3.322 \(\int \frac{\sqrt{x}}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=202 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{5/4}}-\frac{2}{b \sqrt{x}} \]

[Out]

-2/(b*Sqrt[x]) + (c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2
]*b^(5/4)) - (c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^
(5/4)) - (c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2
*Sqrt[2]*b^(5/4)) + (c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x])/(2*Sqrt[2]*b^(5/4))

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Rubi [A]  time = 0.338436, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{5/4}}-\frac{2}{b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(b*x^2 + c*x^4),x]

[Out]

-2/(b*Sqrt[x]) + (c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2
]*b^(5/4)) - (c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^
(5/4)) - (c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2
*Sqrt[2]*b^(5/4)) + (c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x])/(2*Sqrt[2]*b^(5/4))

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Rubi in Sympy [A]  time = 65.8307, size = 190, normalized size = 0.94 \[ - \frac{2}{b \sqrt{x}} - \frac{\sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(b*sqrt(x)) - sqrt(2)*c**(1/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(
b) + sqrt(c)*x)/(4*b**(5/4)) + sqrt(2)*c**(1/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sq
rt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(5/4)) + sqrt(2)*c**(1/4)*atan(1 - sqrt(2)*c*
*(1/4)*sqrt(x)/b**(1/4))/(2*b**(5/4)) - sqrt(2)*c**(1/4)*atan(1 + sqrt(2)*c**(1/
4)*sqrt(x)/b**(1/4))/(2*b**(5/4))

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Mathematica [A]  time = 0.118246, size = 189, normalized size = 0.94 \[ \frac{-\sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+\sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+2 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-2 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{8 \sqrt [4]{b}}{\sqrt{x}}}{4 b^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(b*x^2 + c*x^4),x]

[Out]

((-8*b^(1/4))/Sqrt[x] + 2*Sqrt[2]*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b
^(1/4)] - 2*Sqrt[2]*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - Sqrt
[2]*c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + Sqrt[2]
*c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(4*b^(5/4))

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Maple [A]  time = 0.012, size = 140, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{4\,b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{\sqrt{2}}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{\sqrt{2}}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-2\,{\frac{1}{b\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^4+b*x^2),x)

[Out]

-1/4/b/(b/c)^(1/4)*2^(1/2)*ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x+(b/
c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))-1/2/b/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(b/c)^(1/4)*x^(1/2)+1)-1/2/b/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1
/2)-1)-2/b/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283527, size = 180, normalized size = 0.89 \[ -\frac{4 \, b \sqrt{x} \left (-\frac{c}{b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} \left (-\frac{c}{b^{5}}\right )^{\frac{3}{4}}}{c \sqrt{x} + \sqrt{-b^{3} c \sqrt{-\frac{c}{b^{5}}} + c^{2} x}}\right ) + b \sqrt{x} \left (-\frac{c}{b^{5}}\right )^{\frac{1}{4}} \log \left (b^{4} \left (-\frac{c}{b^{5}}\right )^{\frac{3}{4}} + c \sqrt{x}\right ) - b \sqrt{x} \left (-\frac{c}{b^{5}}\right )^{\frac{1}{4}} \log \left (-b^{4} \left (-\frac{c}{b^{5}}\right )^{\frac{3}{4}} + c \sqrt{x}\right ) + 4}{2 \, b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*b*sqrt(x)*(-c/b^5)^(1/4)*arctan(b^4*(-c/b^5)^(3/4)/(c*sqrt(x) + sqrt(-b^
3*c*sqrt(-c/b^5) + c^2*x))) + b*sqrt(x)*(-c/b^5)^(1/4)*log(b^4*(-c/b^5)^(3/4) +
c*sqrt(x)) - b*sqrt(x)*(-c/b^5)^(1/4)*log(-b^4*(-c/b^5)^(3/4) + c*sqrt(x)) + 4)/
(b*sqrt(x))

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Sympy [A]  time = 86.7168, size = 175, normalized size = 0.87 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: b = 0 \wedge c = 0 \\- \frac{2}{5 c x^{\frac{5}{2}}} & \text{for}\: b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: c = 0 \\- \frac{2}{b \sqrt{x}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{5}{4}} c \left (\frac{1}{c}\right )^{\frac{5}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{5}{4}} c \left (\frac{1}{c}\right )^{\frac{5}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{5}{4}} c \left (\frac{1}{c}\right )^{\frac{5}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(c*x**4+b*x**2),x)

[Out]

Piecewise((zoo/x**(5/2), Eq(b, 0) & Eq(c, 0)), (-2/(5*c*x**(5/2)), Eq(b, 0)), (-
2/(b*sqrt(x)), Eq(c, 0)), (-2/(b*sqrt(x)) + (-1)**(3/4)*log(-(-1)**(1/4)*b**(1/4
)*(1/c)**(1/4) + sqrt(x))/(2*b**(5/4)*c*(1/c)**(5/4)) - (-1)**(3/4)*log((-1)**(1
/4)*b**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(5/4)*c*(1/c)**(5/4)) - (-1)**(3/4)*a
tan((-1)**(3/4)*sqrt(x)/(b**(1/4)*(1/c)**(1/4)))/(b**(5/4)*c*(1/c)**(5/4)), True
))

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GIAC/XCAS [A]  time = 0.277707, size = 257, normalized size = 1.27 \[ -\frac{2}{b \sqrt{x}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c^{2}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(b*sqrt(x)) - 1/2*sqrt(2)*(b*c^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/
4) + 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^2) - 1/2*sqrt(2)*(b*c^3)^(3/4)*arctan(-1/2*s
qrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^2) + 1/4*sqrt(2)*(b
*c^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^2) - 1/4*sqrt
(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^2)