Optimal. Leaf size=787 \[ \frac {\left (-1378263040 b^2 c^5+2684354560 b c^{11}+1665760096 a b^2 c^2 x-4844421120 a b c^8 x-5368709120 a^2 c^{11} x^2+6459228160 a^3 c^8 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (1447176192 b^2 c^4-2013265920 b c^{10}-1873980108 a b^2 c x+4541644800 a b c^7 x+4026531840 a^2 c^{10} x^2-6055526400 a^3 c^7 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-1537624704 b^2 c^3+1761607680 b c^9+2342475135 a b^2 x-4314562560 a b c^6 x-3523215360 a^2 c^9 x^2+5752750080 a^3 c^6 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (1665760096 b^2 c^2-1614807040 b c^8-5368709120 a c^{11} x+6459228160 a^2 c^8 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-1873980108 b^2 c+1513881600 b c^7+4026531840 a c^{10} x-6055526400 a^2 c^7 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (2342475135 b^2-1438187520 b c^6-3523215360 a c^9 x+5752750080 a^2 c^6 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{22052208640 a^2 c^6 x \sqrt {-b+a^2 x^2}+11026104320 a c^6 \left (-b+2 a^2 x^2\right )}+\frac {13923 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}-\frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}}-\frac {13923 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}} \]
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Rubi [F]
time = 0.23, 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 [4]{c+\sqrt [3]{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 [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.78, size = 721, normalized size = 0.92 \begin {gather*} \frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (1310720 a c^6 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-4096 c^5+4928 a c^2 x+3072 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-4620 a c x \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-2688 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+4389 a x \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )+33649 b^2 \left (-40960 c^5+43008 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-45696 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+49504 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )-55692 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}+69615 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/3}\right )-327680 b c^6 \left (-8192 c^5+6144 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-5376 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+4928 c^2 \left (3 a x+\sqrt {-b+a^2 x^2}\right )-4620 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )+4389 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (3 a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{-b+2 a x \left (a x+\sqrt {-b+a^2 x^2}\right )}+2342475135 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )-66156625920 b c^6 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )-2342475135 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+66156625920 b c^6 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{22052208640 a c^{25/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}\, dx\]
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 = 0.50, size = 1060, normalized size = 1.35 \begin {gather*} \frac {2018940 \, a c^{6} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (5070602400912917605986812821504 \, b^{6} c^{36} - 1077239947935646847963781660672 \, b^{7} c^{30} + 95357334105860462596891607040 \, b^{8} c^{24} - 4501885860039249744793436160 \, b^{9} c^{18} + 119552148493435810464399360 \, b^{10} c^{12} - 1693241946893419178360832 \, b^{11} c^{6} + 9992390792252042651841 \, b^{12}\right )} \sqrt {c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}} + {\left (295147905179352825856 \, a^{2} b^{4} c^{37} - 41802411741252943872 \, a^{2} b^{5} c^{31} + 2220210947698458624 \, a^{2} b^{6} c^{25} - 52408849122459648 \, a^{2} b^{7} c^{19} + 463923394732161 \, a^{2} b^{8} c^{13}\right )} \sqrt {\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}}} a c^{6} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {1}{4}} - {\left (2251799813685248 \, a b^{3} c^{24} - 239195318648832 \, a b^{4} c^{18} + 8469432631296 \, a b^{5} c^{12} - 99961946721 \, a b^{6} c^{6}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {1}{4}}}{295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}\right ) + 504735 \, a c^{6} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {1}{4}} \log \left (27 \, a^{3} c^{19} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {3}{4}} + 27 \, {\left (2251799813685248 \, b^{3} c^{18} - 239195318648832 \, b^{4} c^{12} + 8469432631296 \, b^{5} c^{6} - 99961946721 \, b^{6}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 504735 \, a c^{6} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {1}{4}} \log \left (-27 \, a^{3} c^{19} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 41802411741252943872 \, b^{5} c^{18} + 2220210947698458624 \, b^{6} c^{12} - 52408849122459648 \, b^{7} c^{6} + 463923394732161 \, b^{8}}{a^{4} c^{25}}\right )^{\frac {3}{4}} + 27 \, {\left (2251799813685248 \, b^{3} c^{18} - 239195318648832 \, b^{4} c^{12} + 8469432631296 \, b^{5} c^{6} - 99961946721 \, b^{6}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (2684354560 \, c^{11} + 2756526080 \, a^{2} c^{5} x^{2} - 1378263040 \, b c^{5} - 2464 \, {\left (655360 \, a c^{8} + 676039 \, a b c^{2}\right )} x + 21 \, {\left (83886080 \, c^{9} + 146440448 \, a^{2} c^{3} x^{2} - 73220224 \, b c^{3} - 1045 \, {\left (65536 \, a c^{6} + 106743 \, a b\right )} x - 209 \, {\left (327680 \, c^{6} + 700672 \, a c^{3} x - 533715 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 2464 \, {\left (655360 \, c^{8} + 1118720 \, a c^{5} x - 676039 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - 12 \, {\left (167772160 \, c^{10} + 241196032 \, a^{2} c^{4} x^{2} - 120598016 \, b c^{4} - 77 \, {\left (1638400 \, a c^{7} + 2028117 \, a b c\right )} x - 77 \, {\left (1638400 \, c^{7} + 3132416 \, a c^{4} x - 2028117 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}}{44104417280 \, a c^{6}} \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 {a^{2} x^{2} - b}}{\sqrt [4]{c + \sqrt [3]{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 {\sqrt {a^2\,x^2-b}}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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