Optimal. Leaf size=53 \[ \frac {3 a^2 (a+b x)^{4/3}}{4 b^3}-\frac {6 a (a+b x)^{7/3}}{7 b^3}+\frac {3 (a+b x)^{10/3}}{10 b^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, 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^2 (a+b x)^{4/3}}{4 b^3}+\frac {3 (a+b x)^{10/3}}{10 b^3}-\frac {6 a (a+b x)^{7/3}}{7 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^2 \sqrt [3]{a+b x} \, dx &=\int \left (\frac {a^2 \sqrt [3]{a+b x}}{b^2}-\frac {2 a (a+b x)^{4/3}}{b^2}+\frac {(a+b x)^{7/3}}{b^2}\right ) \, dx\\ &=\frac {3 a^2 (a+b x)^{4/3}}{4 b^3}-\frac {6 a (a+b x)^{7/3}}{7 b^3}+\frac {3 (a+b x)^{10/3}}{10 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {3 (a+b x)^{4/3} \left (9 a^2-12 a b x+14 b^2 x^2\right )}{140 b^3} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(53)=106\).
time = 7.13, size = 197, normalized size = 3.72 \begin {gather*} \frac {3 a^{\frac {1}{3}} \left (9 a^6 \left (-1+\left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+3 a^5 b x \left (-9+8 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^4 b^2 x^2 \left (-27+20 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )-9 a^3 b^3 x^3+20 a^3 b^3 x^3 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+45 a^2 b^4 x^4 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+44 a b^5 x^5 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+14 b^6 x^6 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )}{140 b^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 38, normalized size = 0.72
method | result | size |
gosper | \(\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (14 x^{2} b^{2}-12 a b x +9 a^{2}\right )}{140 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {6 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {3 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {6 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {3 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{3}}\) | \(38\) |
trager | \(\frac {3 \left (14 b^{3} x^{3}+2 a \,b^{2} x^{2}-3 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{140 b^{3}}\) | \(43\) |
risch | \(\frac {3 \left (14 b^{3} x^{3}+2 a \,b^{2} x^{2}-3 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{140 b^{3}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 41, normalized size = 0.77 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}}}{10 \, b^{3}} - \frac {6 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2}}{4 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 42, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (14 \, b^{3} x^{3} + 2 \, a b^{2} x^{2} - 3 \, a^{2} b x + 9 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{140 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs.
\(2 (49) = 98\).
time = 0.89, size = 666, normalized size = 12.57 \begin {gather*} \frac {27 a^{\frac {34}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {34}{3}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {31}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {31}{3}} b x}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {28}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {28}{3}} b^{2} x^{2}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {25}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {25}{3}} b^{3} x^{3}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {135 a^{\frac {22}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {132 a^{\frac {19}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {42 a^{\frac {16}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (41) = 82\).
time = 0.00, size = 147, normalized size = 2.77 \begin {gather*} \frac {\frac {3 b \left (\frac {1}{10} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3}-\frac {3}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a+\frac {3}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{2}-\left (a+b x\right )^{\frac {1}{3}} a^{3}\right )}{b^{3}}+\frac {3 a \left (\frac {1}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2}-\frac {1}{2} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a+\left (a+b x\right )^{\frac {1}{3}} a^{2}\right )}{b^{2}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {42\,{\left (a+b\,x\right )}^{10/3}-120\,a\,{\left (a+b\,x\right )}^{7/3}+105\,a^2\,{\left (a+b\,x\right )}^{4/3}}{140\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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