3.32.31 \(\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)
Optimal. Leaf size=787 \[ \frac {13923 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}-\frac {13923 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{131072 a c^{25/4}}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{a \sqrt [4]{c}}+\frac {\sqrt {a^2 x^2-b} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-6055526400 a^2 c^7 x^2+4026531840 a c^{10} x-1873980108 b^2 c+1513881600 b c^7\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5752750080 a^2 c^6 x^2-3523215360 a c^9 x+2342475135 b^2-1438187520 b c^6\right )+\left (6459228160 a^2 c^8 x^2-5368709120 a c^{11} x+1665760096 b^2 c^2-1614807040 b c^8\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}\right )+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-6055526400 a^3 c^7 x^3+4026531840 a^2 c^{10} x^2-1873980108 a b^2 c x+4541644800 a b c^7 x+1447176192 b^2 c^4-2013265920 b c^{10}\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5752750080 a^3 c^6 x^3-3523215360 a^2 c^9 x^2+2342475135 a b^2 x-4314562560 a b c^6 x-1537624704 b^2 c^3+1761607680 b c^9\right )+\left (6459228160 a^3 c^8 x^3-5368709120 a^2 c^{11} x^2+1665760096 a b^2 c^2 x-4844421120 a b c^8 x-1378263040 b^2 c^5+2684354560 b c^{11}\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{11026104320 a c^6 \left (2 a^2 x^2-b\right )+22052208640 a^2 c^6 x \sqrt {a^2 x^2-b}} \]
________________________________________________________________________________________
Rubi [F] time = 0.33, 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*}
________________________________________________________________________________________
Mathematica [B] time = 7.38, size = 2441, normalized size = 3.10 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]
[Out]
-1/44104417280*((c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(35/4)*(1 - b/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^6
+ c^6/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^6 - (6*c^5)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^5 + (15*c^4)/(
c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^4 - (20*c^3)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^3 + (15*c^2)/(c + (a
*x + Sqrt[-b + a^2*x^2])^(1/3))^2 - (6*c)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))*(5752750080*c^(25/4) - (4098
1117100*b^2*c^(21/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^11 - (44104417280*c^(69/4))/(c + (a*x + Sqrt[-b +
a^2*x^2])^(1/3))^11 + (114912007980*b^2*c^(17/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^10 + (359135969280*c
^(65/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^10 - (164814147960*b^2*c^(13/4))/(c + (a*x + Sqrt[-b + a^2*x^2
])^(1/3))^9 - (1348907827200*c^(61/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^9 + (130345727512*b^2*c^(9/4))/(
c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^8 + (3109647810560*c^(57/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^8 -
(54345423132*b^2*c^(5/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^7 - (4920099471360*c^(53/4))/(c + (a*x + Sqrt
[-b + a^2*x^2])^(1/3))^7 + (9369900540*b^2*c^(1/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^6 + (5625764904960*
c^(49/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^6 - (4736415170560*c^(45/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^
(1/3))^5 + (2922027417600*c^(41/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^4 - (1286881935360*c^(37/4))/(c + (
a*x + Sqrt[-b + a^2*x^2])^(1/3))^3 + (383415746560*c^(33/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^2 - (69335
777280*c^(29/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3)) - (1009470*b*(4641*b - 131072*c^6)*(-1 + c/(c + (a*x +
Sqrt[-b + a^2*x^2])^(1/3)))^6*ArcTan[c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[
-b + a^2*x^2])^(1/3))^(23/4) + (504735*b*(4641*b - 131072*c^6)*(-1 + c/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))
^6*Log[1 - c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(23/4
) - (2342475135*b^2*c^6*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a
^2*x^2])^(1/3))^(47/4) + (66156625920*b*c^12*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c
+ (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(47/4) + (14054850810*b^2*c^5*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x
^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(43/4) - (396939755520*b*c^11*Log[1 + c^(1/4)/(c +
(a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(43/4) - (35137127025*b^2*c^4
*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(39/4)
+ (992349388800*b*c^10*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^
2*x^2])^(1/3))^(39/4) + (46849502700*b^2*c^3*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c
+ (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(35/4) - (1323132518400*b*c^9*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x
^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(35/4) - (35137127025*b^2*c^2*Log[1 + c^(1/4)/(c +
(a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(31/4) + (992349388800*b*c^8*
Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(31/4) +
(14054850810*b^2*c*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x
^2])^(1/3))^(27/4) - (396939755520*b*c^7*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (
a*x + Sqrt[-b + a^2*x^2])^(1/3))^(27/4) - (2342475135*b^2*Log[1 + c^(1/4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3
))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(23/4) + (66156625920*b*c^6*Log[1 + c^(1/4)/(c + (a*x + Sqrt
[-b + a^2*x^2])^(1/3))^(1/4)])/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(23/4)))/(a*c^(25/4)*Sqrt[((c + (a*x + S
qrt[-b + a^2*x^2])^(1/3))^6*(1 - b/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^6 + c^6/(c + (a*x + Sqrt[-b + a^2*x^
2])^(1/3))^6 - (6*c^5)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^5 + (15*c^4)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/
3))^4 - (20*c^3)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^3 + (15*c^2)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^2
- (6*c)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))^2)/(-1 + c/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))^6]*(-1 + c/
(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))^9)
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.88, size = 787, normalized size = 1.00 \begin {gather*} \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 \tan ^{-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 \tan ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]
[Out]
((-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^1
1*x^2 + 6459228160*a^3*c^8*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (1447176192*b^2*c^4 - 201326592
0*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)*(a*
x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-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)*(a*x + S
qrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*((1665760096*b^2*c^
2 - 1614807040*b*c^8 - 5368709120*a*c^11*x + 6459228160*a^2*c^8*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3
/4) + (-1873980108*b^2*c + 1513881600*b*c^7 + 4026531840*a*c^10*x - 6055526400*a^2*c^7*x^2)*(a*x + Sqrt[-b + a
^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (2342475135*b^2 - 1438187520*b*c^6 - 3523215360*
a*c^9*x + 5752750080*a^2*c^6*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4
)))/(22052208640*a^2*c^6*x*Sqrt[-b + a^2*x^2] + 11026104320*a*c^6*(-b + 2*a^2*x^2)) + (13923*b^2*ArcTan[(c + (
a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(131072*a*c^(25/4)) - (3*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x
^2])^(1/3))^(1/4)/c^(1/4)])/(a*c^(1/4)) - (13923*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1
/4)])/(131072*a*c^(25/4)) + (3*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(a*c^(1/4))
________________________________________________________________________________________
fricas [A] time = 0.77, size = 1060, normalized size = 1.35
result too large to display
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)
________________________________________________________________________________________
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.
[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
________________________________________________________________________________________
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)
________________________________________________________________________________________
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}{3}}\right )}^{\frac {1}{4}}}\,{d x} \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)
________________________________________________________________________________________
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)
________________________________________________________________________________________
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)
________________________________________________________________________________________