Integrand size = 13, antiderivative size = 51 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{2 b^3}+\frac {3 (a+b x)^{7/3}}{7 b^3} \]
[Out]
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} \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac {3 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{4/3}}{2 b^3} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{b^2 (a+b x)^{2/3}}-\frac {2 a \sqrt [3]{a+b x}}{b^2}+\frac {(a+b x)^{4/3}}{b^2}\right ) \, dx \\ & = \frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{2 b^3}+\frac {3 (a+b x)^{7/3}}{7 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a+b x} \left (9 a^2-3 a b x+2 b^2 x^2\right )}{14 b^3} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (2 b^{2} x^{2}-3 a b x +9 a^{2}\right )}{14 b^{3}}\) | \(32\) |
trager | \(\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (2 b^{2} x^{2}-3 a b x +9 a^{2}\right )}{14 b^{3}}\) | \(32\) |
risch | \(\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (2 b^{2} x^{2}-3 a b x +9 a^{2}\right )}{14 b^{3}}\) | \(32\) |
pseudoelliptic | \(\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (2 b^{2} x^{2}-3 a b x +9 a^{2}\right )}{14 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a \left (b x +a \right )^{\frac {4}{3}}}{2}+3 a^{2} \left (b x +a \right )^{\frac {1}{3}}}{b^{3}}\) | \(37\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a \left (b x +a \right )^{\frac {4}{3}}}{2}+3 a^{2} \left (b x +a \right )^{\frac {1}{3}}}{b^{3}}\) | \(37\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (2 \, b^{2} x^{2} - 3 \, a b x + 9 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{14 \, b^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (48) = 96\).
Time = 1.06 (sec) , antiderivative size = 600, normalized size of antiderivative = 11.76 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {27 a^{\frac {31}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {31}{3}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {28}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {28}{3}} b x}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {25}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {25}{3}} b^{2} x^{2}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {18 a^{\frac {22}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {22}{3}} b^{3} x^{3}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {9 a^{\frac {19}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {6 a^{\frac {16}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}}}{7 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}} a}{2 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}}{b^{3}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )}}{14 \, b^{3}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{(a+b x)^{2/3}} \, dx=\frac {6\,{\left (a+b\,x\right )}^{7/3}-21\,a\,{\left (a+b\,x\right )}^{4/3}+42\,a^2\,{\left (a+b\,x\right )}^{1/3}}{14\,b^3} \]
[In]
[Out]