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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.523117, 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 \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{3/8} c^{13/8}}+\frac{5 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{3/8} c^{13/8}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{3/8} c^{13/8}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{3/8} c^{13/8}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac{x^{5/2}}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(11/2)/(a + c*x^4)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 110.127, size = 289, normalized size = 0.94 \[ - \frac{x^{\frac{5}{2}}}{4 c \left (a + c x^{4}\right )} - \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{13}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{13}{8}} \left (- a\right )^{\frac{3}{8}}} - \frac{5 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{13}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{13}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{13}{8}} \left (- a\right )^{\frac{3}{8}}} - \frac{5 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{13}{8}} \left (- a\right )^{\frac{3}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 1.58524, size = 406, normalized size = 1.32 \[ \frac{-\frac{5 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{3/8}}+\frac{5 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{3/8}}+\frac{5 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{3/8}}-\frac{5 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{3/8}}-\frac{10 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{a^{3/8}}-\frac{10 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{a^{3/8}}-\frac{10 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{a^{3/8}}+\frac{10 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{a^{3/8}}-\frac{8 c^{5/8} x^{5/2}}{a+c x^4}}{32 c^{13/8}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.019, size = 47, normalized size = 0.2 \[ -{\frac{1}{4\,c \left ( c{x}^{4}+a \right ) }{x}^{{\frac{5}{2}}}}+{\frac{5}{32\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)/(c*x^4+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.260771, size = 759, normalized size = 2.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.339623, size = 645, normalized size = 2.09 \[ -\frac{x^{\frac{5}{2}}}{4 \,{\left (c x^{4} + a\right )} c} - \frac{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} - \frac{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} - \frac{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} + \frac{5 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} + \frac{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{5 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x^(5/2)/((c*x^4 + a)*c) - 5/32*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*c) -
5/32*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*c) + 5/32*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*c) + 5/32*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*c) - 5/64*sqrt(-sq
rt(2) + 2)*(a/c)^(5/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4
))/(a*c) + 5/64*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*c) + 5/64*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*c) - 5/64*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*
c)