3.743 \(\int \frac{1}{x^{5/2} \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=299 \[ -\frac{c^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{c^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{2}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.599176, antiderivative size = 299, 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 \[ -\frac{c^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{c^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{2}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 114.322, size = 274, normalized size = 0.92 \[ - \frac{\sqrt{2} c^{\frac{3}{8}} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} c^{\frac{3}{8}} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 \left (- a\right )^{\frac{11}{8}}} - \frac{c^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} c^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} c^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 \left (- a\right )^{\frac{11}{8}}} - \frac{c^{\frac{3}{8}} \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 \left (- a\right )^{\frac{11}{8}}} - \frac{2}{3 a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(c*x**4+a),x)

[Out]

-sqrt(2)*c**(3/8)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)*
*(1/4))/(8*(-a)**(11/8)) + sqrt(2)*c**(3/8)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(
1/8) + c**(1/4)*x + (-a)**(1/4))/(8*(-a)**(11/8)) - c**(3/8)*atan(c**(1/8)*sqrt(
x)/(-a)**(1/8))/(2*(-a)**(11/8)) + sqrt(2)*c**(3/8)*atan(sqrt(2)*c**(1/8)*sqrt(x
)/(-a)**(1/8) - 1)/(4*(-a)**(11/8)) + sqrt(2)*c**(3/8)*atan(sqrt(2)*c**(1/8)*sqr
t(x)/(-a)**(1/8) + 1)/(4*(-a)**(11/8)) - c**(3/8)*atanh(c**(1/8)*sqrt(x)/(-a)**(
1/8))/(2*(-a)**(11/8)) - 2/(3*a*x**(3/2))

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Mathematica [A]  time = 0.349181, size = 437, normalized size = 1.46 \[ \frac{-8 a^{3/8}+3 c^{3/8} x^{3/2} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-3 c^{3/8} x^{3/2} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-3 c^{3/8} x^{3/2} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+3 c^{3/8} x^{3/2} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+6 c^{3/8} x^{3/2} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+6 c^{3/8} x^{3/2} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+6 c^{3/8} x^{3/2} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-6 c^{3/8} x^{3/2} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{12 a^{11/8} x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + c*x^4)),x]

[Out]

(-8*a^(3/8) + 6*c^(3/8)*x^(3/2)*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a
^(1/8)]*Cos[Pi/8] - 6*c^(3/8)*x^(3/2)*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi
/8])/a^(1/8)]*Cos[Pi/8] + 3*c^(3/8)*x^(3/2)*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x -
2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]] - 3*c^(3/8)*x^(3/2)*Cos[Pi/8]*Log[a^(1/4) +
c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]] + 6*c^(3/8)*x^(3/2)*ArcTan[(c^
(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8] + 6*c^(3/8)*x^(3/2)*ArcT
an[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[Pi/8] - 3*c^(3/8)*x^(3/2
)*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8] + 3*c
^(3/8)*x^(3/2)*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]*Si
n[Pi/8])/(12*a^(11/8)*x^(3/2))

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Maple [C]  time = 0.012, size = 38, normalized size = 0.1 \[ -{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }}-{\frac{2}{3\,a}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(c*x^4+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*x^(5/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.261191, size = 710, normalized size = 2.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*x^(5/2)),x, algorithm="fricas")

[Out]

-1/24*sqrt(2)*(12*sqrt(2)*a*x^(3/2)*(-c^3/a^11)^(1/8)*arctan(a^7*(-c^3/a^11)^(5/
8)/(c^2*sqrt(x) + sqrt(-a^3*c^3*(-c^3/a^11)^(1/4) + c^4*x))) - 3*sqrt(2)*a*x^(3/
2)*(-c^3/a^11)^(1/8)*log(a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 3*sqrt(2)*a*x^(3
/2)*(-c^3/a^11)^(1/8)*log(-a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - 12*a*x^(3/2)*(
-c^3/a^11)^(1/8)*arctan(a^7*(-c^3/a^11)^(5/8)/(a^7*(-c^3/a^11)^(5/8) + sqrt(2)*c
^2*sqrt(x) + sqrt(2*sqrt(2)*a^7*c^2*sqrt(x)*(-c^3/a^11)^(5/8) - 2*a^3*c^3*(-c^3/
a^11)^(1/4) + 2*c^4*x))) - 12*a*x^(3/2)*(-c^3/a^11)^(1/8)*arctan(-a^7*(-c^3/a^11
)^(5/8)/(a^7*(-c^3/a^11)^(5/8) - sqrt(2)*c^2*sqrt(x) - sqrt(-2*sqrt(2)*a^7*c^2*s
qrt(x)*(-c^3/a^11)^(5/8) - 2*a^3*c^3*(-c^3/a^11)^(1/4) + 2*c^4*x))) + 3*a*x^(3/2
)*(-c^3/a^11)^(1/8)*log(2*sqrt(2)*a^7*c^2*sqrt(x)*(-c^3/a^11)^(5/8) - 2*a^3*c^3*
(-c^3/a^11)^(1/4) + 2*c^4*x) - 3*a*x^(3/2)*(-c^3/a^11)^(1/8)*log(-2*sqrt(2)*a^7*
c^2*sqrt(x)*(-c^3/a^11)^(5/8) - 2*a^3*c^3*(-c^3/a^11)^(1/4) + 2*c^4*x) + 8*sqrt(
2))/(a*x^(3/2))

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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(1/x**(5/2)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293143, size = 612, normalized size = 2.05 \[ \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} + \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} + \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} - \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} + \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} - \frac{2}{3 \, a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*x^(5/2)),x, algorithm="giac")

[Out]

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