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3.32.31 \(\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3131]

Optimal. Leaf size=787 \[ \frac {\left (-1378263040 b^2 c^5+2684354560 b c^{11}+1665760096 a b^2 c^2 x-4844421120 a b c^8 x-5368709120 a^2 c^{11} x^2+6459228160 a^3 c^8 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (1447176192 b^2 c^4-2013265920 b c^{10}-1873980108 a b^2 c x+4541644800 a b c^7 x+4026531840 a^2 c^{10} x^2-6055526400 a^3 c^7 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-1537624704 b^2 c^3+1761607680 b c^9+2342475135 a b^2 x-4314562560 a b c^6 x-3523215360 a^2 c^9 x^2+5752750080 a^3 c^6 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (1665760096 b^2 c^2-1614807040 b c^8-5368709120 a c^{11} x+6459228160 a^2 c^8 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-1873980108 b^2 c+1513881600 b c^7+4026531840 a c^{10} x-6055526400 a^2 c^7 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (2342475135 b^2-1438187520 b c^6-3523215360 a c^9 x+5752750080 a^2 c^6 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{22052208640 a^2 c^6 x \sqrt {-b+a^2 x^2}+11026104320 a c^6 \left (-b+2 a^2 x^2\right )}+\frac {13923 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}-\frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}}-\frac {13923 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}} \]

[Out]

((-5368709120*a^2*c^11*x^2+6459228160*a^3*c^8*x^3+2684354560*b*c^11-4844421120*a*b*c^8*x-1378263040*b^2*c^5+16
65760096*a*b^2*c^2*x)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(4026531840*a^2*c^10*x^2-6055526400*a^3*c^7*x^3-
2013265920*b*c^10+4541644800*a*b*c^7*x+1447176192*b^2*c^4-1873980108*a*b^2*c*x)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*
(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(-3523215360*a^2*c^9*x^2+5752750080*a^3*c^6*x^3+1761607680*b*c^9-43145
62560*a*b*c^6*x-1537624704*b^2*c^3+2342475135*a*b^2*x)*(a*x+(a^2*x^2-b)^(1/2))^(2/3)*(c+(a*x+(a^2*x^2-b)^(1/2)
)^(1/3))^(3/4)+(a^2*x^2-b)^(1/2)*((-5368709120*a*c^11*x+6459228160*a^2*c^8*x^2-1614807040*b*c^8+1665760096*b^2
*c^2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(4026531840*a*c^10*x-6055526400*a^2*c^7*x^2+1513881600*b*c^7-187
3980108*b^2*c)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(-3523215360*a*c^9*x+5752
750080*a^2*c^6*x^2-1438187520*b*c^6+2342475135*b^2)*(a*x+(a^2*x^2-b)^(1/2))^(2/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(
1/3))^(3/4)))/(22052208640*a^2*c^6*x*(a^2*x^2-b)^(1/2)+11026104320*a*c^6*(2*a^2*x^2-b))+13923/131072*b^2*arcta
n((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(25/4)-3*b*arctan((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/
4)/c^(1/4))/a/c^(1/4)-13923/131072*b^2*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(25/4)+3*b
*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(1/4)

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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.

[In]

Int[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]

[Out]

Defer[Int][Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4), x]

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.

[In]

Integrate[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]

[Out]

((2*c^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)*(1310720*a*c^6*x*(a*x + Sqrt[-b + a^2*x^2])*(-4096*c^
5 + 4928*a*c^2*x + 3072*c^4*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 4620*a*c*x*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 2
688*c^3*(a*x + Sqrt[-b + a^2*x^2])^(2/3) + 4389*a*x*(a*x + Sqrt[-b + a^2*x^2])^(2/3)) + 33649*b^2*(-40960*c^5
+ 43008*c^4*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 45696*c^3*(a*x + Sqrt[-b + a^2*x^2])^(2/3) + 49504*c^2*(a*x + S
qrt[-b + a^2*x^2]) - 55692*c*(a*x + Sqrt[-b + a^2*x^2])^(4/3) + 69615*(a*x + Sqrt[-b + a^2*x^2])^(5/3)) - 3276
80*b*c^6*(-8192*c^5 + 6144*c^4*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 5376*c^3*(a*x + Sqrt[-b + a^2*x^2])^(2/3) +
4928*c^2*(3*a*x + Sqrt[-b + a^2*x^2]) - 4620*c*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(3*a*x + Sqrt[-b + a^2*x^2]) +
 4389*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(3*a*x + Sqrt[-b + a^2*x^2]))))/(-b + 2*a*x*(a*x + Sqrt[-b + a^2*x^2]))
 + 2342475135*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)] - 66156625920*b*c^6*ArcTan[(c +
 (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)] - 2342475135*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/
3))^(1/4)/c^(1/4)] + 66156625920*b*c^6*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2205220
8640*a*c^(25/4))

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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.

[In]

int((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x)

[Out]

int((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x)

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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.

[In]

integrate((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*x^2 - b)/(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4), x)

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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.

[In]

integrate((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x, algorithm="fricas")

[Out]

1/44104417280*(2018940*a*c^6*((295147905179352825856*b^4*c^24 - 41802411741252943872*b^5*c^18 + 22202109476984
58624*b^6*c^12 - 52408849122459648*b^7*c^6 + 463923394732161*b^8)/(a^4*c^25))^(1/4)*arctan((sqrt((507060240091
2917605986812821504*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)*sqrt(c + (a*x + sqrt(a^2*x^2 - b))^(1/3)) + (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)*sqrt((295147905179352825856*b^4*c^24 - 41802411741252943872*b^5*c^18 + 22
20210947698458624*b^6*c^12 - 52408849122459648*b^7*c^6 + 463923394732161*b^8)/(a^4*c^25)))*a*c^6*((29514790517
9352825856*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))^(1/4) - (2251799813685248*a*b^3*c^24 - 239195318648832*a*b^4*c^18 + 846943
2631296*a*b^5*c^12 - 99961946721*a*b^6*c^6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)*((29514790517935282585
6*b^4*c^24 - 41802411741252943872*b^5*c^18 + 2220210947698458624*b^6*c^12 - 52408849122459648*b^7*c^6 + 463923
394732161*b^8)/(a^4*c^25))^(1/4))/(295147905179352825856*b^4*c^24 - 41802411741252943872*b^5*c^18 + 2220210947
698458624*b^6*c^12 - 52408849122459648*b^7*c^6 + 463923394732161*b^8)) + 504735*a*c^6*((295147905179352825856*
b^4*c^24 - 41802411741252943872*b^5*c^18 + 2220210947698458624*b^6*c^12 - 52408849122459648*b^7*c^6 + 46392339
4732161*b^8)/(a^4*c^25))^(1/4)*log(27*a^3*c^19*((295147905179352825856*b^4*c^24 - 41802411741252943872*b^5*c^1
8 + 2220210947698458624*b^6*c^12 - 52408849122459648*b^7*c^6 + 463923394732161*b^8)/(a^4*c^25))^(3/4) + 27*(22
51799813685248*b^3*c^18 - 239195318648832*b^4*c^12 + 8469432631296*b^5*c^6 - 99961946721*b^6)*(c + (a*x + sqrt
(a^2*x^2 - b))^(1/3))^(1/4)) - 504735*a*c^6*((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))^(1/4)*log(-27*a^3
*c^19*((295147905179352825856*b^4*c^24 - 41802411741252943872*b^5*c^18 + 2220210947698458624*b^6*c^12 - 524088
49122459648*b^7*c^6 + 463923394732161*b^8)/(a^4*c^25))^(3/4) + 27*(2251799813685248*b^3*c^18 - 239195318648832
*b^4*c^12 + 8469432631296*b^5*c^6 - 99961946721*b^6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 4*(2684354
560*c^11 + 2756526080*a^2*c^5*x^2 - 1378263040*b*c^5 - 2464*(655360*a*c^8 + 676039*a*b*c^2)*x + 21*(83886080*c
^9 + 146440448*a^2*c^3*x^2 - 73220224*b*c^3 - 1045*(65536*a*c^6 + 106743*a*b)*x - 209*(327680*c^6 + 700672*a*c
^3*x - 533715*b)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3) - 2464*(655360*c^8 + 1118720*a*c^5*x - 676
039*b*c^2)*sqrt(a^2*x^2 - b) - 12*(167772160*c^10 + 241196032*a^2*c^4*x^2 - 120598016*b*c^4 - 77*(1638400*a*c^
7 + 2028117*a*b*c)*x - 77*(1638400*c^7 + 3132416*a*c^4*x - 2028117*b*c)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2
 - b))^(1/3))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(3/4))/(a*c^6)

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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.

[In]

integrate((a**2*x**2-b)**(1/2)/(c+(a*x+(a**2*x**2-b)**(1/2))**(1/3))**(1/4),x)

[Out]

Integral(sqrt(a**2*x**2 - b)/(c + (a*x + sqrt(a**2*x**2 - b))**(1/3))**(1/4), x)

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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.

[In]

integrate((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x, algorithm="giac")

[Out]

Timed out

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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.

[In]

int((a^2*x^2 - b)^(1/2)/(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/3))^(1/4),x)

[Out]

int((a^2*x^2 - b)^(1/2)/(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/3))^(1/4), x)

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