Optimal. Leaf size=72 \[ -\frac {3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac {9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac {9 a (a+b x)^{13/3}}{13 b^4}+\frac {3 (a+b x)^{16/3}}{16 b^4} \]
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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)^{7/3}}{7 b^4}+\frac {9 a^2 (a+b x)^{10/3}}{10 b^4}+\frac {3 (a+b x)^{16/3}}{16 b^4}-\frac {9 a (a+b x)^{13/3}}{13 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^3 (a+b x)^{4/3} \, dx &=\int \left (-\frac {a^3 (a+b x)^{4/3}}{b^3}+\frac {3 a^2 (a+b x)^{7/3}}{b^3}-\frac {3 a (a+b x)^{10/3}}{b^3}+\frac {(a+b x)^{13/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac {9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac {9 a (a+b x)^{13/3}}{13 b^4}+\frac {3 (a+b x)^{16/3}}{16 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.64 \begin {gather*} \frac {3 (a+b x)^{7/3} \left (-81 a^3+189 a^2 b x-315 a b^2 x^2+455 b^3 x^3\right )}{7280 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 50, normalized size = 0.69
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (-455 b^{3} x^{3}+315 a \,b^{2} x^{2}-189 a^{2} b x +81 a^{3}\right )}{7280 b^{4}}\) | \(43\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {16}{3}}}{16}-\frac {9 a \left (b x +a \right )^{\frac {13}{3}}}{13}+\frac {9 a^{2} \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a^{3} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{4}}\) | \(50\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {16}{3}}}{16}-\frac {9 a \left (b x +a \right )^{\frac {13}{3}}}{13}+\frac {9 a^{2} \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a^{3} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{4}}\) | \(50\) |
trager | \(-\frac {3 \left (-455 b^{5} x^{5}-595 a \,b^{4} x^{4}-14 a^{2} b^{3} x^{3}+18 a^{3} b^{2} x^{2}-27 a^{4} b x +81 a^{5}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7280 b^{4}}\) | \(65\) |
risch | \(-\frac {3 \left (-455 b^{5} x^{5}-595 a \,b^{4} x^{4}-14 a^{2} b^{3} x^{3}+18 a^{3} b^{2} x^{2}-27 a^{4} b x +81 a^{5}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7280 b^{4}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 56, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {16}{3}}}{16 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {13}{3}} a}{13 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {10}{3}} a^{2}}{10 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{3}}{7 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.68, size = 64, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (455 \, b^{5} x^{5} + 595 \, a b^{4} x^{4} + 14 \, a^{2} b^{3} x^{3} - 18 \, a^{3} b^{2} x^{2} + 27 \, a^{4} b x - 81 \, a^{5}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{7280 \, b^{4}} \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. 1844 vs.
\(2 (68) = 136\).
time = 1.40, size = 1844, normalized size = 25.61 \begin {gather*} - \frac {243 a^{\frac {76}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {243 a^{\frac {76}{3}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} - \frac {1377 a^{\frac {73}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {1458 a^{\frac {73}{3}} b x}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} - \frac {3213 a^{\frac {70}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {3645 a^{\frac {70}{3}} b^{2} x^{2}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} - \frac {3927 a^{\frac {67}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {4860 a^{\frac {67}{3}} b^{3} x^{3}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} - \frac {798 a^{\frac {64}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {3645 a^{\frac {64}{3}} b^{4} x^{4}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {11382 a^{\frac {61}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {1458 a^{\frac {61}{3}} b^{5} x^{5}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {35238 a^{\frac {58}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {243 a^{\frac {58}{3}} b^{6} x^{6}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {56562 a^{\frac {55}{3}} b^{7} x^{7} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {54273 a^{\frac {52}{3}} b^{8} x^{8} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {31227 a^{\frac {49}{3}} b^{9} x^{9} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {9975 a^{\frac {46}{3}} b^{10} x^{10} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} + \frac {1365 a^{\frac {43}{3}} b^{11} x^{11} \sqrt [3]{1 + \frac {b x}{a}}}{7280 a^{20} b^{4} + 43680 a^{19} b^{5} x + 109200 a^{18} b^{6} x^{2} + 145600 a^{17} b^{7} x^{3} + 109200 a^{16} b^{8} x^{4} + 43680 a^{15} b^{9} x^{5} + 7280 a^{14} b^{10} x^{6}} \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. 193 vs.
\(2 (56) = 112\).
time = 0.85, size = 193, normalized size = 2.68 \begin {gather*} \frac {3 \, {\left (\frac {52 \, {\left (14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}\right )} a^{2}}{b^{3}} + \frac {32 \, {\left (35 \, {\left (b x + a\right )}^{\frac {13}{3}} - 182 \, {\left (b x + a\right )}^{\frac {10}{3}} a + 390 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2} - 455 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{3} + 455 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{4}\right )} a}{b^{3}} + \frac {5 \, {\left (91 \, {\left (b x + a\right )}^{\frac {16}{3}} - 560 \, {\left (b x + a\right )}^{\frac {13}{3}} a + 1456 \, {\left (b x + a\right )}^{\frac {10}{3}} a^{2} - 2080 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{3} + 1820 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{4} - 1456 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{5}\right )}}{b^{3}}\right )}}{7280 \, b} \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 = 0.78 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{16/3}}{16\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{7/3}}{7\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{10/3}}{10\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{13/3}}{13\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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