Integrand size = 13, antiderivative size = 49 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=-\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {3 (a+b x)^{5/3}}{5 b^3} \]
[Out]
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} \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=-\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {3 (a+b x)^{5/3}}{5 b^3} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{b^2 (a+b x)^{4/3}}-\frac {2 a}{b^2 \sqrt [3]{a+b x}}+\frac {(a+b x)^{2/3}}{b^2}\right ) \, dx \\ & = -\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {3 (a+b x)^{5/3}}{5 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=\frac {3 \left (-9 a^2-3 a b x+b^2 x^2\right )}{5 b^3 \sqrt [3]{a+b x}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {\frac {3}{5} b^{2} x^{2}-\frac {9}{5} a b x -\frac {27}{5} a^{2}}{\left (b x +a \right )^{\frac {1}{3}} b^{3}}\) | \(31\) |
gosper | \(-\frac {3 \left (-b^{2} x^{2}+3 a b x +9 a^{2}\right )}{5 \left (b x +a \right )^{\frac {1}{3}} b^{3}}\) | \(32\) |
trager | \(-\frac {3 \left (-b^{2} x^{2}+3 a b x +9 a^{2}\right )}{5 \left (b x +a \right )^{\frac {1}{3}} b^{3}}\) | \(32\) |
risch | \(-\frac {3 \left (-b x +4 a \right ) \left (b x +a \right )^{\frac {2}{3}}}{5 b^{3}}-\frac {3 a^{2}}{b^{3} \left (b x +a \right )^{\frac {1}{3}}}\) | \(37\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-3 a \left (b x +a \right )^{\frac {2}{3}}-\frac {3 a^{2}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-3 a \left (b x +a \right )^{\frac {2}{3}}-\frac {3 a^{2}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{3}}\) | \(38\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=\frac {3 \, {\left (b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{5 \, {\left (b^{4} x + a b^{3}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (46) = 92\).
Time = 1.51 (sec) , antiderivative size = 534, normalized size of antiderivative = 10.90 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=- \frac {27 a^{\frac {29}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {27 a^{\frac {29}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac {63 a^{\frac {26}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {81 a^{\frac {26}{3}} b x}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac {42 a^{\frac {23}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {81 a^{\frac {23}{3}} b^{2} x^{2}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac {3 a^{\frac {20}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {27 a^{\frac {20}{3}} b^{3} x^{3}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {3 a^{\frac {17}{3}} b^{4} x^{4} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}}}{5 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{b^{3}} - \frac {3 \, a^{2}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{3}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=-\frac {3 \, a^{2}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{\frac {5}{3}} b^{12} - 5 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{12}\right )}}{5 \, b^{15}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{(a+b x)^{4/3}} \, dx=-\frac {15\,a\,\left (a+b\,x\right )-3\,{\left (a+b\,x\right )}^2+15\,a^2}{5\,b^3\,{\left (a+b\,x\right )}^{1/3}} \]
[In]
[Out]