3.741 \(\int \frac{1}{\sqrt{x} \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=287 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}} \]

[Out]

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

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Rubi [A]  time = 0.465634, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(a + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 104.998, size = 264, normalized size = 0.92 \[ \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 \sqrt [8]{c} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 \sqrt [8]{c} \left (- a\right )^{\frac{7}{8}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 \sqrt [8]{c} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 \sqrt [8]{c} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 \sqrt [8]{c} \left (- a\right )^{\frac{7}{8}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 \sqrt [8]{c} \left (- a\right )^{\frac{7}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.289531, size = 348, normalized size = 1.21 \[ \frac{-\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{4 a^{7/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] + 2*ArcTan[
(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - Cos[Pi/8]*Log[a^(1/
4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]] + Cos[Pi/8]*Log[a^(1/4) +
c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]] - 2*ArcTan[Cot[Pi/8] - (c^(1/8)
*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + 2*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*C
sc[Pi/8])/a^(1/8)]*Sin[Pi/8] - Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[
x]*Sin[Pi/8]]*Sin[Pi/8] + Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Si
n[Pi/8]]*Sin[Pi/8])/(4*a^(7/8)*c^(1/8))

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Maple [C]  time = 0.008, size = 29, normalized size = 0.1 \[{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+a)/x^(1/2),x)

[Out]

1/4/c*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. \[ -c \int \frac{x^{\frac{7}{2}}}{a c x^{4} + a^{2}}\,{d x} + \frac{2 \, \sqrt{x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*integrate(x^(7/2)/(a*c*x^4 + a^2), x) + 2*sqrt(x)/a

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Fricas [A]  time = 0.259465, size = 541, normalized size = 1.89 \[ -\frac{1}{8} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}}}{\sqrt{a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + x} + \sqrt{x}}\right ) - \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \log \left (a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \log \left (-a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + 4 \, \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}}}{a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{x} + \sqrt{2 \, a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + 2 \, \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + 2 \, x}}\right ) + 4 \, \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \arctan \left (-\frac{a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}}}{a \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} - \sqrt{2} \sqrt{x} - \sqrt{2 \, a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} - 2 \, \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + 2 \, x}}\right ) - \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \log \left (2 \, a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + 2 \, \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + 2 \, x\right ) + \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \log \left (2 \, a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} - 2 \, \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + 2 \, x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.271278, size = 590, normalized size = 2.06 \[ \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqr
t(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a + 1/4*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*arc
tan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)
))/a + 1/4*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 + 1/4*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 + 1/8*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 - 1/8*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 + 1/8*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 - 1/8*sqr
t(-sqrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/
c)^(1/4))/a