3.748 \(\int \frac{x^{5/2}}{\left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=308 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{9/8} c^{7/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{9/8} c^{7/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{9/8} c^{7/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{9/8} c^{7/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac{x^{7/2}}{4 a \left (a+c x^4\right )} \]

[Out]

x^(7/2)/(4*a*(a + c*x^4)) + ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(16
*Sqrt[2]*(-a)^(9/8)*c^(7/8)) - ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/
(16*Sqrt[2]*(-a)^(9/8)*c^(7/8)) - ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(16*(-a)^
(9/8)*c^(7/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(16*(-a)^(9/8)*c^(7/8)) -
Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(32*Sqrt[2]*(-
a)^(9/8)*c^(7/8)) + Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4
)*x]/(32*Sqrt[2]*(-a)^(9/8)*c^(7/8))

_______________________________________________________________________________________

Rubi [A]  time = 0.530903, 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 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{9/8} c^{7/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{9/8} c^{7/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{9/8} c^{7/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{9/8} c^{7/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac{x^{7/2}}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + c*x^4)^2,x]

[Out]

x^(7/2)/(4*a*(a + c*x^4)) + ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(16
*Sqrt[2]*(-a)^(9/8)*c^(7/8)) - ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/
(16*Sqrt[2]*(-a)^(9/8)*c^(7/8)) - ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(16*(-a)^
(9/8)*c^(7/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(16*(-a)^(9/8)*c^(7/8)) -
Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(32*Sqrt[2]*(-
a)^(9/8)*c^(7/8)) + Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4
)*x]/(32*Sqrt[2]*(-a)^(9/8)*c^(7/8))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 113.475, size = 279, normalized size = 0.91 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{7}{8}} \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{7}{8}} \left (- a\right )^{\frac{9}{8}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{7}{8}} \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{7}{8}} \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{7}{8}} \left (- a\right )^{\frac{9}{8}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{7}{8}} \left (- a\right )^{\frac{9}{8}}} + \frac{x^{\frac{7}{2}}}{4 a \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(c*x**4+a)**2,x)

[Out]

-sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(
64*c**(7/8)*(-a)**(9/8)) + sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c*
*(1/4)*x + (-a)**(1/4))/(64*c**(7/8)*(-a)**(9/8)) - atan(c**(1/8)*sqrt(x)/(-a)**
(1/8))/(16*c**(7/8)*(-a)**(9/8)) - sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(
1/8) - 1)/(32*c**(7/8)*(-a)**(9/8)) - sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)
**(1/8) + 1)/(32*c**(7/8)*(-a)**(9/8)) + atanh(c**(1/8)*sqrt(x)/(-a)**(1/8))/(16
*c**(7/8)*(-a)**(9/8)) + x**(7/2)/(4*a*(a + c*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 1.69989, size = 404, normalized size = 1.31 \[ \frac{\frac{\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}-\frac{\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}+\frac{\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}-\frac{\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}+\frac{2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{c^{7/8}}+\frac{2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{c^{7/8}}-\frac{2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{c^{7/8}}+\frac{2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{c^{7/8}}+\frac{8 \sqrt [8]{a} x^{7/2}}{a+c x^4}}{32 a^{9/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(a + c*x^4)^2,x]

[Out]

((8*a^(1/8)*x^(7/2))/(a + c*x^4) + (2*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8)
- Tan[Pi/8]]*Cos[Pi/8])/c^(7/8) + (2*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8)
+ Tan[Pi/8]]*Cos[Pi/8])/c^(7/8) + (Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8
)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/c^(7/8) - (Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a
^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/c^(7/8) - (2*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt
[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/c^(7/8) + (2*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt
[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/c^(7/8) + (Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8
)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/c^(7/8) - (Log[a^(1/4) + c^(1/4)*x + 2*a
^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/c^(7/8))/(32*a^(9/8))

_______________________________________________________________________________________

Maple [C]  time = 0.018, size = 50, normalized size = 0.2 \[{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }{x}^{{\frac{7}{2}}}}+{\frac{1}{32\,ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(c*x^4+a)^2,x)

[Out]

1/4*x^(7/2)/a/(c*x^4+a)+1/32/a/c*sum(1/_R*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{\frac{7}{2}}}{4 \,{\left (a c x^{4} + a^{2}\right )}} + \int \frac{x^{\frac{5}{2}}}{8 \,{\left (a c x^{4} + a^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

1/4*x^(7/2)/(a*c*x^4 + a^2) + integrate(1/8*x^(5/2)/(a*c*x^4 + a^2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.267931, 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^(5/2)/(c*x^4 + a)^2,x, algorithm="fricas")

[Out]

1/64*sqrt(2)*(8*sqrt(2)*x^(7/2) + 4*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^9*c^7))^(1/8)
*arctan(a^8*c^6*(-1/(a^9*c^7))^(7/8)/(sqrt(-a^7*c^5*(-1/(a^9*c^7))^(3/4) + x) +
sqrt(x))) + sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^9*c^7))^(1/8)*log(a^8*c^6*(-1/(a^9*c^
7))^(7/8) + sqrt(x)) - sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^9*c^7))^(1/8)*log(-a^8*c^6
*(-1/(a^9*c^7))^(7/8) + sqrt(x)) + 4*(a*c*x^4 + a^2)*(-1/(a^9*c^7))^(1/8)*arctan
(a^8*c^6*(-1/(a^9*c^7))^(7/8)/(a^8*c^6*(-1/(a^9*c^7))^(7/8) + sqrt(2)*sqrt(x) +
sqrt(2*sqrt(2)*a^8*c^6*sqrt(x)*(-1/(a^9*c^7))^(7/8) - 2*a^7*c^5*(-1/(a^9*c^7))^(
3/4) + 2*x))) + 4*(a*c*x^4 + a^2)*(-1/(a^9*c^7))^(1/8)*arctan(-a^8*c^6*(-1/(a^9*
c^7))^(7/8)/(a^8*c^6*(-1/(a^9*c^7))^(7/8) - sqrt(2)*sqrt(x) - sqrt(-2*sqrt(2)*a^
8*c^6*sqrt(x)*(-1/(a^9*c^7))^(7/8) - 2*a^7*c^5*(-1/(a^9*c^7))^(3/4) + 2*x))) + (
a*c*x^4 + a^2)*(-1/(a^9*c^7))^(1/8)*log(2*sqrt(2)*a^8*c^6*sqrt(x)*(-1/(a^9*c^7))
^(7/8) - 2*a^7*c^5*(-1/(a^9*c^7))^(3/4) + 2*x) - (a*c*x^4 + a^2)*(-1/(a^9*c^7))^
(1/8)*log(-2*sqrt(2)*a^8*c^6*sqrt(x)*(-1/(a^9*c^7))^(7/8) - 2*a^7*c^5*(-1/(a^9*c
^7))^(3/4) + 2*x))/(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**(5/2)/(c*x**4+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.332132, size = 613, normalized size = 1.99 \[ \frac{x^{\frac{7}{2}}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + a)^2,x, algorithm="giac")

[Out]

1/4*x^(7/2)/((c*x^4 + a)*a) + 1/32*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*arctan((sqrt(-s
qrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 1/32
*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(
x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 1/32*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*a
rctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8
)))/a^2 + 1/32*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(
1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^2 - 1/64*sqrt(sqrt(2) + 2)
*(a/c)^(7/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 + 1
/64*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x
+ (a/c)^(1/4))/a^2 - 1/64*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*ln(sqrt(x)*sqrt(-sqrt(2
) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 + 1/64*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*
ln(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2