Optimal. Leaf size=803 \[ \frac {\left (363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{1917583360 a c^7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/3}}-\frac {49725 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}+\frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}}+\frac {49725 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}} \]
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Rubi [F]
time = 0.78, 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]{a x+\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 [3]{a x+\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 [3]{a x+\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 = 2.63, size = 803, normalized size = 1.00 \begin {gather*} \frac {\left (363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 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 (-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 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 (238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 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 (290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 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 )}{1917583360 a c^7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/3}}-\frac {49725 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}+\frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}}+\frac {49725 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}\, dx\]
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 = 0.44, size = 1121, normalized size = 1.40 \begin {gather*} -\frac {87780 \, a b c^{7} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (5070602400912917605986812821504 \, b^{6} c^{36} - 3847285528341595885584934502400 \, b^{7} c^{30} + 1216292526860465084144025600000 \, b^{8} c^{24} - 205078619717531420590080000000 \, b^{9} c^{18} + 19450253230007648256000000000 \, b^{10} c^{12} - 983849936790090240000000000 \, b^{11} c^{6} + 20735820391713136962890625 \, 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^{39} - 149294327647331942400 \, a^{2} b^{5} c^{33} + 28319017190031360000 \, a^{2} b^{6} c^{27} - 2387429351424000000 \, a^{2} b^{7} c^{21} + 75476916312890625 \, a^{2} b^{8} c^{15}\right )} \sqrt {\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}}} a c^{7} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {1}{4}} - {\left (2251799813685248 \, a b^{3} c^{25} - 854268995174400 \, a b^{4} c^{19} + 108028477440000 \, a b^{5} c^{13} - 4553660109375 \, a b^{6} c^{7}\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} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {1}{4}}}{295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}\right ) + 21945 \, a b c^{7} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {1}{4}} \log \left (27 \, a^{3} c^{22} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {3}{4}} + 27 \, {\left (2251799813685248 \, b^{3} c^{18} - 854268995174400 \, b^{4} c^{12} + 108028477440000 \, b^{5} c^{6} - 4553660109375 \, b^{6}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 21945 \, a b c^{7} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {1}{4}} \log \left (-27 \, a^{3} c^{22} \left (\frac {295147905179352825856 \, b^{4} c^{24} - 149294327647331942400 \, b^{5} c^{18} + 28319017190031360000 \, b^{6} c^{12} - 2387429351424000000 \, b^{7} c^{6} + 75476916312890625 \, b^{8}}{a^{4} c^{29}}\right )^{\frac {3}{4}} + 27 \, {\left (2251799813685248 \, b^{3} c^{18} - 854268995174400 \, b^{4} c^{12} + 108028477440000 \, b^{5} c^{6} - 4553660109375 \, 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 (536870912 \, b c^{11} + 428032000 \, a^{2} b c^{5} x^{2} - 214016000 \, b^{2} c^{5} - 2464 \, {\left (131072 \, a b c^{8} + 104975 \, a b^{2} c^{2}\right )} x - 3 \, {\left (273940480 \, a^{3} c^{6} x^{3} - 117440512 \, b c^{9} - 159174400 \, a^{2} b c^{3} x^{2} + 79587200 \, b^{2} c^{3} - 17765 \, {\left (65536 \, a b c^{6} - 6825 \, a b^{2}\right )} x - 1045 \, {\left (262144 \, a^{2} c^{6} x^{2} - 983040 \, b c^{6} - 152320 \, a b c^{3} x + 116025 \, b^{2}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 352 \, {\left (917504 \, b c^{8} + 1216000 \, a b c^{5} x - 734825 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b} - 12 \, {\left (33554432 \, b c^{10} + 37452800 \, a^{2} b c^{4} x^{2} - 18726400 \, b^{2} c^{4} - 385 \, {\left (65536 \, a b c^{7} + 62985 \, a b^{2} c\right )} x - 385 \, {\left (65536 \, b c^{7} + 97280 \, a b c^{4} x - 62985 \, b^{2} 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}}}{7670333440 \, a b c^{7}} \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}}} \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 (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,{\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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