Optimal. Leaf size=73 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1+\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \]
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Rubi [A]
time = 0.14, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2165, 212}
\begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (x \sqrt [3]{\frac {b}{a}}+1\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2165
Rubi steps
\begin {align*} \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}}\right )}{\sqrt [3]{\frac {b}{a}}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1+\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}}\\ \end {align*}
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Mathematica [A]
time = 7.42, size = 82, normalized size = 1.12 \begin {gather*} -\frac {2 \sqrt [6]{\frac {b}{a}} \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {b}{a}} \sqrt {b+\frac {2 b}{\sqrt {3}}} \sqrt {a+b x^3}}{a \left (\frac {b}{a}\right )^{2/3}+b x}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\sqrt {3}}{\left (1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}\right ) \sqrt {b \,x^{3}+a}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.78, size = 1273, normalized size = 17.44 \begin {gather*} \left [\frac {1}{2} \, \sqrt {\frac {1}{3}} \sqrt {\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}} \log \left (\frac {b^{8} x^{24} - 1840 \, a b^{7} x^{21} + 67264 \, a^{2} b^{6} x^{18} - 58624 \, a^{3} b^{5} x^{15} + 504064 \, a^{4} b^{4} x^{12} + 2140160 \, a^{5} b^{3} x^{9} + 3100672 \, a^{6} b^{2} x^{6} + 1089536 \, a^{7} b x^{3} + 28672 \, a^{8} + 4 \, \sqrt {\frac {1}{3}} {\left (486 \, a b^{7} x^{20} - 28512 \, a^{2} b^{6} x^{17} + 86832 \, a^{3} b^{5} x^{14} - 145152 \, a^{4} b^{4} x^{11} - 238464 \, a^{5} b^{3} x^{8} - 414720 \, a^{6} b^{2} x^{5} - 82944 \, a^{7} b x^{2} + {\left (3 \, a b^{7} x^{22} - 2688 \, a^{2} b^{6} x^{19} + 56952 \, a^{3} b^{5} x^{16} - 93504 \, a^{4} b^{4} x^{13} - 63552 \, a^{5} b^{3} x^{10} - 377856 \, a^{6} b^{2} x^{7} - 314880 \, a^{7} b x^{4} - 24576 \, a^{8} x - 2 \, \sqrt {3} {\left (a b^{7} x^{22} - 764 \, a^{2} b^{6} x^{19} + 16860 \, a^{3} b^{5} x^{16} - 19792 \, a^{4} b^{4} x^{13} + 42368 \, a^{5} b^{3} x^{10} + 104448 \, a^{6} b^{2} x^{7} + 90880 \, a^{7} b x^{4} + 7168 \, a^{8} x\right )}\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} - 6 \, \sqrt {3} {\left (47 \, a b^{7} x^{20} - 2724 \, a^{2} b^{6} x^{17} + 8976 \, a^{3} b^{5} x^{14} - 4928 \, a^{4} b^{4} x^{11} + 32448 \, a^{5} b^{3} x^{8} + 37632 \, a^{6} b^{2} x^{5} + 8192 \, a^{7} b x^{2}\right )} - 2 \, {\left (30 \, a b^{7} x^{21} - 5010 \, a^{2} b^{6} x^{18} + 44640 \, a^{3} b^{5} x^{15} - 21360 \, a^{4} b^{4} x^{12} + 79872 \, a^{5} b^{3} x^{9} + 233856 \, a^{6} b^{2} x^{6} + 86016 \, a^{7} b x^{3} + 3072 \, a^{8} - \sqrt {3} {\left (17 \, a b^{7} x^{21} - 2920 \, a^{2} b^{6} x^{18} + 24864 \, a^{3} b^{5} x^{15} - 26576 \, a^{4} b^{4} x^{12} - 56000 \, a^{5} b^{3} x^{9} - 115968 \, a^{6} b^{2} x^{6} - 56320 \, a^{7} b x^{3} - 1024 \, a^{8}\right )}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \sqrt {b x^{3} + a} \sqrt {\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}} - 8 \, {\left (3 \, a b^{7} x^{23} - 1077 \, a^{2} b^{6} x^{20} + 13320 \, a^{3} b^{5} x^{17} - 19200 \, a^{4} b^{4} x^{14} - 111360 \, a^{5} b^{3} x^{11} - 345024 \, a^{6} b^{2} x^{8} - 328704 \, a^{7} b x^{5} - 61440 \, a^{8} x^{2} - 2 \, \sqrt {3} {\left (a b^{7} x^{23} - 299 \, a^{2} b^{6} x^{20} + 4260 \, a^{3} b^{5} x^{17} + 1520 \, a^{4} b^{4} x^{14} + 26720 \, a^{5} b^{3} x^{11} + 105024 \, a^{6} b^{2} x^{8} + 93184 \, a^{7} b x^{5} + 17920 \, a^{8} x^{2}\right )}\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + 32 \, \sqrt {3} {\left (35 \, a b^{7} x^{21} - 1141 \, a^{2} b^{6} x^{18} + 2544 \, a^{3} b^{5} x^{15} + 6760 \, a^{4} b^{4} x^{12} + 39520 \, a^{5} b^{3} x^{9} + 55680 \, a^{6} b^{2} x^{6} + 19712 \, a^{7} b x^{3} + 512 \, a^{8}\right )} + 32 \, {\left (9 \, a b^{7} x^{22} - 846 \, a^{2} b^{6} x^{19} + 4617 \, a^{3} b^{5} x^{16} + 5472 \, a^{4} b^{4} x^{13} + 43776 \, a^{5} b^{3} x^{10} + 98496 \, a^{6} b^{2} x^{7} + 59328 \, a^{7} b x^{4} + 4608 \, a^{8} x - \sqrt {3} {\left (5 \, a b^{7} x^{22} - 505 \, a^{2} b^{6} x^{19} + 2130 \, a^{3} b^{5} x^{16} - 4928 \, a^{4} b^{4} x^{13} - 28688 \, a^{5} b^{3} x^{10} - 53760 \, a^{6} b^{2} x^{7} - 35200 \, a^{7} b x^{4} - 2560 \, a^{8} x\right )}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b^{8} x^{24} + 80 \, a b^{7} x^{21} + 2368 \, a^{2} b^{6} x^{18} + 30080 \, a^{3} b^{5} x^{15} + 121984 \, a^{4} b^{4} x^{12} - 240640 \, a^{5} b^{3} x^{9} + 151552 \, a^{6} b^{2} x^{6} - 40960 \, a^{7} b x^{3} + 4096 \, a^{8}}\right ), \sqrt {\frac {1}{3}} \sqrt {-\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b x^{2} + 2 \, {\left (\sqrt {3} a x - 2 \, a x\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + 2 \, {\left (\sqrt {3} a - a\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}}}{2 \, \sqrt {b x^{3} + a}}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt [3]{\frac {b}{a}} + 1 + \sqrt {3}}{\sqrt {a + b x^{3}} \left (x \sqrt [3]{\frac {b}{a}} - \sqrt {3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {b\,x^3+a}\,\left (x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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