3.755 \(\int \frac{x^{11/2}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=332 \[ \frac{15 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{11/8} c^{13/8}}-\frac{15 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{11/8} c^{13/8}}+\frac{15 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{11/8} c^{13/8}}-\frac{15 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{11/8} c^{13/8}}+\frac{15 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{11/8} c^{13/8}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{11/8} c^{13/8}}+\frac{5 x^{5/2}}{64 a c \left (a+c x^4\right )}-\frac{x^{5/2}}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x^(5/2)/(8*c*(a + c*x^4)^2) + (5*x^(5/2))/(64*a*c*(a + c*x^4)) + (15*ArcTan[1 -
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(11/8)*c^(13/8)) - (15
*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(11/8)*c^(1
3/8)) + (15*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(11/8)*c^(13/8)) + (
15*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(11/8)*c^(13/8)) + (15*Log[(
-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(
11/8)*c^(13/8)) - (15*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1
/4)*x])/(512*Sqrt[2]*(-a)^(11/8)*c^(13/8))

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Rubi [A]  time = 0.592031, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.867 \[ \frac{15 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{11/8} c^{13/8}}-\frac{15 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{11/8} c^{13/8}}+\frac{15 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{11/8} c^{13/8}}-\frac{15 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{11/8} c^{13/8}}+\frac{15 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{11/8} c^{13/8}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{11/8} c^{13/8}}+\frac{5 x^{5/2}}{64 a c \left (a+c x^4\right )}-\frac{x^{5/2}}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(11/2)/(a + c*x^4)^3,x]

[Out]

-x^(5/2)/(8*c*(a + c*x^4)^2) + (5*x^(5/2))/(64*a*c*(a + c*x^4)) + (15*ArcTan[1 -
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(11/8)*c^(13/8)) - (15
*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(11/8)*c^(1
3/8)) + (15*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(11/8)*c^(13/8)) + (
15*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(11/8)*c^(13/8)) + (15*Log[(
-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(
11/8)*c^(13/8)) - (15*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1
/4)*x])/(512*Sqrt[2]*(-a)^(11/8)*c^(13/8))

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Rubi in Sympy [A]  time = 124.944, size = 309, normalized size = 0.93 \[ - \frac{x^{\frac{5}{2}}}{8 c \left (a + c x^{4}\right )^{2}} + \frac{15 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{13}{8}} \left (- a\right )^{\frac{11}{8}}} - \frac{15 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{13}{8}} \left (- a\right )^{\frac{11}{8}}} + \frac{15 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{13}{8}} \left (- a\right )^{\frac{11}{8}}} - \frac{15 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 c^{\frac{13}{8}} \left (- a\right )^{\frac{11}{8}}} - \frac{15 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 c^{\frac{13}{8}} \left (- a\right )^{\frac{11}{8}}} + \frac{15 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{13}{8}} \left (- a\right )^{\frac{11}{8}}} + \frac{5 x^{\frac{5}{2}}}{64 a c \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(11/2)/(c*x**4+a)**3,x)

[Out]

-x**(5/2)/(8*c*(a + c*x**4)**2) + 15*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)*
*(1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(13/8)*(-a)**(11/8)) - 15*sqrt(2)*lo
g(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(13/
8)*(-a)**(11/8)) + 15*atan(c**(1/8)*sqrt(x)/(-a)**(1/8))/(256*c**(13/8)*(-a)**(1
1/8)) - 15*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) - 1)/(512*c**(13/8)
*(-a)**(11/8)) - 15*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(512*
c**(13/8)*(-a)**(11/8)) + 15*atanh(c**(1/8)*sqrt(x)/(-a)**(1/8))/(256*c**(13/8)*
(-a)**(11/8)) + 5*x**(5/2)/(64*a*c*(a + c*x**4))

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Mathematica [A]  time = 0.972584, size = 430, normalized size = 1.3 \[ \frac{-\frac{15 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{11/8}}+\frac{15 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{11/8}}+\frac{15 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{11/8}}-\frac{15 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{11/8}}-\frac{30 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{a^{11/8}}-\frac{30 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{a^{11/8}}-\frac{30 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{a^{11/8}}+\frac{30 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{a^{11/8}}+\frac{40 c^{5/8} x^{5/2}}{a^2+a c x^4}-\frac{64 c^{5/8} x^{5/2}}{\left (a+c x^4\right )^2}}{512 c^{13/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(11/2)/(a + c*x^4)^3,x]

[Out]

((-64*c^(5/8)*x^(5/2))/(a + c*x^4)^2 + (40*c^(5/8)*x^(5/2))/(a^2 + a*c*x^4) - (3
0*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8])/a^(11/8) +
(30*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8])/a^(11/8)
- (15*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]])/
a^(11/8) + (15*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin
[Pi/8]])/a^(11/8) - (30*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*
Sin[Pi/8])/a^(11/8) - (30*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]
]*Sin[Pi/8])/a^(11/8) + (15*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*
Cos[Pi/8]]*Sin[Pi/8])/a^(11/8) - (15*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)
*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8])/a^(11/8))/(512*c^(13/8))

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Maple [C]  time = 0.027, size = 61, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{3\,{x}^{5/2}}{128\,c}}+{\frac{5\,{x}^{13/2}}{128\,a}} \right ) }+{\frac{15}{512\,{c}^{2}a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)/(c*x^4+a)^3,x)

[Out]

2*(-3/128/c*x^(5/2)+5/128/a*x^(13/2))/(c*x^4+a)^2+15/512/c^2/a*sum(1/_R^3*ln(x^(
1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 \, c x^{\frac{13}{2}} - 3 \, a x^{\frac{5}{2}}}{64 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} + 15 \, \int \frac{x^{\frac{3}{2}}}{128 \,{\left (a c^{2} x^{4} + a^{2} c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

1/64*(5*c*x^(13/2) - 3*a*x^(5/2))/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c) + 15*integ
rate(1/128*x^(3/2)/(a*c^2*x^4 + a^2*c), x)

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Fricas [A]  time = 0.268926, size = 941, normalized size = 2.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

1/1024*sqrt(2)*(60*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^11*c^13))^
(1/8)*arctan(a^7*c^8*(-1/(a^11*c^13))^(5/8)/(sqrt(-a^3*c^3*(-1/(a^11*c^13))^(1/4
) + x) + sqrt(x))) - 15*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^11*c^
13))^(1/8)*log(a^7*c^8*(-1/(a^11*c^13))^(5/8) + sqrt(x)) + 15*sqrt(2)*(a*c^3*x^8
+ 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^11*c^13))^(1/8)*log(-a^7*c^8*(-1/(a^11*c^13))^(
5/8) + sqrt(x)) - 60*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^11*c^13))^(1/8)*
arctan(a^7*c^8*(-1/(a^11*c^13))^(5/8)/(a^7*c^8*(-1/(a^11*c^13))^(5/8) + sqrt(2)*
sqrt(x) + sqrt(2*sqrt(2)*a^7*c^8*sqrt(x)*(-1/(a^11*c^13))^(5/8) - 2*a^3*c^3*(-1/
(a^11*c^13))^(1/4) + 2*x))) - 60*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^11*c
^13))^(1/8)*arctan(-a^7*c^8*(-1/(a^11*c^13))^(5/8)/(a^7*c^8*(-1/(a^11*c^13))^(5/
8) - sqrt(2)*sqrt(x) - sqrt(-2*sqrt(2)*a^7*c^8*sqrt(x)*(-1/(a^11*c^13))^(5/8) -
2*a^3*c^3*(-1/(a^11*c^13))^(1/4) + 2*x))) + 15*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*
c)*(-1/(a^11*c^13))^(1/8)*log(2*sqrt(2)*a^7*c^8*sqrt(x)*(-1/(a^11*c^13))^(5/8) -
2*a^3*c^3*(-1/(a^11*c^13))^(1/4) + 2*x) - 15*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c
)*(-1/(a^11*c^13))^(1/8)*log(-2*sqrt(2)*a^7*c^8*sqrt(x)*(-1/(a^11*c^13))^(5/8) -
2*a^3*c^3*(-1/(a^11*c^13))^(1/4) + 2*x) + 8*sqrt(2)*(5*c*x^6 - 3*a*x^2)*sqrt(x)
)/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)

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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**(11/2)/(c*x**4+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.356975, size = 663, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

-15/512*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) +
2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c) - 15/512*sqrt(-sqrt(2) + 2)*
(a/c)^(5/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) +
2)*(a/c)^(1/8)))/(a^2*c) + 15/512*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*arctan((sqrt(sq
rt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c) +
15/512*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*
sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c) - 15/1024*sqrt(-sqrt(2) + 2)*
(a/c)^(5/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c)
+ 15/1024*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/
8) + x + (a/c)^(1/4))/(a^2*c) + 15/1024*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*ln(sqrt(x)
*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c) - 15/1024*sqrt(sqrt(2
) + 2)*(a/c)^(5/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))
/(a^2*c) + 1/64*(5*c*x^(13/2) - 3*a*x^(5/2))/((c*x^4 + a)^2*a*c)