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

Optimal. Leaf size=308 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{13/8} c^{3/8}}-\frac{5 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{13/8} c^{3/8}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{13/8} c^{3/8}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{13/8} c^{3/8}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac{x^{3/2}}{4 a \left (a+c x^4\right )} \]

[Out]

x^(3/2)/(4*a*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])
/(16*Sqrt[2]*(-a)^(13/8)*c^(3/8)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)
^(1/8)])/(16*Sqrt[2]*(-a)^(13/8)*c^(3/8)) - (5*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(16*(-a)^(13/8)*c^(3/8)) + (5*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-
a)^(13/8)*c^(3/8)) + (5*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^
(1/4)*x])/(32*Sqrt[2]*(-a)^(13/8)*c^(3/8)) - (5*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1
/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(13/8)*c^(3/8))

_______________________________________________________________________________________

Rubi [A]  time = 0.563296, 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 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{13/8} c^{3/8}}-\frac{5 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{13/8} c^{3/8}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{13/8} c^{3/8}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{13/8} c^{3/8}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac{x^{3/2}}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + c*x^4)^2,x]

[Out]

x^(3/2)/(4*a*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])
/(16*Sqrt[2]*(-a)^(13/8)*c^(3/8)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)
^(1/8)])/(16*Sqrt[2]*(-a)^(13/8)*c^(3/8)) - (5*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(16*(-a)^(13/8)*c^(3/8)) + (5*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-
a)^(13/8)*c^(3/8)) + (5*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^
(1/4)*x])/(32*Sqrt[2]*(-a)^(13/8)*c^(3/8)) - (5*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1
/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(13/8)*c^(3/8))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 120.903, size = 289, normalized size = 0.94 \[ \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{3}{8}} \left (- a\right )^{\frac{13}{8}}} - \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{3}{8}} \left (- a\right )^{\frac{13}{8}}} - \frac{5 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{3}{8}} \left (- a\right )^{\frac{13}{8}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{3}{8}} \left (- a\right )^{\frac{13}{8}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{3}{8}} \left (- a\right )^{\frac{13}{8}}} + \frac{5 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{3}{8}} \left (- a\right )^{\frac{13}{8}}} + \frac{x^{\frac{3}{2}}}{4 a \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/
(64*c**(3/8)*(-a)**(13/8)) - 5*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8)
+ c**(1/4)*x + (-a)**(1/4))/(64*c**(3/8)*(-a)**(13/8)) - 5*atan(c**(1/8)*sqrt(x)
/(-a)**(1/8))/(16*c**(3/8)*(-a)**(13/8)) + 5*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(
x)/(-a)**(1/8) - 1)/(32*c**(3/8)*(-a)**(13/8)) + 5*sqrt(2)*atan(sqrt(2)*c**(1/8)
*sqrt(x)/(-a)**(1/8) + 1)/(32*c**(3/8)*(-a)**(13/8)) + 5*atanh(c**(1/8)*sqrt(x)/
(-a)**(1/8))/(16*c**(3/8)*(-a)**(13/8)) + x**(3/2)/(4*a*(a + c*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 2.03238, size = 406, normalized size = 1.32 \[ \frac{\frac{8 a^{5/8} x^{3/2}}{a+c x^4}+\frac{5 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}-\frac{5 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}-\frac{5 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}+\frac{5 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}-\frac{10 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{c^{3/8}}-\frac{10 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{c^{3/8}}-\frac{10 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{c^{3/8}}+\frac{10 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{c^{3/8}}}{32 a^{13/8}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^(5/8)*x^(3/2))/(a + c*x^4) - (10*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[P
i/8])/a^(1/8)]*Cos[Pi/8])/c^(3/8) + (10*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[
Pi/8])/a^(1/8)]*Cos[Pi/8])/c^(3/8) + (5*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^
(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]])/c^(3/8) - (5*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x
+ 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]])/c^(3/8) - (10*ArcTan[(c^(1/8)*Sqrt[x]*S
ec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8])/c^(3/8) - (10*ArcTan[(c^(1/8)*Sqrt[x]*
Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[Pi/8])/c^(3/8) - (5*Log[a^(1/4) + c^(1/4)*x
- 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8])/c^(3/8) + (5*Log[a^(1/4) + c^(
1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8])/c^(3/8))/(32*a^(13/8))

_______________________________________________________________________________________

Maple [C]  time = 0.019, size = 50, normalized size = 0.2 \[{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{5}{32\,ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^4+a)^2,x)

[Out]

1/4*x^(3/2)/a/(c*x^4+a)+5/32/a/c*sum(1/_R^5*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{\frac{3}{2}}}{4 \,{\left (a c x^{4} + a^{2}\right )}} + 5 \, \int \frac{\sqrt{x}}{8 \,{\left (a c x^{4} + a^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^(3/2)/(a*c*x^4 + a^2) + 5*integrate(1/8*sqrt(x)/(a*c*x^4 + a^2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.261596, size = 730, normalized size = 2.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*sqrt(2)*(20*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^13*c^3))^(1/8)*arctan(a^5*c*(-1
/(a^13*c^3))^(3/8)/(sqrt(a^10*c^2*(-1/(a^13*c^3))^(3/4) + x) + sqrt(x))) + 5*sqr
t(2)*(a*c*x^4 + a^2)*(-1/(a^13*c^3))^(1/8)*log(a^5*c*(-1/(a^13*c^3))^(3/8) + sqr
t(x)) - 5*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^13*c^3))^(1/8)*log(-a^5*c*(-1/(a^13*c^3
))^(3/8) + sqrt(x)) - 20*(a*c*x^4 + a^2)*(-1/(a^13*c^3))^(1/8)*arctan(a^5*c*(-1/
(a^13*c^3))^(3/8)/(a^5*c*(-1/(a^13*c^3))^(3/8) + sqrt(2)*sqrt(x) + sqrt(2*a^10*c
^2*(-1/(a^13*c^3))^(3/4) + 2*sqrt(2)*a^5*c*sqrt(x)*(-1/(a^13*c^3))^(3/8) + 2*x))
) - 20*(a*c*x^4 + a^2)*(-1/(a^13*c^3))^(1/8)*arctan(-a^5*c*(-1/(a^13*c^3))^(3/8)
/(a^5*c*(-1/(a^13*c^3))^(3/8) - sqrt(2)*sqrt(x) - sqrt(2*a^10*c^2*(-1/(a^13*c^3)
)^(3/4) - 2*sqrt(2)*a^5*c*sqrt(x)*(-1/(a^13*c^3))^(3/8) + 2*x))) - 5*(a*c*x^4 +
a^2)*(-1/(a^13*c^3))^(1/8)*log(2*a^10*c^2*(-1/(a^13*c^3))^(3/4) + 2*sqrt(2)*a^5*
c*sqrt(x)*(-1/(a^13*c^3))^(3/8) + 2*x) + 5*(a*c*x^4 + a^2)*(-1/(a^13*c^3))^(1/8)
*log(2*a^10*c^2*(-1/(a^13*c^3))^(3/4) - 2*sqrt(2)*a^5*c*sqrt(x)*(-1/(a^13*c^3))^
(3/8) + 2*x) - 8*sqrt(2)*x^(3/2))/(a*c*x^4 + a^2)

_______________________________________________________________________________________

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(x**(1/2)/(c*x**4+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.327839, size = 613, normalized size = 1.99 \[ -\frac{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{x^{\frac{3}{2}}}{4 \,{\left (c x^{4} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/32*sqrt(-sqrt(2) + 2)*(a/c)^(3/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*
sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 - 5/32*sqrt(-sqrt(2) + 2)*(a/c)^(3
/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c
)^(1/8)))/a^2 + 5/32*sqrt(sqrt(2) + 2)*(a/c)^(3/8)*arctan((sqrt(sqrt(2) + 2)*(a/
c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 5/32*sqrt(sqrt(2)
+ 2)*(a/c)^(3/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt
(2) + 2)*(a/c)^(1/8)))/a^2 + 5/64*sqrt(-sqrt(2) + 2)*(a/c)^(3/8)*ln(sqrt(x)*sqrt
(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 - 5/64*sqrt(-sqrt(2) + 2)*(a/c)
^(3/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 - 5/64*s
qrt(sqrt(2) + 2)*(a/c)^(3/8)*ln(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/
c)^(1/4))/a^2 + 5/64*sqrt(sqrt(2) + 2)*(a/c)^(3/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2
)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 + 1/4*x^(3/2)/((c*x^4 + a)*a)