Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{5/3}}{5 b^2}+\frac {3 (a+b x)^{8/3}}{8 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {3 (a+b x)^{8/3}}{8 b^2}-\frac {3 a (a+b x)^{5/3}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int x (a+b x)^{2/3} \, dx &=\int \left (-\frac {a (a+b x)^{2/3}}{b}+\frac {(a+b x)^{5/3}}{b}\right ) \, dx\\ &=-\frac {3 a (a+b x)^{5/3}}{5 b^2}+\frac {3 (a+b x)^{8/3}}{8 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 35, normalized size = 1.03 \begin {gather*} \frac {3 (a+b x)^{2/3} \left (-3 a^2+2 a b x+5 b^2 x^2\right )}{40 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(34)=68\).
time = 3.52, size = 96, normalized size = 2.82 \begin {gather*} \frac {3 a^{\frac {2}{3}} \left (3 a^3 \left (1-\left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+a^2 b x \left (3-\left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+7 a b^2 x^2 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}+5 b^3 x^3 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )}{40 b^2 \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 26, normalized size = 0.76
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (-5 b x +3 a \right )}{40 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{2}}\) | \(26\) |
trager | \(-\frac {3 \left (-5 x^{2} b^{2}-2 a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {2}{3}}}{40 b^{2}}\) | \(32\) |
risch | \(-\frac {3 \left (-5 x^{2} b^{2}-2 a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {2}{3}}}{40 b^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 26, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}}}{8 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}} a}{5 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 31, normalized size = 0.91 \begin {gather*} \frac {3 \, {\left (5 \, b^{2} x^{2} + 2 \, a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{40 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs.
\(2 (31) = 62\).
time = 0.61, size = 202, normalized size = 5.94 \begin {gather*} - \frac {9 a^{\frac {14}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {9 a^{\frac {14}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} - \frac {3 a^{\frac {11}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {9 a^{\frac {11}{3}} b x}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {21 a^{\frac {8}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {15 a^{\frac {5}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (26) = 52\).
time = 0.00, size = 113, normalized size = 3.32 \begin {gather*} \frac {\frac {3 b \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^{2}}+\frac {3 a \left (\frac {1}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )-\frac {1}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a\right )}{b}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {24\,a\,{\left (a+b\,x\right )}^{5/3}-15\,{\left (a+b\,x\right )}^{8/3}}{40\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________