Integrand size = 16, antiderivative size = 63 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 a (A b-a B)}{b^3 \sqrt {a+b x}}+\frac {2 (A b-2 a B) \sqrt {a+b x}}{b^3}+\frac {2 B (a+b x)^{3/2}}{3 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 63, 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} \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x} (A b-2 a B)}{b^3}+\frac {2 a (A b-a B)}{b^3 \sqrt {a+b x}}+\frac {2 B (a+b x)^{3/2}}{3 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^{3/2}}+\frac {A b-2 a B}{b^2 \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b^2}\right ) \, dx \\ & = \frac {2 a (A b-a B)}{b^3 \sqrt {a+b x}}+\frac {2 (A b-2 a B) \sqrt {a+b x}}{b^3}+\frac {2 B (a+b x)^{3/2}}{3 b^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.75 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \left (-8 a^2 B+b^2 x (3 A+B x)+a (6 A b-4 b B x)\right )}{3 b^3 \sqrt {a+b x}} \]
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Time = 0.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {2 x \left (\frac {B x}{3}+A \right ) b^{2}+4 a \left (-\frac {2 B x}{3}+A \right ) b -\frac {16 a^{2} B}{3}}{\sqrt {b x +a}\, b^{3}}\) | \(41\) |
gosper | \(\frac {\frac {2}{3} b^{2} B \,x^{2}+2 A \,b^{2} x -\frac {8}{3} B a b x +4 a b A -\frac {16}{3} a^{2} B}{\sqrt {b x +a}\, b^{3}}\) | \(46\) |
trager | \(\frac {\frac {2}{3} b^{2} B \,x^{2}+2 A \,b^{2} x -\frac {8}{3} B a b x +4 a b A -\frac {16}{3} a^{2} B}{\sqrt {b x +a}\, b^{3}}\) | \(46\) |
risch | \(\frac {2 \left (b B x +3 A b -5 B a \right ) \sqrt {b x +a}}{3 b^{3}}+\frac {2 a \left (A b -B a \right )}{b^{3} \sqrt {b x +a}}\) | \(48\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A b \sqrt {b x +a}-4 B a \sqrt {b x +a}+\frac {2 a \left (A b -B a \right )}{\sqrt {b x +a}}}{b^{3}}\) | \(55\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A b \sqrt {b x +a}-4 B a \sqrt {b x +a}+\frac {2 a \left (A b -B a \right )}{\sqrt {b x +a}}}{b^{3}}\) | \(55\) |
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none
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (B b^{2} x^{2} - 8 \, B a^{2} + 6 \, A a b - {\left (4 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x + a b^{3}\right )}} \]
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Time = 0.96 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.21 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {3}{2}}}{3 b} - \frac {a \left (- A b + B a\right )}{b \sqrt {a + b x}} + \frac {\sqrt {a + b x} \left (A b - 2 B a\right )}{b}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{3}}{3}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (b x + a\right )}^{\frac {3}{2}} B - 3 \, {\left (2 \, B a - A b\right )} \sqrt {b x + a}}{b} - \frac {3 \, {\left (B a^{2} - A a b\right )}}{\sqrt {b x + a} b}\right )}}{3 \, b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (B a^{2} - A a b\right )}}{\sqrt {b x + a} b^{3}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} B b^{6} - 6 \, \sqrt {b x + a} B a b^{6} + 3 \, \sqrt {b x + a} A b^{7}\right )}}{3 \, b^{9}} \]
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Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2\,B\,{\left (a+b\,x\right )}^2-6\,B\,a^2+6\,A\,a\,b+6\,A\,b\,\left (a+b\,x\right )-12\,B\,a\,\left (a+b\,x\right )}{3\,b^3\,\sqrt {a+b\,x}} \]
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