3.761 \(\int \frac{1}{\sqrt{x} \left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=329 \[ \frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{105 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \]

[Out]

Sqrt[x]/(8*a*(a + c*x^4)^2) + (15*Sqrt[x])/(64*a^2*(a + c*x^4)) + (105*ArcTan[1
- (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(23/8)*c^(1/8)) - (10
5*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(23/8)*c^(
1/8)) - (105*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) - (
105*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) + (105*Log[
(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^
(23/8)*c^(1/8)) - (105*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(
1/4)*x])/(512*Sqrt[2]*(-a)^(23/8)*c^(1/8))

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Rubi [A]  time = 0.626389, 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 \[ \frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{105 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/(8*a*(a + c*x^4)^2) + (15*Sqrt[x])/(64*a^2*(a + c*x^4)) + (105*ArcTan[1
- (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(23/8)*c^(1/8)) - (10
5*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(23/8)*c^(
1/8)) - (105*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) - (
105*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) + (105*Log[
(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^
(23/8)*c^(1/8)) - (105*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(
1/4)*x])/(512*Sqrt[2]*(-a)^(23/8)*c^(1/8))

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Rubi in Sympy [A]  time = 133.098, size = 309, normalized size = 0.94 \[ \frac{105 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 \sqrt [8]{c} \left (- a\right )^{\frac{23}{8}}} - \frac{105 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 \sqrt [8]{c} \left (- a\right )^{\frac{23}{8}}} - \frac{105 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 \sqrt [8]{c} \left (- a\right )^{\frac{23}{8}}} - \frac{105 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 \sqrt [8]{c} \left (- a\right )^{\frac{23}{8}}} - \frac{105 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 \sqrt [8]{c} \left (- a\right )^{\frac{23}{8}}} - \frac{105 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 \sqrt [8]{c} \left (- a\right )^{\frac{23}{8}}} + \frac{\sqrt{x}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{15 \sqrt{x}}{64 a^{2} \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

105*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4)
)/(1024*c**(1/8)*(-a)**(23/8)) - 105*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**
(1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(1/8)*(-a)**(23/8)) - 105*atan(c**(1/
8)*sqrt(x)/(-a)**(1/8))/(256*c**(1/8)*(-a)**(23/8)) - 105*sqrt(2)*atan(sqrt(2)*c
**(1/8)*sqrt(x)/(-a)**(1/8) - 1)/(512*c**(1/8)*(-a)**(23/8)) - 105*sqrt(2)*atan(
sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(512*c**(1/8)*(-a)**(23/8)) - 105*atan
h(c**(1/8)*sqrt(x)/(-a)**(1/8))/(256*c**(1/8)*(-a)**(23/8)) + sqrt(x)/(8*a*(a +
c*x**4)**2) + 15*sqrt(x)/(64*a**2*(a + c*x**4))

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Mathematica [A]  time = 0.874826, size = 427, normalized size = 1.3 \[ \frac{\frac{64 a^{15/8} \sqrt{x}}{\left (a+c x^4\right )^2}+\frac{120 a^{7/8} \sqrt{x}}{a+c x^4}-\frac{105 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}+\frac{105 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}-\frac{105 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}+\frac{105 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{c}}+\frac{210 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{c}}+\frac{210 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{c}}-\frac{210 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{\sqrt [8]{c}}+\frac{210 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{c}}}{512 a^{23/8}} \]

Antiderivative was successfully verified.

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

[Out]

((64*a^(15/8)*Sqrt[x])/(a + c*x^4)^2 + (120*a^(7/8)*Sqrt[x])/(a + c*x^4) + (210*
ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8])/c^(1/8) + (21
0*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8])/c^(1/8) - (
105*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) + (105*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[P
i/8]])/c^(1/8) - (210*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Si
n[Pi/8])/c^(1/8) + (210*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*
Sin[Pi/8])/c^(1/8) - (105*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Si
n[Pi/8]]*Sin[Pi/8])/c^(1/8) + (105*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*S
qrt[x]*Sin[Pi/8]]*Sin[Pi/8])/c^(1/8))/(512*a^(23/8))

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Maple [C]  time = 0.026, size = 62, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{23\,\sqrt{x}}{128\,a}}+{\frac{15\,c{x}^{9/2}}{128\,{a}^{2}}} \right ) }+{\frac{105}{512\,{a}^{2}c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+a)^3/x^(1/2),x)

[Out]

2*(23/128*x^(1/2)/a+15/128/a^2*c*x^(9/2))/(c*x^4+a)^2+105/512/a^2/c*sum(1/_R^7*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. \[ -105 \, c \int \frac{x^{\frac{7}{2}}}{128 \,{\left (a^{3} c x^{4} + a^{4}\right )}}\,{d x} + \frac{105 \, c^{2} x^{\frac{17}{2}} + 225 \, a c x^{\frac{9}{2}} + 128 \, a^{2} \sqrt{x}}{64 \,{\left (a^{3} c^{2} x^{8} + 2 \, a^{4} c x^{4} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*sqrt(x)),x, algorithm="maxima")

[Out]

-105*c*integrate(1/128*x^(7/2)/(a^3*c*x^4 + a^4), x) + 1/64*(105*c^2*x^(17/2) +
225*a*c*x^(9/2) + 128*a^2*sqrt(x))/(a^3*c^2*x^8 + 2*a^4*c*x^4 + a^5)

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Fricas [A]  time = 0.258798, size = 849, normalized size = 2.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*sqrt(x)),x, algorithm="fricas")

[Out]

-1/1024*sqrt(2)*(420*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/
8)*arctan(a^3*(-1/(a^23*c))^(1/8)/(sqrt(a^6*(-1/(a^23*c))^(1/4) + x) + sqrt(x)))
- 105*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*log(a^3*(-1
/(a^23*c))^(1/8) + sqrt(x)) + 105*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/
(a^23*c))^(1/8)*log(-a^3*(-1/(a^23*c))^(1/8) + sqrt(x)) + 420*(a^2*c^2*x^8 + 2*a
^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*arctan(a^3*(-1/(a^23*c))^(1/8)/(a^3*(-1/(a^2
3*c))^(1/8) + sqrt(2)*sqrt(x) + sqrt(2*a^6*(-1/(a^23*c))^(1/4) + 2*sqrt(2)*a^3*s
qrt(x)*(-1/(a^23*c))^(1/8) + 2*x))) + 420*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/
(a^23*c))^(1/8)*arctan(-a^3*(-1/(a^23*c))^(1/8)/(a^3*(-1/(a^23*c))^(1/8) - sqrt(
2)*sqrt(x) - sqrt(2*a^6*(-1/(a^23*c))^(1/4) - 2*sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c)
)^(1/8) + 2*x))) - 105*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*log
(2*a^6*(-1/(a^23*c))^(1/4) + 2*sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c))^(1/8) + 2*x) +
105*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*log(2*a^6*(-1/(a^23*c)
)^(1/4) - 2*sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c))^(1/8) + 2*x) - 8*sqrt(2)*(15*c*x^4
+ 23*a)*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(1/(c*x**4+a)**3/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.335583, size = 626, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*sqrt(x)),x, algorithm="giac")

[Out]

105/512*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^3 + 105/512*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^3 + 105/512*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^3 + 105/512*sqr
t(-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^3 + 105/1024*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*l
n(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 105/1024*sqrt(s
qrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1
/4))/a^3 + 105/1024*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^3 - 105/1024*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^3 + 1/64*(15*c*x^(
9/2) + 23*a*sqrt(x))/((c*x^4 + a)^2*a^2)