Optimal. Leaf size=59 \[ \frac {3 a^2 (-a+b x)^{2/3}}{2 b^3}+\frac {6 a (-a+b x)^{5/3}}{5 b^3}+\frac {3 (-a+b x)^{8/3}}{8 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {3 a^2 (b x-a)^{2/3}}{2 b^3}+\frac {3 (b x-a)^{8/3}}{8 b^3}+\frac {6 a (b x-a)^{5/3}}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt [3]{-a+b x}} \, dx &=\int \left (\frac {a^2}{b^2 \sqrt [3]{-a+b x}}+\frac {2 a (-a+b x)^{2/3}}{b^2}+\frac {(-a+b x)^{5/3}}{b^2}\right ) \, dx\\ &=\frac {3 a^2 (-a+b x)^{2/3}}{2 b^3}+\frac {6 a (-a+b x)^{5/3}}{5 b^3}+\frac {3 (-a+b x)^{8/3}}{8 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 37, normalized size = 0.63 \begin {gather*} \frac {3 (-a+b x)^{2/3} \left (9 a^2+6 a b x+5 b^2 x^2\right )}{40 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 6.98, size = 837, normalized size = 14.19 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {3 a^{\frac {2}{3}} \left (9 a^5 \left (-1^{\frac {2}{3}}+\left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )+3 a^4 b x \left (9-1^{\frac {2}{3}}-7 \left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )+a^3 b^2 x^2 \left (-27-1^{\frac {2}{3}}+14 \left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )+9 a b^3 x^3 \left (-1^{\frac {2}{3}} a+b x \left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )-6 a^2 b^3 x^3 \left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}-5 b^5 x^5 \left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )}{40 b^3 \left (a^3-3 a^2 b x+3 a b^2 x^2-b^3 x^3\right )},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {-27 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {32}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}+\frac {27 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {32}{3}} \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}-\frac {63 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {29}{3}} b x \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}+\frac {81 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {29}{3}} b x}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}-\frac {81 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {26}{3}} b^2 x^2}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}+\frac {42 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {26}{3}} b^2 x^2 \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}-\frac {18 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {23}{3}} b^3 x^3 \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}+\frac {27 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {23}{3}} b^3 x^3}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}+\frac {27 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {20}{3}} b^4 x^4 \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}-\frac {15 E^{\frac {I 2 \text {Pi}}{3}} a^{\frac {17}{3}} b^5 x^5 \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^8 b^3-120 a^7 b^4 x+120 a^6 b^5 x^2-40 a^5 b^6 x^3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 44, normalized size = 0.75
method | result | size |
gosper | \(\frac {3 \left (5 x^{2} b^{2}+6 a b x +9 a^{2}\right ) \left (b x -a \right )^{\frac {2}{3}}}{40 b^{3}}\) | \(34\) |
trager | \(\frac {3 \left (5 x^{2} b^{2}+6 a b x +9 a^{2}\right ) \left (b x -a \right )^{\frac {2}{3}}}{40 b^{3}}\) | \(34\) |
risch | \(-\frac {3 \left (-b x +a \right ) \left (5 x^{2} b^{2}+6 a b x +9 a^{2}\right )}{40 b^{3} \left (b x -a \right )^{\frac {1}{3}}}\) | \(40\) |
derivativedivides | \(\frac {\frac {3 \left (b x -a \right )^{\frac {8}{3}}}{8}+\frac {6 a \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a^{2} \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{3}}\) | \(44\) |
default | \(\frac {\frac {3 \left (b x -a \right )^{\frac {8}{3}}}{8}+\frac {6 a \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a^{2} \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{3}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 47, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x - a\right )}^{\frac {8}{3}}}{8 \, b^{3}} + \frac {6 \, {\left (b x - a\right )}^{\frac {5}{3}} a}{5 \, b^{3}} + \frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}} a^{2}}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.29, size = 33, normalized size = 0.56 \begin {gather*} \frac {3 \, {\left (5 \, b^{2} x^{2} + 6 \, a b x + 9 \, a^{2}\right )} {\left (b x - a\right )}^{\frac {2}{3}}}{40 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.91, size = 1326, normalized size = 22.47
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 73, normalized size = 1.24 \begin {gather*} \frac {3 \left (\frac {1}{8} \left (\left (-a+b x\right )^{\frac {1}{3}}\right )^{2} \left (-a+b x\right )^{2}+\frac {2}{5} \left (\left (-a+b x\right )^{\frac {1}{3}}\right )^{2} \left (-a+b x\right ) a+\frac {1}{2} \left (\left (-a+b x\right )^{\frac {1}{3}}\right )^{2} a^{2}\right )}{b b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 43, normalized size = 0.73 \begin {gather*} \frac {48\,a\,{\left (b\,x-a\right )}^{5/3}+15\,{\left (b\,x-a\right )}^{8/3}+60\,a^2\,{\left (b\,x-a\right )}^{2/3}}{40\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________