Optimal. Leaf size=278 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.31, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2166, 224,
2165, 212} \begin {gather*} \frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 224
Rule 2165
Rule 2166
Rubi steps
\begin {align*} \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx &=\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \left (-22 a b+\left (1-\sqrt {3}\right )^3 a b\right )+6 a b^{4/3} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx}{6 \left (3+\sqrt {3}\right ) a b^{4/3}}+\frac {\left (2+\sqrt {3}\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{\left (3+\sqrt {3}\right ) \sqrt [3]{b}}\\ &=\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\left (2 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {a+b x^3}}\right )}{\left (3+\sqrt {3}\right ) b^{2/3}}\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.81, size = 427, normalized size = 1.54 \begin {gather*} -\frac {4 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-\frac {i \sqrt [4]{3} \left (\left ((-2-i)+\sqrt {3}\right ) \sqrt [3]{a}+\left ((1+2 i)-i \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {i+\sqrt {3}-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{2 \sqrt {2}}+i \left (-1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (3-(2-i) \sqrt {3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right ) \sqrt {b \,x^{3}+a}}\, dx\]
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.85, size = 1288, normalized size = 4.63 \begin {gather*} \left [\frac {\sqrt {2} a^{\frac {1}{3}} b^{\frac {4}{3}} \sqrt {\frac {\sqrt {3}}{a}} \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} - 2 \, \sqrt {2} {\left (26 \, a b^{7} x^{21} - 4180 \, a^{2} b^{6} x^{18} + 39552 \, a^{3} b^{5} x^{15} + 10432 \, a^{4} b^{4} x^{12} + 271744 \, a^{5} b^{3} x^{9} + 699648 \, a^{6} b^{2} x^{6} + 284672 \, a^{7} b x^{3} + 8192 \, a^{8} - {\left (b^{7} x^{22} - 1160 \, a b^{6} x^{19} + 23232 \, a^{2} b^{5} x^{16} - 53920 \, a^{3} b^{4} x^{13} - 148288 \, a^{4} b^{3} x^{10} - 586752 \, a^{5} b^{2} x^{7} - 496640 \, a^{6} b x^{4} - 38912 \, a^{7} x - \sqrt {3} {\left (b^{7} x^{22} - 632 \, a b^{6} x^{19} + 14736 \, a^{2} b^{5} x^{16} - 8416 \, a^{3} b^{4} x^{13} + 105920 \, a^{4} b^{3} x^{10} + 334848 \, a^{5} b^{2} x^{7} + 286720 \, a^{6} b x^{4} + 22528 \, a^{7} x\right )}\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} - 12 \, {\left (17 \, a b^{6} x^{20} - 1014 \, a^{2} b^{5} x^{17} + 2748 \, a^{3} b^{4} x^{14} - 9632 \, a^{4} b^{3} x^{11} - 36096 \, a^{5} b^{2} x^{8} - 53376 \, a^{6} b x^{5} - 11008 \, a^{7} x^{2} - 2 \, \sqrt {3} {\left (5 \, a b^{6} x^{20} - 285 \, a^{2} b^{5} x^{17} + 1038 \, a^{3} b^{4} x^{14} + 784 \, a^{4} b^{3} x^{11} + 11424 \, a^{5} b^{2} x^{8} + 15168 \, a^{6} b x^{5} + 3200 \, a^{7} x^{2}\right )}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} - 2 \, \sqrt {3} {\left (7 \, a b^{7} x^{21} - 1250 \, a^{2} b^{6} x^{18} + 9984 \, a^{3} b^{5} x^{15} - 19456 \, a^{4} b^{4} x^{12} - 82624 \, a^{5} b^{3} x^{9} - 193920 \, a^{6} b^{2} x^{6} - 84992 \, a^{7} b x^{3} - 2048 \, a^{8}\right )}\right )} \sqrt {b x^{3} + a} \sqrt {\frac {\sqrt {3}}{a}} + 32 \, {\left (9 \, b^{7} x^{22} - 846 \, a b^{6} x^{19} + 4617 \, a^{2} b^{5} x^{16} + 5472 \, a^{3} b^{4} x^{13} + 43776 \, a^{4} b^{3} x^{10} + 98496 \, a^{5} b^{2} x^{7} + 59328 \, a^{6} b x^{4} + 4608 \, a^{7} x - \sqrt {3} {\left (5 \, b^{7} x^{22} - 505 \, a b^{6} x^{19} + 2130 \, a^{2} b^{5} x^{16} - 4928 \, a^{3} b^{4} x^{13} - 28688 \, a^{4} b^{3} x^{10} - 53760 \, a^{5} b^{2} x^{7} - 35200 \, a^{6} b x^{4} - 2560 \, a^{7} x\right )}\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} - 8 \, {\left (3 \, b^{7} x^{23} - 1077 \, a b^{6} x^{20} + 13320 \, a^{2} b^{5} x^{17} - 19200 \, a^{3} b^{4} x^{14} - 111360 \, a^{4} b^{3} x^{11} - 345024 \, a^{5} b^{2} x^{8} - 328704 \, a^{6} b x^{5} - 61440 \, a^{7} x^{2} - 2 \, \sqrt {3} {\left (b^{7} x^{23} - 299 \, a b^{6} x^{20} + 4260 \, a^{2} b^{5} x^{17} + 1520 \, a^{3} b^{4} x^{14} + 26720 \, a^{4} b^{3} x^{11} + 105024 \, a^{5} b^{2} x^{8} + 93184 \, a^{6} b x^{5} + 17920 \, a^{7} x^{2}\right )}\right )} a^{\frac {1}{3}} b^{\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 )}}{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 ) + 4 \, b^{\frac {7}{6}} {\left (\sqrt {3} + 3\right )} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )}{12 \, b^{2}}, \frac {\sqrt {2} a^{\frac {1}{3}} b^{\frac {4}{3}} \sqrt {-\frac {\sqrt {3}}{a}} \arctan \left (\frac {2 \, \sqrt {2} {\left (\sqrt {3} x - 3 \, x\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {-\frac {\sqrt {3}}{a}} + \sqrt {2} {\left (\sqrt {3} x^{2} + 3 \, x^{2}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} \sqrt {-\frac {\sqrt {3}}{a}} + 4 \, \sqrt {3} \sqrt {2} a \sqrt {-\frac {\sqrt {3}}{a}}}{12 \, \sqrt {b x^{3} + a}}\right ) + 2 \, b^{\frac {7}{6}} {\left (\sqrt {3} + 3\right )} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )}{6 \, b^{2}}\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 {a + b x^{3}} \left (- \sqrt {3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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