Optimal. Leaf size=70 \[ \frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 70, 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}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^{4/3}} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^{4/3}}+\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {(a+b x)^{5/3}}{b^3}\right ) \, dx\\ &=\frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.66 \begin {gather*} \frac {3 \left (81 a^3+27 a^2 b x-9 a b^2 x^2+5 b^3 x^3\right )}{40 b^4 \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 49, normalized size = 0.70
method | result | size |
gosper | \(\frac {\frac {3}{8} b^{3} x^{3}-\frac {27}{40} a \,b^{2} x^{2}+\frac {81}{40} a^{2} b x +\frac {243}{40} a^{3}}{\left (b x +a \right )^{\frac {1}{3}} b^{4}}\) | \(43\) |
trager | \(\frac {\frac {3}{8} b^{3} x^{3}-\frac {27}{40} a \,b^{2} x^{2}+\frac {81}{40} a^{2} b x +\frac {243}{40} a^{3}}{\left (b x +a \right )^{\frac {1}{3}} b^{4}}\) | \(43\) |
risch | \(\frac {3 \left (5 x^{2} b^{2}-14 a b x +41 a^{2}\right ) \left (b x +a \right )^{\frac {2}{3}}}{40 b^{4}}+\frac {3 a^{3}}{b^{4} \left (b x +a \right )^{\frac {1}{3}}}\) | \(48\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {9 a \left (b x +a \right )^{\frac {5}{3}}}{5}+\frac {9 a^{2} \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a^{3}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{4}}\) | \(49\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {9 a \left (b x +a \right )^{\frac {5}{3}}}{5}+\frac {9 a^{2} \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a^{3}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{4}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 56, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}}}{8 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {5}{3}} a}{5 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2}}{2 \, b^{4}} + \frac {3 \, a^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 52, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (5 \, b^{3} x^{3} - 9 \, a b^{2} x^{2} + 27 \, a^{2} b x + 81 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{40 \, {\left (b^{5} x + a b^{4}\right )}} \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. 1538 vs.
\(2 (66) = 132\).
time = 1.27, size = 1538, normalized size = 21.97 \begin {gather*} \frac {243 a^{\frac {68}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {243 a^{\frac {68}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {1296 a^{\frac {65}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {1458 a^{\frac {65}{3}} b x}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {2808 a^{\frac {62}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {3645 a^{\frac {62}{3}} b^{2} x^{2}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {3120 a^{\frac {59}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {4860 a^{\frac {59}{3}} b^{3} x^{3}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {1830 a^{\frac {56}{3}} b^{4} x^{4} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {3645 a^{\frac {56}{3}} b^{4} x^{4}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {528 a^{\frac {53}{3}} b^{5} x^{5} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {1458 a^{\frac {53}{3}} b^{5} x^{5}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {96 a^{\frac {50}{3}} b^{6} x^{6} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} - \frac {243 a^{\frac {50}{3}} b^{6} x^{6}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {48 a^{\frac {47}{3}} b^{7} x^{7} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} + \frac {15 a^{\frac {44}{3}} b^{8} x^{8} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{20} b^{4} + 240 a^{19} b^{5} x + 600 a^{18} b^{6} x^{2} + 800 a^{17} b^{7} x^{3} + 600 a^{16} b^{8} x^{4} + 240 a^{15} b^{9} x^{5} + 40 a^{14} b^{10} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 62, normalized size = 0.89 \begin {gather*} \frac {3 \, a^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{4}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {8}{3}} b^{28} - 24 \, {\left (b x + a\right )}^{\frac {5}{3}} a b^{28} + 60 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2} b^{28}\right )}}{40 \, b^{32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 56, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{8/3}}{8\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{2/3}}{2\,b^4}+\frac {3\,a^3}{b^4\,{\left (a+b\,x\right )}^{1/3}}-\frac {9\,a\,{\left (a+b\,x\right )}^{5/3}}{5\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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