Optimal. Leaf size=1225 \[ \frac {\left (-3272028760 b^3-319355415288 b^2 c^8+1032601284 a b^2 c^4 x+5423886846 a b c^{12} x+6544057520 a^2 b^2 x^2+470457082368 a^2 b c^8 x^2-7231849128 a^3 c^{12} x^3+176421405888 a^4 c^8 x^4\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-748701954 b^2 c^7-10460353203 b c^{15}-1204701498 a b^2 c^3 x-4519905705 a b c^{11} x+20920706406 a^2 c^{15} x^2+6026540940 a^3 c^{11} x^3\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (820006902 b^2 c^6+3486784401 b c^{14}+1472412942 a b^2 c^2 x+3917251611 a b c^{10} x-6973568802 a^2 c^{14} x^2-5223002148 a^3 c^{10} x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-911118780 b^2 c^5-2324522934 b c^{13}-1963217256 a b^2 c x-3482001432 a b c^9 x+4649045868 a^2 c^{13} x^2+4642668576 a^3 c^9 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (1032601284 b^2 c^4+1807962282 b c^{12}+6544057520 a b^2 x+558667785312 a b c^8 x-7231849128 a^2 c^{12} x^2+176421405888 a^3 c^8 x^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-1204701498 b^2 c^3-1506635235 b c^{11}+20920706406 a c^{15} x+6026540940 a^2 c^{11} x^2\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (1472412942 b^2 c^2+1305750537 b c^{10}-6973568802 a c^{14} x-5223002148 a^2 c^{10} x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-1963217256 b^2 c-1160667144 b c^9+4649045868 a c^{13} x+4642668576 a^2 c^9 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{161719622064 a c^8 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-\frac {21505 b^2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{531441 \sqrt {3} a c^{26/3}}+\frac {2 b \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a c^{2/3}}+\frac {21505 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{1594323 a c^{26/3}}-\frac {2 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{3 a c^{2/3}}-\frac {21505 b^2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3188646 a c^{26/3}}+\frac {b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3 a c^{2/3}} \]
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Rubi [F]
time = 0.73, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx &=\int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\\ \end {align*}
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Mathematica [A]
time = 4.29, size = 1219, normalized size = 1.00 \begin {gather*} \frac {\frac {3 c^{2/3} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \left (-3272028760 b^3+494 b^2 \left (-646468452 c^8-1515591 c^7 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1659933 c^6 \sqrt {a x+\sqrt {-b+a^2 x^2}}-1844370 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+2090286 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )+13247080 a x \left (a x+\sqrt {-b+a^2 x^2}\right )-2438667 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+2980593 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}-3974124 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}\right )+3188646 a c^8 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (55328 a^2 x^2+729 c^5 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (9 c^2-3 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+2 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+14 a c x \left (-162 c^3+135 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-117 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+104 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )+531441 b c^8 \left (885248 a^2 x^2+7 a x \left (1458 c^4+150176 \sqrt {-b+a^2 x^2}-1215 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1053 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-936 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )-3 c \left (-1134 c^3 \sqrt {-b+a^2 x^2}+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+945 c^2 \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-2187 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}-819 c \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}+1458 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+728 \sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-6544057520 \sqrt {3} b^2 \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+323439244128 \sqrt {3} b c^8 \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+6544057520 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-323439244128 b c^8 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-3272028760 b^2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )+161719622064 b c^8 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{485158866192 a c^{26/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}\, dx\]
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 = 0.47, size = 719, normalized size = 0.59 \begin {gather*} \frac {304304 \, \sqrt {3} {\left (1062882 \, b^{2} c^{9} - 21505 \, b^{3} c\right )} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} c \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} - 2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}}}{3 \, c^{2}}\right ) + 152152 \, {\left (1062882 \, b^{2} c^{8} - 21505 \, b^{3}\right )} \left (-c^{2}\right )^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c - \left (-c^{2}\right )^{\frac {1}{3}} c + \left (-c^{2}\right )^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 304304 \, {\left (1062882 \, b^{2} c^{8} - 21505 \, b^{3}\right )} \left (-c^{2}\right )^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - \left (-c^{2}\right )^{\frac {2}{3}}\right ) + 3 \, {\left (10460353203 \, b c^{17} - 1497403908 \, a^{2} b c^{9} x^{2} + 748701954 \, b^{2} c^{9} + 567 \, {\left (2657205 \, a b c^{13} - 2124694 \, a b^{2} c^{5}\right )} x - 2 \, {\left (35937693792 \, a^{3} c^{10} x^{3} + 903981141 \, b c^{14} - 1032601284 \, a^{2} b c^{6} x^{2} + 516300642 \, b^{2} c^{6} - 6916 \, {\left (28874961 \, a b c^{10} + 236555 \, a b^{2} c^{2}\right )} x - 988 \, {\left (36374184 \, a^{2} c^{10} x^{2} - 161617113 \, b c^{10} - 1045143 \, a b c^{6} x - 1655885 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 567 \, {\left (2657205 \, b c^{13} + 2640924 \, a b c^{9} x + 2124694 \, b^{2} c^{5}\right )} \sqrt {a^{2} x^{2} - b} + 6 \, {\left (387420489 \, b c^{15} - 303706260 \, a^{2} b c^{7} x^{2} + 151853130 \, b^{2} c^{7} + 364 \, {\left (531441 \, a b c^{11} - 898909 \, a b^{2} c^{3}\right )} x + 52 \, {\left (3720087 \, b c^{11} + 5840505 \, a b c^{7} x + 6292363 \, b^{2} c^{3}\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (387420489 \, b c^{16} - 182223756 \, a^{2} b c^{8} x^{2} + 91111878 \, b^{2} c^{8} + 91 \, {\left (1594323 \, a b c^{12} - 1797818 \, a b^{2} c^{4}\right )} x + 13 \, {\left (11160261 \, b c^{12} + 14017212 \, a b c^{8} x + 12584726 \, b^{2} c^{4}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{485158866192 \, a b c^{10}} \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 {\sqrt [3]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a^{2} x^{2} - b}}{\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}\, 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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}\,\sqrt {a^2\,x^2-b}}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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