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]

3.32.31.1 Optimal result

Integrand size = 45, antiderivative size = 787 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (-1378263040 b^2 c^5+2684354560 b c^{11}[phone] a b^2 c^2 x-4844421120 a b c^8 x-5368709120 a^2 c^{11} x^2+6459228160 a^3 c^8 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (1447176192 b^2 c^4-2013265920 b c^{10}-1873980108 a b^2 c x+4541644800 a b c^7 x+4026531840 a^2 c^{10} x^2-6055526400 a^3 c^7 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-1537624704 b^2 c^3+1761607680 b c^9+2342475135 a b^2 x-4314562560 a b c^6 x-3523215360 a^2 c^9 x^2+5752750080 a^3 c^6 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (1665760096 b^2 c^2-1614807040 b c^8-5368709120 a c^{11} x+6459228160 a^2 c^8 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-1873980108 b^2 c+1513881600 b c^7+4026531840 a c^{10} x-6055526400 a^2 c^7 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (2342475135 b^2-1438187520 b c^6-3523215360 a c^9 x+5752750080 a^2 c^6 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{22052208640 a^2 c^6 x \sqrt {-b+a^2 x^2}[phone] a c^6 \left (-b+2 a^2 x^2\right )}+\frac {13923 b^2 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}-\frac {3 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}}-\frac {13923 b^2 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}} \]

output

((-5368709120*a^2*c^11*x^2+6459228160*a^3*c^8*x^3+2684354560*b*c^11-484442 
1120*a*b*c^8*x-1378263040*b^2*c^5+1665760096*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^3-20 
13265920*b*c^10+4541644800*a*b*c^7*x[phone]*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^9-431456256 
0*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)*((-536870 
9120*a*c^11*x[phone]*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^7-1873980108*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[phone]*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)[phone]*a*c^6*(2*a^2*x^2-b))+13923/131072*b^2*arctan((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)
 

3.32.31.2 Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 721, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (1310720 a c^6 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-4096 c^5+4928 a c^2 x+3072 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-4620 a c x \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-2688 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+4389 a x \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )+33649 b^2 \left (-40960 c^5+43008 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-45696 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+49504 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )-55692 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}+69615 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/3}\right )-327680 b c^6 \left (-8192 c^5+6144 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-5376 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+4928 c^2 \left (3 a x+\sqrt {-b+a^2 x^2}\right )-4620 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )+4389 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (3 a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{-b+2 a x \left (a x+\sqrt {-b+a^2 x^2}\right )}[phone] b^2 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )-66156625920 b c^6 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )-2342475135 b^2 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )[phone] b c^6 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{22052208640 a c^{25/4}} \]

input

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

output

((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 + Sqr 
t[-b + a^2*x^2])^(1/3) - 4620*a*c*x*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 268 
8*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 + Sqrt[ 
-b + a^2*x^2]) - 55692*c*(a*x + Sqrt[-b + a^2*x^2])^(4/3) + 69615*(a*x + S 
qrt[-b + a^2*x^2])^(5/3)) - 327680*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])) + 23424 
75135*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)] - 6 
6156625920*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)])/(22052208640*a*c^(25/4))
 

3.32.31.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.32.31.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.32.31.4 Maple [F]

\[\int \frac {\sqrt {a^{2} x^{2}-b}}{{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )}^{\frac {1}{4}}}d x\]

input

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

output

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

3.32.31.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 948, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Too large to display} \]

input

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

output

1/44104417280*(504735*a*c^6*((295147905179352825856*b^4*c^24 - 41802411741 
252943872*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*((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))^(3 
/4) + 27*(2251799813685248*b^3*c^18 - 239195318648832*b^4*c^12 + 846943263 
1296*b^5*c^6 - 99961946721*b^6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4 
)) - 504735*I*a*c^6*((295147905179352825856*b^4*c^24 - 4180241174125294387 
2*b^5*c^18 + 2220210947698458624*b^6*c^12 - 52408849122459648*b^7*c^6 + 46 
3923394732161*b^8)/(a^4*c^25))^(1/4)*log(27*I*a^3*c^19*((29514790517935282 
5856*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))^(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)) + 5 
04735*I*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*I*a^3*c^19*((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))^(3/4) + 27*(2 
251799813685248*b^3*c^18 - 239195318648832*b^4*c^12 + 8469432631296*b^5...
 

3.32.31.6 Sympy [F]

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

input

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

output

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

3.32.31.7 Maxima [F]

\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b}}{{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.32.31.8 Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.32.31.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b}}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \]

input

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

output

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