Integrand size = 13, antiderivative size = 49 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=-\frac {2 a^2}{b^3 \sqrt {a+b x}}-\frac {4 a \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{3/2}}{3 b^3} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=-\frac {2 a^2}{b^3 \sqrt {a+b x}}-\frac {4 a \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{3/2}}{3 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{b^2 (a+b x)^{3/2}}-\frac {2 a}{b^2 \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b^2}\right ) \, dx \\ & = -\frac {2 a^2}{b^3 \sqrt {a+b x}}-\frac {4 a \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{3/2}}{3 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=\frac {2 \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt {a+b x}} \]
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Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {\frac {2}{3} b^{2} x^{2}-\frac {8}{3} a b x -\frac {16}{3} a^{2}}{\sqrt {b x +a}\, b^{3}}\) | \(31\) |
gosper | \(-\frac {2 \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right )}{3 \sqrt {b x +a}\, b^{3}}\) | \(32\) |
trager | \(-\frac {2 \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right )}{3 \sqrt {b x +a}\, b^{3}}\) | \(32\) |
risch | \(-\frac {2 \left (-b x +5 a \right ) \sqrt {b x +a}}{3 b^{3}}-\frac {2 a^{2}}{b^{3} \sqrt {b x +a}}\) | \(37\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-4 a \sqrt {b x +a}-\frac {2 a^{2}}{\sqrt {b x +a}}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-4 a \sqrt {b x +a}-\frac {2 a^{2}}{\sqrt {b x +a}}}{b^{3}}\) | \(38\) |
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none
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x + a b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (46) = 92\).
Time = 1.46 (sec) , antiderivative size = 534, normalized size of antiderivative = 10.90 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=- \frac {16 a^{\frac {19}{2}} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {16 a^{\frac {19}{2}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac {40 a^{\frac {17}{2}} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {48 a^{\frac {17}{2}} b x}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac {30 a^{\frac {15}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {48 a^{\frac {15}{2}} b^{2} x^{2}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac {4 a^{\frac {13}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {16 a^{\frac {13}{2}} b^{3} x^{3}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {2 a^{\frac {11}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b^{3}} - \frac {4 \, \sqrt {b x + a} a}{b^{3}} - \frac {2 \, a^{2}}{\sqrt {b x + a} b^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=-\frac {2 \, a^{2}}{\sqrt {b x + a} b^{3}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x + a} a b^{6}\right )}}{3 \, b^{9}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{(a+b x)^{3/2}} \, dx=-\frac {12\,a\,\left (a+b\,x\right )-2\,{\left (a+b\,x\right )}^2+6\,a^2}{3\,b^3\,\sqrt {a+b\,x}} \]
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