Integrand size = 13, antiderivative size = 51 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {2 a^2 \sqrt {a+b x}}{b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3} \]
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Time = 0.01 (sec) , antiderivative size = 51, 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} \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {2 a^2 \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{b^2 \sqrt {a+b x}}-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {(a+b x)^{3/2}}{b^2}\right ) \, dx \\ & = \frac {2 a^2 \sqrt {a+b x}}{b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3} \]
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Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {2 \sqrt {b x +a}\, \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\) | \(32\) |
trager | \(\frac {2 \sqrt {b x +a}\, \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\) | \(32\) |
risch | \(\frac {2 \sqrt {b x +a}\, \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\) | \(32\) |
pseudoelliptic | \(\frac {2 \sqrt {b x +a}\, \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {4 a \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {b x +a}}{b^{3}}\) | \(37\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {4 a \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {b x +a}}{b^{3}}\) | \(37\) |
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Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt {b x + a}}{15 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (48) = 96\).
Time = 1.45 (sec) , antiderivative size = 600, normalized size of antiderivative = 11.76 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {16 a^{\frac {21}{2}} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {21}{2}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {19}{2}} b x \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {19}{2}} b x}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {17}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {17}{2}} b^{2} x^{2}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {10 a^{\frac {15}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {15}{2}} b^{3} x^{3}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {10 a^{\frac {13}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {6 a^{\frac {11}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{3 \, b^{3}} + \frac {2 \, \sqrt {b x + a} a^{2}}{b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )}}{15 \, b^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {a+b x}} \, dx=\frac {6\,{\left (a+b\,x\right )}^{5/2}-20\,a\,{\left (a+b\,x\right )}^{3/2}+30\,a^2\,\sqrt {a+b\,x}}{15\,b^3} \]
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