Optimal. Leaf size=1225 \[ -\frac {21505 \tan ^{-1}\left (\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right ) b^2}{531441 \sqrt {3} a c^{26/3}}+\frac {21505 \log \left (\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}-\sqrt [3]{c}\right ) b^2}{1594323 a c^{26/3}}-\frac {21505 \log \left (c^{2/3}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \sqrt [3]{c}+\left (c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\right )^{2/3}\right ) b^2}{3188646 a c^{26/3}}+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right ) b}{\sqrt {3} a c^{2/3}}-\frac {2 \log \left (\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}-\sqrt [3]{c}\right ) b}{3 a c^{2/3}}+\frac {\log \left (c^{2/3}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \sqrt [3]{c}+\left (c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\right )^{2/3}\right ) b}{3 a c^{2/3}}+\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \left (4649045868 a^2 x^2 c^{13}-2324522934 b c^{13}+4642668576 a^3 x^3 c^9-3482001432 a b x c^9-911118780 b^2 c^5-1963217256 a b^2 x c\right )+\sqrt {a^2 x^2-b} \left (\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \left (4649045868 a x c^{13}+4642668576 a^2 x^2 c^9-1160667144 b c^9-1963217256 b^2 c\right )+\left (-7231849128 a^2 x^2 c^{12}+1807962282 b c^{12}+176421405888 a^3 x^3 c^8+558667785312 a b x c^8+1032601284 b^2 c^4+6544057520 a b^2 x\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-6973568802 a x c^{14}-5223002148 a^2 x^2 c^{10}+1305750537 b c^{10}+1472412942 b^2 c^2\right ) \sqrt {a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (20920706406 a x c^{15}+6026540940 a^2 x^2 c^{11}-1506635235 b c^{11}-1204701498 b^2 c^3\right ) \sqrt [4]{a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}\right )+\left (-7231849128 a^3 x^3 c^{12}+5423886846 a b x c^{12}+176421405888 a^4 x^4 c^8-319355415288 b^2 c^8+470457082368 a^2 b x^2 c^8+1032601284 a b^2 x c^4-3272028760 b^3+6544057520 a^2 b^2 x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-6973568802 a^2 x^2 c^{14}+3486784401 b c^{14}-5223002148 a^3 x^3 c^{10}+3917251611 a b x c^{10}+820006902 b^2 c^6+1472412942 a b^2 x c^2\right ) \sqrt {a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (20920706406 a^2 x^2 c^{15}-10460353203 b c^{15}+6026540940 a^3 x^3 c^{11}-4519905705 a b x c^{11}-748701954 b^2 c^7-1204701498 a b^2 x c^3\right ) \sqrt [4]{a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{161719622064 a c^8 \left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}} \]
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Rubi [F] time = 0.95, 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 [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.09, size = 1225, normalized size = 1.00 \begin {gather*} \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 \tan ^{-1}\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 \tan ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, 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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giac [F(-1)] 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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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*} \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}}}\,{d x} \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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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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