3.744 \(\int \frac{x^{13/2}}{\left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=308 \[ \frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} \sqrt [8]{-a} c^{15/8}}-\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} \sqrt [8]{-a} c^{15/8}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} \sqrt [8]{-a} c^{15/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} \sqrt [8]{-a} c^{15/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt [8]{-a} c^{15/8}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt [8]{-a} c^{15/8}}-\frac{x^{7/2}}{4 c \left (a+c x^4\right )} \]

[Out]

-x^(7/2)/(4*c*(a + c*x^4)) - (7*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(16*Sqrt[2]*(-a)^(1/8)*c^(15/8)) + (7*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a
)^(1/8)])/(16*Sqrt[2]*(-a)^(1/8)*c^(15/8)) + (7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1
/8)])/(16*(-a)^(1/8)*c^(15/8)) - (7*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(
-a)^(1/8)*c^(15/8)) + (7*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c
^(1/4)*x])/(32*Sqrt[2]*(-a)^(1/8)*c^(15/8)) - (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(
1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(1/8)*c^(15/8))

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Rubi [A]  time = 0.56899, 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 \[ \frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} \sqrt [8]{-a} c^{15/8}}-\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} \sqrt [8]{-a} c^{15/8}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} \sqrt [8]{-a} c^{15/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} \sqrt [8]{-a} c^{15/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt [8]{-a} c^{15/8}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt [8]{-a} c^{15/8}}-\frac{x^{7/2}}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/(a + c*x^4)^2,x]

[Out]

-x^(7/2)/(4*c*(a + c*x^4)) - (7*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(16*Sqrt[2]*(-a)^(1/8)*c^(15/8)) + (7*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a
)^(1/8)])/(16*Sqrt[2]*(-a)^(1/8)*c^(15/8)) + (7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1
/8)])/(16*(-a)^(1/8)*c^(15/8)) - (7*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(
-a)^(1/8)*c^(15/8)) + (7*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c
^(1/4)*x])/(32*Sqrt[2]*(-a)^(1/8)*c^(15/8)) - (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(
1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(1/8)*c^(15/8))

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Rubi in Sympy [A]  time = 109.696, size = 289, normalized size = 0.94 \[ - \frac{x^{\frac{7}{2}}}{4 c \left (a + c x^{4}\right )} + \frac{7 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{15}{8}} \sqrt [8]{- a}} - \frac{7 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{15}{8}} \sqrt [8]{- a}} + \frac{7 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{15}{8}} \sqrt [8]{- a}} + \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{15}{8}} \sqrt [8]{- a}} + \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{15}{8}} \sqrt [8]{- a}} - \frac{7 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{15}{8}} \sqrt [8]{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)/(c*x**4+a)**2,x)

[Out]

-x**(7/2)/(4*c*(a + c*x**4)) + 7*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/
8) + c**(1/4)*x + (-a)**(1/4))/(64*c**(15/8)*(-a)**(1/8)) - 7*sqrt(2)*log(sqrt(2
)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(64*c**(15/8)*(-a)**(
1/8)) + 7*atan(c**(1/8)*sqrt(x)/(-a)**(1/8))/(16*c**(15/8)*(-a)**(1/8)) + 7*sqrt
(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) - 1)/(32*c**(15/8)*(-a)**(1/8)) +
7*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(32*c**(15/8)*(-a)**(1/
8)) - 7*atanh(c**(1/8)*sqrt(x)/(-a)**(1/8))/(16*c**(15/8)*(-a)**(1/8))

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Mathematica [A]  time = 1.72581, size = 406, normalized size = 1.32 \[ \frac{-\frac{8 c^{7/8} x^{7/2}}{a+c x^4}+\frac{7 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{a}}-\frac{7 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{a}}+\frac{7 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{a}}-\frac{7 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\sqrt [8]{a}}+\frac{14 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{a}}+\frac{14 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{a}}-\frac{14 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{\sqrt [8]{a}}+\frac{14 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{\sqrt [8]{a}}}{32 c^{15/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/(a + c*x^4)^2,x]

[Out]

((-8*c^(7/8)*x^(7/2))/(a + c*x^4) + (14*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/
8) - Tan[Pi/8]]*Cos[Pi/8])/a^(1/8) + (14*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1
/8) + Tan[Pi/8]]*Cos[Pi/8])/a^(1/8) + (7*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a
^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/a^(1/8) - (7*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*
x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/a^(1/8) - (14*ArcTan[Cot[Pi/8] - (c^(1
/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/a^(1/8) + (14*ArcTan[Cot[Pi/8] + (c^(
1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/a^(1/8) + (7*Log[a^(1/4) + c^(1/4)*x
- 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/a^(1/8) - (7*Log[a^(1/4) + c^
(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/a^(1/8))/(32*c^(15/8))

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Maple [C]  time = 0.02, size = 47, normalized size = 0.2 \[ -{\frac{1}{4\,c \left ( c{x}^{4}+a \right ) }{x}^{{\frac{7}{2}}}}+{\frac{7}{32\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/(c*x^4+a)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.263735, size = 729, normalized size = 2.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + a)^2,x, algorithm="fricas")

[Out]

-1/64*sqrt(2)*(8*sqrt(2)*x^(7/2) - 28*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8
)*arctan(a*c^13*(-1/(a*c^15))^(7/8)/(sqrt(-a*c^11*(-1/(a*c^15))^(3/4) + x) + sqr
t(x))) - 7*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8)*log(a*c^13*(-1/(a*c^15))^
(7/8) + sqrt(x)) + 7*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8)*log(-a*c^13*(-1
/(a*c^15))^(7/8) + sqrt(x)) - 28*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8)*arctan(a*c^
13*(-1/(a*c^15))^(7/8)/(a*c^13*(-1/(a*c^15))^(7/8) + sqrt(2)*sqrt(x) + sqrt(2*sq
rt(2)*a*c^13*sqrt(x)*(-1/(a*c^15))^(7/8) - 2*a*c^11*(-1/(a*c^15))^(3/4) + 2*x)))
- 28*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8)*arctan(-a*c^13*(-1/(a*c^15))^(7/8)/(a*
c^13*(-1/(a*c^15))^(7/8) - sqrt(2)*sqrt(x) - sqrt(-2*sqrt(2)*a*c^13*sqrt(x)*(-1/
(a*c^15))^(7/8) - 2*a*c^11*(-1/(a*c^15))^(3/4) + 2*x))) - 7*(c^2*x^4 + a*c)*(-1/
(a*c^15))^(1/8)*log(2*sqrt(2)*a*c^13*sqrt(x)*(-1/(a*c^15))^(7/8) - 2*a*c^11*(-1/
(a*c^15))^(3/4) + 2*x) + 7*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8)*log(-2*sqrt(2)*a*
c^13*sqrt(x)*(-1/(a*c^15))^(7/8) - 2*a*c^11*(-1/(a*c^15))^(3/4) + 2*x))/(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**(13/2)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.330762, size = 645, normalized size = 2.09 \[ -\frac{x^{\frac{7}{2}}}{4 \,{\left (c x^{4} + a\right )} c} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} - \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + a)^2,x, algorithm="giac")

[Out]

-1/4*x^(7/2)/((c*x^4 + a)*c) + 7/32*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*c) + 7
/32*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sq
rt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c) + 7/32*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*c) + 7/32*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*c) - 7/64*sqrt(sqr
t(2) + 2)*(a/c)^(7/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4)
)/(a*c) + 7/64*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*c) - 7/64*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*c) + 7/64*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*
c)