b^3}-\frac {(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{5/2}}{a b (a+b x)}-\frac {(a (3 A b-5 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^3} \\ & = \frac {(3 A b-5 a B) \sqrt {x}}{b^3}-\frac {(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{5/2}}{a b (a+b x)}-\frac {\sqrt {a} (3 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=\frac {\sqrt {x} \left (-15 a^2 B+a b (9 A-10 B x)+2 b^2 x (3 A+B x)\right )}{3 b^3 (a+b x)}+\frac {\sqrt {a} (-3 A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \]
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Time = 1.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {2 \left (b B x +3 A b -6 B a \right ) \sqrt {x}}{3 b^{3}}-\frac {a \left (\frac {2 \left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{3}}\) | \(77\) |
derivativedivides | \(\frac {\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-4 B a \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}\) | \(82\) |
default | \(\frac {\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-4 B a \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}\) | \(82\) |
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Time = 0.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.14 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=\left [-\frac {3 \, {\left (5 \, B a^{2} - 3 \, A a b + {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \, {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt {x}}{6 \, {\left (b^{4} x + a b^{3}\right )}}, \frac {3 \, {\left (5 \, B a^{2} - 3 \, A a b + {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \, {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt {x}}{3 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (95) = 190\).
Time = 4.09 (sec) , antiderivative size = 762, normalized size of antiderivative = 7.06 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{a^{2}} & \text {for}\: b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{b^{2}} & \text {for}\: a = 0 \\- \frac {9 A a^{2} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {9 A a^{2} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {18 A a b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} - \frac {9 A a b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {9 A a b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {12 A b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {15 B a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} - \frac {15 B a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} - \frac {30 B a^{2} b \sqrt {x} \sqrt {- \frac {a}{b}}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {15 B a^{2} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} - \frac {15 B a^{2} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} - \frac {20 B a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} + \frac {4 B b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{6 a b^{4} \sqrt {- \frac {a}{b}} + 6 b^{5} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=-\frac {{\left (B a^{2} - A a b\right )} \sqrt {x}}{b^{4} x + a b^{3}} + \frac {{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (B b x^{\frac {3}{2}} - 3 \, {\left (2 \, B a - A b\right )} \sqrt {x}\right )}}{3 \, b^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=\frac {{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} - \frac {B a^{2} \sqrt {x} - A a b \sqrt {x}}{{\left (b x + a\right )} b^{3}} + \frac {2 \, {\left (B b^{4} x^{\frac {3}{2}} - 6 \, B a b^{3} \sqrt {x} + 3 \, A b^{4} \sqrt {x}\right )}}{3 \, b^{6}} \]
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Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^2} \, dx=\sqrt {x}\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )-\frac {\sqrt {x}\,\left (B\,a^2-A\,a\,b\right )}{x\,b^4+a\,b^3}+\frac {2\,B\,x^{3/2}}{3\,b^2}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\sqrt {x}\,\left (3\,A\,b-5\,B\,a\right )}{5\,B\,a^2-3\,A\,a\,b}\right )\,\left (3\,A\,b-5\,B\,a\right )}{b^{7/2}} \]
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