3.760 \(\int \frac{\sqrt{x}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=329 \[ \frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}-\frac{65 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}-\frac{65 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}+\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2} \]

[Out]

x^(3/2)/(8*a*(a + c*x^4)^2) + (13*x^(3/2))/(64*a^2*(a + c*x^4)) + (65*ArcTan[1 -
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(21/8)*c^(3/8)) - (65*
ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(21/8)*c^(3/
8)) + (65*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*
ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*Log[(-a)^
(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(21/8
)*c^(3/8)) + (65*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x
])/(512*Sqrt[2]*(-a)^(21/8)*c^(3/8))

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Rubi [A]  time = 0.651351, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}-\frac{65 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}-\frac{65 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}+\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + c*x^4)^3,x]

[Out]

x^(3/2)/(8*a*(a + c*x^4)^2) + (13*x^(3/2))/(64*a^2*(a + c*x^4)) + (65*ArcTan[1 -
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(21/8)*c^(3/8)) - (65*
ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(21/8)*c^(3/
8)) + (65*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*
ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*Log[(-a)^
(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(21/8
)*c^(3/8)) + (65*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x
])/(512*Sqrt[2]*(-a)^(21/8)*c^(3/8))

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Rubi in Sympy [A]  time = 135.674, size = 309, normalized size = 0.94 \[ - \frac{65 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{3}{8}} \left (- a\right )^{\frac{21}{8}}} + \frac{65 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{3}{8}} \left (- a\right )^{\frac{21}{8}}} + \frac{65 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{3}{8}} \left (- a\right )^{\frac{21}{8}}} - \frac{65 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 c^{\frac{3}{8}} \left (- a\right )^{\frac{21}{8}}} - \frac{65 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 c^{\frac{3}{8}} \left (- a\right )^{\frac{21}{8}}} - \frac{65 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{3}{8}} \left (- a\right )^{\frac{21}{8}}} + \frac{x^{\frac{3}{2}}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{13 x^{\frac{3}{2}}}{64 a^{2} \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(c*x**4+a)**3,x)

[Out]

-65*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4)
)/(1024*c**(3/8)*(-a)**(21/8)) + 65*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(
1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(3/8)*(-a)**(21/8)) + 65*atan(c**(1/8)
*sqrt(x)/(-a)**(1/8))/(256*c**(3/8)*(-a)**(21/8)) - 65*sqrt(2)*atan(sqrt(2)*c**(
1/8)*sqrt(x)/(-a)**(1/8) - 1)/(512*c**(3/8)*(-a)**(21/8)) - 65*sqrt(2)*atan(sqrt
(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(512*c**(3/8)*(-a)**(21/8)) - 65*atanh(c**
(1/8)*sqrt(x)/(-a)**(1/8))/(256*c**(3/8)*(-a)**(21/8)) + x**(3/2)/(8*a*(a + c*x*
*4)**2) + 13*x**(3/2)/(64*a**2*(a + c*x**4))

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Mathematica [A]  time = 1.08447, size = 427, normalized size = 1.3 \[ \frac{\frac{64 a^{13/8} x^{3/2}}{\left (a+c x^4\right )^2}+\frac{104 a^{5/8} x^{3/2}}{a+c x^4}+\frac{65 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}-\frac{65 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}-\frac{65 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}+\frac{65 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{3/8}}-\frac{130 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{c^{3/8}}-\frac{130 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{c^{3/8}}-\frac{130 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{c^{3/8}}+\frac{130 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{c^{3/8}}}{512 a^{21/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(a + c*x^4)^3,x]

[Out]

((64*a^(13/8)*x^(3/2))/(a + c*x^4)^2 + (104*a^(5/8)*x^(3/2))/(a + c*x^4) - (130*
ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8])/c^(3/8) + (13
0*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8])/c^(3/8) + (
65*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) - (65*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) - (130*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[
Pi/8])/c^(3/8) - (130*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Si
n[Pi/8])/c^(3/8) - (65*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[P
i/8]]*Sin[Pi/8])/c^(3/8) + (65*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))/(512*a^(21/8))

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Maple [C]  time = 0.027, size = 62, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{21\,{x}^{3/2}}{128\,a}}+{\frac{13\,c{x}^{11/2}}{128\,{a}^{2}}} \right ) }+{\frac{65}{512\,{a}^{2}c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^4+a)^3,x)

[Out]

2*(21/128*x^(3/2)/a+13/128/a^2*c*x^(11/2))/(c*x^4+a)^2+65/512/a^2/c*sum(1/_R^5*l
n(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{13 \, c x^{\frac{11}{2}} + 21 \, a x^{\frac{3}{2}}}{64 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + 65 \, \int \frac{\sqrt{x}}{128 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

1/64*(13*c*x^(11/2) + 21*a*x^(3/2))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 65*integ
rate(1/128*sqrt(x)/(a^2*c*x^4 + a^3), x)

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Fricas [A]  time = 0.270768, size = 886, normalized size = 2.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

-1/1024*sqrt(2)*(260*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(
1/8)*arctan(a^8*c*(-1/(a^21*c^3))^(3/8)/(sqrt(a^16*c^2*(-1/(a^21*c^3))^(3/4) + x
) + sqrt(x))) + 65*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/
8)*log(a^8*c*(-1/(a^21*c^3))^(3/8) + sqrt(x)) - 65*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*
c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(-a^8*c*(-1/(a^21*c^3))^(3/8) + sqrt(x)) -
260*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*arctan(a^8*c*(-1/(a
^21*c^3))^(3/8)/(a^8*c*(-1/(a^21*c^3))^(3/8) + sqrt(2)*sqrt(x) + sqrt(2*a^16*c^2
*(-1/(a^21*c^3))^(3/4) + 2*sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + 2*x)))
- 260*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*arctan(-a^8*c*(-1/
(a^21*c^3))^(3/8)/(a^8*c*(-1/(a^21*c^3))^(3/8) - sqrt(2)*sqrt(x) - sqrt(2*a^16*c
^2*(-1/(a^21*c^3))^(3/4) - 2*sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + 2*x))
) - 65*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(2*a^16*c^2*(-
1/(a^21*c^3))^(3/4) + 2*sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + 2*x) + 65*
(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(2*a^16*c^2*(-1/(a^21
*c^3))^(3/4) - 2*sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + 2*x) - 8*sqrt(2)*
(13*c*x^5 + 21*a*x)*sqrt(x))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.351052, size = 626, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

-65/512*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^3 - 65/512*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^3 + 65/512*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^3 + 65/512*sqrt(
sqrt(2) + 2)*(a/c)^(3/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sq
rt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 65/1024*sqrt(-sqrt(2) + 2)*(a/c)^(3/8)*ln(s
qrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 65/1024*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^3 - 65/1024*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^3 + 65/1024*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^3 + 1/64*(13*c*x^(11/2) +
21*a*x^(3/2))/((c*x^4 + a)^2*a^2)