3.746 \(\int \frac{x^{9/2}}{\left (a+c x^4\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.544083, 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 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{5/8} c^{11/8}}+\frac{3 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{5/8} c^{11/8}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{5/8} c^{11/8}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{5/8} c^{11/8}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac{x^{3/2}}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(9/2)/(a + c*x^4)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 117.273, size = 289, normalized size = 0.94 \[ - \frac{x^{\frac{3}{2}}}{4 c \left (a + c x^{4}\right )} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{11}{8}} \left (- a\right )^{\frac{5}{8}}} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{11}{8}} \left (- a\right )^{\frac{5}{8}}} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{11}{8}} \left (- a\right )^{\frac{5}{8}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{11}{8}} \left (- a\right )^{\frac{5}{8}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{11}{8}} \left (- a\right )^{\frac{5}{8}}} - \frac{3 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{11}{8}} \left (- a\right )^{\frac{5}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)/(c*x**4+a)**2,x)

[Out]

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

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Mathematica [A]  time = 1.94538, size = 406, normalized size = 1.32 \[ \frac{\frac{3 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{5/8}}-\frac{3 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{5/8}}-\frac{3 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{5/8}}+\frac{3 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{5/8}}-\frac{6 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{a^{5/8}}-\frac{6 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{a^{5/8}}-\frac{6 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{a^{5/8}}+\frac{6 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{a^{5/8}}-\frac{8 c^{3/8} x^{3/2}}{a+c x^4}}{32 c^{11/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(9/2)/(a + c*x^4)^2,x]

[Out]

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

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Maple [C]  time = 0.02, size = 47, normalized size = 0.2 \[ -{\frac{1}{4\,c \left ( c{x}^{4}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{3}{32\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)/(c*x^4+a)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.277675, size = 771, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + a)^2,x, algorithm="fricas")

[Out]

-1/64*sqrt(2)*(12*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*arctan(a^2*c^4*(
-1/(a^5*c^11))^(3/8)/(sqrt(a^4*c^8*(-1/(a^5*c^11))^(3/4) + x) + sqrt(x))) + 3*sq
rt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(a^2*c^4*(-1/(a^5*c^11))^(3/8) +
sqrt(x)) - 3*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(-a^2*c^4*(-1/(a^5
*c^11))^(3/8) + sqrt(x)) - 12*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*arctan(a^2*c
^4*(-1/(a^5*c^11))^(3/8)/(a^2*c^4*(-1/(a^5*c^11))^(3/8) + sqrt(2)*sqrt(x) + sqrt
(2*a^4*c^8*(-1/(a^5*c^11))^(3/4) + 2*sqrt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/
8) + 2*x))) - 12*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*arctan(-a^2*c^4*(-1/(a^5*
c^11))^(3/8)/(a^2*c^4*(-1/(a^5*c^11))^(3/8) - sqrt(2)*sqrt(x) - sqrt(2*a^4*c^8*(
-1/(a^5*c^11))^(3/4) - 2*sqrt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/8) + 2*x)))
- 3*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(2*a^4*c^8*(-1/(a^5*c^11))^(3/4) +
2*sqrt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/8) + 2*x) + 3*(c^2*x^4 + a*c)*(-1/(
a^5*c^11))^(1/8)*log(2*a^4*c^8*(-1/(a^5*c^11))^(3/4) - 2*sqrt(2)*a^2*c^4*sqrt(x)
*(-1/(a^5*c^11))^(3/8) + 2*x) + 8*sqrt(2)*x^(3/2))/(c^2*x^4 + a*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**(9/2)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.333066, size = 645, normalized size = 2.09 \[ -\frac{3 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} - \frac{3 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{3 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{3 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{3 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{3 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{3 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} + \frac{3 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{x^{\frac{3}{2}}}{4 \,{\left (c x^{4} + a\right )} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + a)^2,x, algorithm="giac")

[Out]

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