3.747 \(\int \frac{x^{7/2}}{\left (a+c x^4\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.528013, 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 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{7/8} c^{9/8}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{7/8} c^{9/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{7/8} c^{9/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{7/8} c^{9/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac{\sqrt{x}}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(a + c*x^4)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 114.957, size = 279, normalized size = 0.91 \[ - \frac{\sqrt{x}}{4 c \left (a + c x^{4}\right )} + \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{9}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 c^{\frac{9}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{9}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 c^{\frac{9}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 c^{\frac{9}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 c^{\frac{9}{8}} \left (- a\right )^{\frac{7}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(c*x**4+a)**2,x)

[Out]

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

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Mathematica [A]  time = 1.55356, size = 404, normalized size = 1.31 \[ \frac{-\frac{\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}+\frac{\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}-\frac{\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}+\frac{\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}+\frac{2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{a^{7/8}}+\frac{2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{a^{7/8}}-\frac{2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{a^{7/8}}+\frac{2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{a^{7/8}}-\frac{8 \sqrt [8]{c} \sqrt{x}}{a+c x^4}}{32 c^{9/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(a + c*x^4)^2,x]

[Out]

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

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Maple [C]  time = 0.018, size = 47, normalized size = 0.2 \[ -{\frac{1}{4\,c \left ( c{x}^{4}+a \right ) }\sqrt{x}}+{\frac{1}{32\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(c*x^4+a)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{\frac{9}{2}}}{4 \,{\left (a c x^{4} + a^{2}\right )}} - \int \frac{x^{\frac{7}{2}}}{8 \,{\left (a c x^{4} + a^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

1/4*x^(9/2)/(a*c*x^4 + a^2) - integrate(1/8*x^(7/2)/(a*c*x^4 + a^2), x)

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Fricas [A]  time = 0.261808, size = 709, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.30362, size = 645, normalized size = 2.09 \[ \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a c} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a c} - \frac{\sqrt{x}}{4 \,{\left (c x^{4} + a\right )} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a)^2,x, algorithm="giac")

[Out]

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