3.751 \(\int \frac{1}{\sqrt{x} \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=308 \[ \frac{\sqrt{x}}{4 a \left (a+c x^4\right )}-\frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}} \]

[Out]

Sqrt[x]/(4*a*(a + c*x^4)) - (7*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])
/(16*Sqrt[2]*(-a)^(15/8)*c^(1/8)) + (7*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)
^(1/8)])/(16*Sqrt[2]*(-a)^(15/8)*c^(1/8)) + (7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(16*(-a)^(15/8)*c^(1/8)) + (7*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-
a)^(15/8)*c^(1/8)) - (7*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^
(1/4)*x])/(32*Sqrt[2]*(-a)^(15/8)*c^(1/8)) + (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1
/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(15/8)*c^(1/8))

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Rubi [A]  time = 0.577137, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{\sqrt{x}}{4 a \left (a+c x^4\right )}-\frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/(4*a*(a + c*x^4)) - (7*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])
/(16*Sqrt[2]*(-a)^(15/8)*c^(1/8)) + (7*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)
^(1/8)])/(16*Sqrt[2]*(-a)^(15/8)*c^(1/8)) + (7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(16*(-a)^(15/8)*c^(1/8)) + (7*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-
a)^(15/8)*c^(1/8)) - (7*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^
(1/4)*x])/(32*Sqrt[2]*(-a)^(15/8)*c^(1/8)) + (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1
/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(15/8)*c^(1/8))

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Rubi in Sympy [A]  time = 119.262, size = 289, normalized size = 0.94 \[ - \frac{7 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 \sqrt [8]{c} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 \sqrt [8]{c} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 \sqrt [8]{c} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 \sqrt [8]{c} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 \sqrt [8]{c} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 \sqrt [8]{c} \left (- a\right )^{\frac{15}{8}}} + \frac{\sqrt{x}}{4 a \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**4+a)**2/x**(1/2),x)

[Out]

-7*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))
/(64*c**(1/8)*(-a)**(15/8)) + 7*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8)
+ c**(1/4)*x + (-a)**(1/4))/(64*c**(1/8)*(-a)**(15/8)) + 7*atan(c**(1/8)*sqrt(x
)/(-a)**(1/8))/(16*c**(1/8)*(-a)**(15/8)) + 7*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt
(x)/(-a)**(1/8) - 1)/(32*c**(1/8)*(-a)**(15/8)) + 7*sqrt(2)*atan(sqrt(2)*c**(1/8
)*sqrt(x)/(-a)**(1/8) + 1)/(32*c**(1/8)*(-a)**(15/8)) + 7*atanh(c**(1/8)*sqrt(x)
/(-a)**(1/8))/(16*c**(1/8)*(-a)**(15/8)) + sqrt(x)/(4*a*(a + c*x**4))

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Mathematica [A]  time = 2.07623, size = 406, normalized size = 1.32 \[ \frac{\frac{8 a^{7/8} \sqrt{x}}{a+c x^4}-\frac{7 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}+\frac{7 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}-\frac{7 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}+\frac{7 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}+\frac{14 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{c}}+\frac{14 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{c}}-\frac{14 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{\sqrt [8]{c}}+\frac{14 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{c}}}{32 a^{15/8}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^(7/8)*Sqrt[x])/(a + c*x^4) + (14*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8
) - Tan[Pi/8]]*Cos[Pi/8])/c^(1/8) + (14*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/
8) + Tan[Pi/8]]*Cos[Pi/8])/c^(1/8) - (7*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^
(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/c^(1/8) + (7*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x
+ 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/c^(1/8) - (14*ArcTan[Cot[Pi/8] - (c^(1/
8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/c^(1/8) + (14*ArcTan[Cot[Pi/8] + (c^(1
/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/c^(1/8) - (7*Log[a^(1/4) + c^(1/4)*x
- 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/c^(1/8) + (7*Log[a^(1/4) + c^(
1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/c^(1/8))/(32*a^(15/8))

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Maple [C]  time = 0.019, size = 50, normalized size = 0.2 \[{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }\sqrt{x}}+{\frac{7}{32\,ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+a)^2/x^(1/2),x)

[Out]

1/4*x^(1/2)/a/(c*x^4+a)+7/32/a/c*sum(1/_R^7*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -7 \, c \int \frac{x^{\frac{7}{2}}}{8 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} + \frac{7 \, c x^{\frac{9}{2}} + 8 \, a \sqrt{x}}{4 \,{\left (a^{2} c x^{4} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*sqrt(x)),x, algorithm="maxima")

[Out]

-7*c*integrate(1/8*x^(7/2)/(a^2*c*x^4 + a^3), x) + 1/4*(7*c*x^(9/2) + 8*a*sqrt(x
))/(a^2*c*x^4 + a^3)

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Fricas [A]  time = 0.26749, size = 695, normalized size = 2.26 \[ -\frac{\sqrt{2}{\left (28 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}}}{\sqrt{a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} + x} + \sqrt{x}}\right ) - 7 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + 7 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (-a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + 28 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}}}{a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{x} + \sqrt{2 \, a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} + 2 \, \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + 2 \, x}}\right ) + 28 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}}}{a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} - \sqrt{2} \sqrt{x} - \sqrt{2 \, a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} - 2 \, \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + 2 \, x}}\right ) - 7 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (2 \, a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} + 2 \, \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + 2 \, x\right ) + 7 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (2 \, a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} - 2 \, \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + 2 \, x\right ) - 8 \, \sqrt{2} \sqrt{x}\right )}}{64 \,{\left (a c x^{4} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*sqrt(x)),x, algorithm="fricas")

[Out]

-1/64*sqrt(2)*(28*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^15*c))^(1/8)*arctan(a^2*(-1/(a^
15*c))^(1/8)/(sqrt(a^4*(-1/(a^15*c))^(1/4) + x) + sqrt(x))) - 7*sqrt(2)*(a*c*x^4
+ a^2)*(-1/(a^15*c))^(1/8)*log(a^2*(-1/(a^15*c))^(1/8) + sqrt(x)) + 7*sqrt(2)*(
a*c*x^4 + a^2)*(-1/(a^15*c))^(1/8)*log(-a^2*(-1/(a^15*c))^(1/8) + sqrt(x)) + 28*
(a*c*x^4 + a^2)*(-1/(a^15*c))^(1/8)*arctan(a^2*(-1/(a^15*c))^(1/8)/(a^2*(-1/(a^1
5*c))^(1/8) + sqrt(2)*sqrt(x) + sqrt(2*a^4*(-1/(a^15*c))^(1/4) + 2*sqrt(2)*a^2*s
qrt(x)*(-1/(a^15*c))^(1/8) + 2*x))) + 28*(a*c*x^4 + a^2)*(-1/(a^15*c))^(1/8)*arc
tan(-a^2*(-1/(a^15*c))^(1/8)/(a^2*(-1/(a^15*c))^(1/8) - sqrt(2)*sqrt(x) - sqrt(2
*a^4*(-1/(a^15*c))^(1/4) - 2*sqrt(2)*a^2*sqrt(x)*(-1/(a^15*c))^(1/8) + 2*x))) -
7*(a*c*x^4 + a^2)*(-1/(a^15*c))^(1/8)*log(2*a^4*(-1/(a^15*c))^(1/4) + 2*sqrt(2)*
a^2*sqrt(x)*(-1/(a^15*c))^(1/8) + 2*x) + 7*(a*c*x^4 + a^2)*(-1/(a^15*c))^(1/8)*l
og(2*a^4*(-1/(a^15*c))^(1/4) - 2*sqrt(2)*a^2*sqrt(x)*(-1/(a^15*c))^(1/8) + 2*x)
- 8*sqrt(2)*sqrt(x))/(a*c*x^4 + a^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**4+a)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296256, size = 613, normalized size = 1.99 \[ \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} - \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} - \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} + \frac{\sqrt{x}}{4 \,{\left (c x^{4} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*sqrt(x)),x, algorithm="giac")

[Out]

7/32*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sq
rt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 7/32*sqrt(sqrt(2) + 2)*(a/c)^(1/8)
*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(
1/8)))/a^2 + 7/32*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)
^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 7/32*sqrt(-sqrt(2) +
2)*(a/c)^(1/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(
2) + 2)*(a/c)^(1/8)))/a^2 + 7/64*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*ln(sqrt(x)*sqrt(s
qrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 - 7/64*sqrt(sqrt(2) + 2)*(a/c)^(1
/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 + 7/64*sqrt
(-sqrt(2) + 2)*(a/c)^(1/8)*ln(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)
^(1/4))/a^2 - 7/64*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2)
*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 + 1/4*sqrt(x)/((c*x^4 + a)*a)