Integrand size = 16, antiderivative size = 65 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=-\frac {2 a (A b-a B) \sqrt {a+b x}}{b^3}+\frac {2 (A b-2 a B) (a+b x)^{3/2}}{3 b^3}+\frac {2 B (a+b x)^{5/2}}{5 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 (a+b x)^{3/2} (A b-2 a B)}{3 b^3}-\frac {2 a \sqrt {a+b x} (A b-a B)}{b^3}+\frac {2 B (a+b x)^{5/2}}{5 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B)}{b^2 \sqrt {a+b x}}+\frac {(A b-2 a B) \sqrt {a+b x}}{b^2}+\frac {B (a+b x)^{3/2}}{b^2}\right ) \, dx \\ & = -\frac {2 a (A b-a B) \sqrt {a+b x}}{b^3}+\frac {2 (A b-2 a B) (a+b x)^{3/2}}{3 b^3}+\frac {2 B (a+b x)^{5/2}}{5 b^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (8 a^2 B-2 a b (5 A+2 B x)+b^2 x (5 A+3 B x)\right )}{15 b^3} \]
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Time = 0.53 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(-\frac {4 \left (-\frac {\left (\frac {3 B x}{5}+A \right ) x \,b^{2}}{2}+a \left (\frac {2 B x}{5}+A \right ) b -\frac {4 a^{2} B}{5}\right ) \sqrt {b x +a}}{3 b^{3}}\) | \(41\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +4 B a b x +10 a b A -8 a^{2} B \right )}{15 b^{3}}\) | \(47\) |
trager | \(-\frac {2 \sqrt {b x +a}\, \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +4 B a b x +10 a b A -8 a^{2} B \right )}{15 b^{3}}\) | \(47\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +4 B a b x +10 a b A -8 a^{2} B \right )}{15 b^{3}}\) | \(47\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \left (A b -B a \right ) \sqrt {b x +a}}{b^{3}}\) | \(52\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \left (A b -B a \right ) \sqrt {b x +a}}{b^{3}}\) | \(52\) |
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none
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 10 \, A a b - {\left (4 \, B a b - 5 \, A b^{2}\right )} x\right )} \sqrt {b x + a}}{15 \, b^{3}} \]
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Time = 0.55 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.23 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {5}{2}}}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A b - 2 B a\right )}{3 b} + \frac {\sqrt {a + b x} \left (- A a b + B a^{2}\right )}{b}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{3}}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} B - 5 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (B a^{2} - A a b\right )} \sqrt {b x + a}\right )}}{15 \, b^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B}{b^{2}}\right )}}{15 \, b} \]
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Time = 0.38 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2\,\sqrt {a+b\,x}\,\left (15\,B\,a^2+3\,B\,{\left (a+b\,x\right )}^2-15\,A\,a\,b+5\,A\,b\,\left (a+b\,x\right )-10\,B\,a\,\left (a+b\,x\right )\right )}{15\,b^3} \]
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