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

Optimal. Leaf size=308 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{x^{5/2}}{4 a \left (a+c x^4\right )} \]

[Out]

x^(5/2)/(4*a*(a + c*x^4)) + (3*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])
/(16*Sqrt[2]*(-a)^(11/8)*c^(5/8)) - (3*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)
^(1/8)])/(16*Sqrt[2]*(-a)^(11/8)*c^(5/8)) + (3*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(16*(-a)^(11/8)*c^(5/8)) + (3*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-
a)^(11/8)*c^(5/8)) + (3*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^
(1/4)*x])/(32*Sqrt[2]*(-a)^(11/8)*c^(5/8)) - (3*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1
/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(11/8)*c^(5/8))

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Rubi [A]  time = 0.531817, 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 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{x^{5/2}}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + c*x^4)^2,x]

[Out]

x^(5/2)/(4*a*(a + c*x^4)) + (3*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])
/(16*Sqrt[2]*(-a)^(11/8)*c^(5/8)) - (3*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)
^(1/8)])/(16*Sqrt[2]*(-a)^(11/8)*c^(5/8)) + (3*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(16*(-a)^(11/8)*c^(5/8)) + (3*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-
a)^(11/8)*c^(5/8)) + (3*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^
(1/4)*x])/(32*Sqrt[2]*(-a)^(11/8)*c^(5/8)) - (3*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1
/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(11/8)*c^(5/8))

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Rubi in Sympy [A]  time = 113.347, size = 289, normalized size = 0.94 \[ \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{5}{8}} \left (- a\right )^{\frac{11}{8}}} - \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{5}{8}} \left (- a\right )^{\frac{11}{8}}} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{5}{8}} \left (- a\right )^{\frac{11}{8}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{5}{8}} \left (- a\right )^{\frac{11}{8}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{5}{8}} \left (- a\right )^{\frac{11}{8}}} + \frac{3 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{5}{8}} \left (- a\right )^{\frac{11}{8}}} + \frac{x^{\frac{5}{2}}}{4 a \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/
(64*c**(5/8)*(-a)**(11/8)) - 3*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8)
+ c**(1/4)*x + (-a)**(1/4))/(64*c**(5/8)*(-a)**(11/8)) + 3*atan(c**(1/8)*sqrt(x)
/(-a)**(1/8))/(16*c**(5/8)*(-a)**(11/8)) - 3*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(
x)/(-a)**(1/8) - 1)/(32*c**(5/8)*(-a)**(11/8)) - 3*sqrt(2)*atan(sqrt(2)*c**(1/8)
*sqrt(x)/(-a)**(1/8) + 1)/(32*c**(5/8)*(-a)**(11/8)) + 3*atanh(c**(1/8)*sqrt(x)/
(-a)**(1/8))/(16*c**(5/8)*(-a)**(11/8)) + x**(5/2)/(4*a*(a + c*x**4))

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Mathematica [A]  time = 1.95493, size = 406, normalized size = 1.32 \[ \frac{\frac{8 a^{3/8} x^{5/2}}{a+c x^4}-\frac{3 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}+\frac{3 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}+\frac{3 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}-\frac{3 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}-\frac{6 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{c^{5/8}}-\frac{6 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{c^{5/8}}-\frac{6 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{c^{5/8}}+\frac{6 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{c^{5/8}}}{32 a^{11/8}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^(3/8)*x^(5/2))/(a + c*x^4) - (6*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi
/8])/a^(1/8)]*Cos[Pi/8])/c^(5/8) + (6*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi
/8])/a^(1/8)]*Cos[Pi/8])/c^(5/8) - (3*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1
/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]])/c^(5/8) + (3*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x +
2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]])/c^(5/8) - (6*ArcTan[(c^(1/8)*Sqrt[x]*Sec[
Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8])/c^(5/8) - (6*ArcTan[(c^(1/8)*Sqrt[x]*Sec[
Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[Pi/8])/c^(5/8) + (3*Log[a^(1/4) + c^(1/4)*x - 2*
a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8])/c^(5/8) - (3*Log[a^(1/4) + c^(1/4)
*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8])/c^(5/8))/(32*a^(11/8))

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Maple [C]  time = 0.018, size = 50, normalized size = 0.2 \[{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }{x}^{{\frac{5}{2}}}}+{\frac{3}{32\,ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(c*x^4+a)^2,x)

[Out]

1/4*x^(5/2)/a/(c*x^4+a)+3/32/a/c*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. \[ \frac{x^{\frac{5}{2}}}{4 \,{\left (a c x^{4} + a^{2}\right )}} + 3 \, \int \frac{x^{\frac{3}{2}}}{8 \,{\left (a c x^{4} + a^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270533, size = 748, normalized size = 2.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*sqrt(2)*(8*sqrt(2)*x^(5/2) + 12*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^11*c^5))^(1/
8)*arctan(a^7*c^3*(-1/(a^11*c^5))^(5/8)/(sqrt(-a^3*c*(-1/(a^11*c^5))^(1/4) + x)
+ sqrt(x))) - 3*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^11*c^5))^(1/8)*log(a^7*c^3*(-1/(a
^11*c^5))^(5/8) + sqrt(x)) + 3*sqrt(2)*(a*c*x^4 + a^2)*(-1/(a^11*c^5))^(1/8)*log
(-a^7*c^3*(-1/(a^11*c^5))^(5/8) + sqrt(x)) - 12*(a*c*x^4 + a^2)*(-1/(a^11*c^5))^
(1/8)*arctan(a^7*c^3*(-1/(a^11*c^5))^(5/8)/(a^7*c^3*(-1/(a^11*c^5))^(5/8) + sqrt
(2)*sqrt(x) + sqrt(2*sqrt(2)*a^7*c^3*sqrt(x)*(-1/(a^11*c^5))^(5/8) - 2*a^3*c*(-1
/(a^11*c^5))^(1/4) + 2*x))) - 12*(a*c*x^4 + a^2)*(-1/(a^11*c^5))^(1/8)*arctan(-a
^7*c^3*(-1/(a^11*c^5))^(5/8)/(a^7*c^3*(-1/(a^11*c^5))^(5/8) - sqrt(2)*sqrt(x) -
sqrt(-2*sqrt(2)*a^7*c^3*sqrt(x)*(-1/(a^11*c^5))^(5/8) - 2*a^3*c*(-1/(a^11*c^5))^
(1/4) + 2*x))) + 3*(a*c*x^4 + a^2)*(-1/(a^11*c^5))^(1/8)*log(2*sqrt(2)*a^7*c^3*s
qrt(x)*(-1/(a^11*c^5))^(5/8) - 2*a^3*c*(-1/(a^11*c^5))^(1/4) + 2*x) - 3*(a*c*x^4
+ a^2)*(-1/(a^11*c^5))^(1/8)*log(-2*sqrt(2)*a^7*c^3*sqrt(x)*(-1/(a^11*c^5))^(5/
8) - 2*a^3*c*(-1/(a^11*c^5))^(1/4) + 2*x))/(a*c*x^4 + a^2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.335713, size = 613, normalized size = 1.99 \[ \frac{x^{\frac{5}{2}}}{4 \,{\left (c x^{4} + a\right )} a} - \frac{3 \, \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^{2}} - \frac{3 \, \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^{2}} + \frac{3 \, \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^{2}} + \frac{3 \, \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^{2}} - \frac{3 \, \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^{2}} + \frac{3 \, \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^{2}} + \frac{3 \, \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^{2}} - \frac{3 \, \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^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^(5/2)/((c*x^4 + a)*a) - 3/32*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^2 - 3/3
2*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqr
t(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 3/32*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^2 + 3/32*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^2 - 3/64*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^2 +
3/64*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^2 + 3/64*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^2 - 3/64*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^2