Optimal. Leaf size=72 \[ -\frac {3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac {9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac {9 a (a+b x)^{11/3}}{11 b^4}+\frac {3 (a+b x)^{14/3}}{14 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} -\frac {3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac {9 a^2 (a+b x)^{8/3}}{8 b^4}+\frac {3 (a+b x)^{14/3}}{14 b^4}-\frac {9 a (a+b x)^{11/3}}{11 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int x^3 (a+b x)^{2/3} \, dx &=\int \left (-\frac {a^3 (a+b x)^{2/3}}{b^3}+\frac {3 a^2 (a+b x)^{5/3}}{b^3}-\frac {3 a (a+b x)^{8/3}}{b^3}+\frac {(a+b x)^{11/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac {9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac {9 a (a+b x)^{11/3}}{11 b^4}+\frac {3 (a+b x)^{14/3}}{14 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 46, normalized size = 0.64 \begin {gather*} \frac {3 (a+b x)^{5/3} \left (-81 a^3+135 a^2 b x-180 a b^2 x^2+220 b^3 x^3\right )}{3080 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 50, normalized size = 0.69
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (-220 b^{3} x^{3}+180 a \,b^{2} x^{2}-135 a^{2} b x +81 a^{3}\right )}{3080 b^{4}}\) | \(43\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {14}{3}}}{14}-\frac {9 a \left (b x +a \right )^{\frac {11}{3}}}{11}+\frac {9 a^{2} \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a^{3} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{4}}\) | \(50\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {14}{3}}}{14}-\frac {9 a \left (b x +a \right )^{\frac {11}{3}}}{11}+\frac {9 a^{2} \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a^{3} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{4}}\) | \(50\) |
trager | \(-\frac {3 \left (-220 b^{4} x^{4}-40 a \,b^{3} x^{3}+45 a^{2} b^{2} x^{2}-54 a^{3} b x +81 a^{4}\right ) \left (b x +a \right )^{\frac {2}{3}}}{3080 b^{4}}\) | \(54\) |
risch | \(-\frac {3 \left (-220 b^{4} x^{4}-40 a \,b^{3} x^{3}+45 a^{2} b^{2} x^{2}-54 a^{3} b x +81 a^{4}\right ) \left (b x +a \right )^{\frac {2}{3}}}{3080 b^{4}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 56, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {14}{3}}}{14 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {11}{3}} a}{11 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {8}{3}} a^{2}}{8 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{3}}{5 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 53, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (220 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} - 45 \, a^{2} b^{2} x^{2} + 54 \, a^{3} b x - 81 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3080 \, b^{4}} \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. 1742 vs.
\(2 (68) = 136\).
time = 1.38, size = 1742, normalized size = 24.19
result too large to display
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. 117 vs.
\(2 (56) = 112\).
time = 0.00, size = 217, normalized size = 3.01 \begin {gather*} \frac {\frac {3 b \left (\frac {1}{14} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{4}-\frac {4}{11} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{3} a+\frac {3}{4} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{2} a^{2}-\frac {4}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right ) a^{3}+\frac {1}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a^{4}\right )}{b^{4}}+\frac {3 a \left (\frac {1}{11} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{3}-\frac {3}{8} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{2} a+\frac {3}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right ) a^{2}-\frac {1}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a^{3}\right )}{b^{3}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 56, normalized size = 0.78 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{14/3}}{14\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{5/3}}{5\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{8/3}}{8\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{11/3}}{11\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________