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

Optimal. Leaf size=243 \[ \frac{9 c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{9 c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{9 c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{13/4}}+\frac{9 c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{13/4}}+\frac{9 c}{2 b^3 \sqrt{x}}-\frac{9}{10 b^2 x^{5/2}}+\frac{1}{2 b x^{5/2} \left (b+c x^2\right )} \]

[Out]

-9/(10*b^2*x^(5/2)) + (9*c)/(2*b^3*Sqrt[x]) + 1/(2*b*x^(5/2)*(b + c*x^2)) - (9*c
^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (9*
c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (9
*c^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*
b^(13/4)) - (9*c^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/(8*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.430819, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{9 c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{9 c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{9 c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{13/4}}+\frac{9 c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{13/4}}+\frac{9 c}{2 b^3 \sqrt{x}}-\frac{9}{10 b^2 x^{5/2}}+\frac{1}{2 b x^{5/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-9/(10*b^2*x^(5/2)) + (9*c)/(2*b^3*Sqrt[x]) + 1/(2*b*x^(5/2)*(b + c*x^2)) - (9*c
^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (9*
c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (9
*c^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*
b^(13/4)) - (9*c^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/(8*Sqrt[2]*b^(13/4))

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Rubi in Sympy [A]  time = 79.7024, size = 231, normalized size = 0.95 \[ \frac{1}{2 b x^{\frac{5}{2}} \left (b + c x^{2}\right )} - \frac{9}{10 b^{2} x^{\frac{5}{2}}} + \frac{9 c}{2 b^{3} \sqrt{x}} + \frac{9 \sqrt{2} c^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{13}{4}}} - \frac{9 \sqrt{2} c^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{13}{4}}} - \frac{9 \sqrt{2} c^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{13}{4}}} + \frac{9 \sqrt{2} c^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*b*x**(5/2)*(b + c*x**2)) - 9/(10*b**2*x**(5/2)) + 9*c/(2*b**3*sqrt(x)) + 9*
sqrt(2)*c**(5/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(
16*b**(13/4)) - 9*sqrt(2)*c**(5/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(
b) + sqrt(c)*x)/(16*b**(13/4)) - 9*sqrt(2)*c**(5/4)*atan(1 - sqrt(2)*c**(1/4)*sq
rt(x)/b**(1/4))/(8*b**(13/4)) + 9*sqrt(2)*c**(5/4)*atan(1 + sqrt(2)*c**(1/4)*sqr
t(x)/b**(1/4))/(8*b**(13/4))

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Mathematica [A]  time = 0.405078, size = 227, normalized size = 0.93 \[ \frac{-\frac{32 b^{5/4}}{x^{5/2}}+45 \sqrt{2} c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-45 \sqrt{2} c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-90 \sqrt{2} c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{40 \sqrt [4]{b} c^2 x^{3/2}}{b+c x^2}+\frac{320 \sqrt [4]{b} c}{\sqrt{x}}}{80 b^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*b^(5/4))/x^(5/2) + (320*b^(1/4)*c)/Sqrt[x] + (40*b^(1/4)*c^2*x^(3/2))/(b +
c*x^2) - 90*Sqrt[2]*c^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 90*
Sqrt[2]*c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 45*Sqrt[2]*c^(5/
4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 45*Sqrt[2]*c^(5/
4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(80*b^(13/4))

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Maple [A]  time = 0.025, size = 172, normalized size = 0.7 \[{\frac{{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{9\,c\sqrt{2}}{16\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{2}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}+4\,{\frac{c}{{b}^{3}\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^4+b*x^2)^2,x)

[Out]

1/2/b^3*c^2*x^(3/2)/(c*x^2+b)+9/16/b^3*c/(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)))+9/8/b^3
*c/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+9/8/b^3*c/(b/c)^(1/
4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-2/5/b^2/x^(5/2)+4*c/b^3/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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284435, size = 325, normalized size = 1.34 \[ \frac{180 \, c^{2} x^{4} + 144 \, b c x^{2} + 180 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )} \sqrt{x} \left (-\frac{c^{5}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{729 \, b^{10} \left (-\frac{c^{5}}{b^{13}}\right )^{\frac{3}{4}}}{729 \, c^{4} \sqrt{x} + \sqrt{-531441 \, b^{7} c^{5} \sqrt{-\frac{c^{5}}{b^{13}}} + 531441 \, c^{8} x}}\right ) + 45 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )} \sqrt{x} \left (-\frac{c^{5}}{b^{13}}\right )^{\frac{1}{4}} \log \left (729 \, b^{10} \left (-\frac{c^{5}}{b^{13}}\right )^{\frac{3}{4}} + 729 \, c^{4} \sqrt{x}\right ) - 45 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )} \sqrt{x} \left (-\frac{c^{5}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-729 \, b^{10} \left (-\frac{c^{5}}{b^{13}}\right )^{\frac{3}{4}} + 729 \, c^{4} \sqrt{x}\right ) - 16 \, b^{2}}{40 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*(180*c^2*x^4 + 144*b*c*x^2 + 180*(b^3*c*x^4 + b^4*x^2)*sqrt(x)*(-c^5/b^13)^
(1/4)*arctan(729*b^10*(-c^5/b^13)^(3/4)/(729*c^4*sqrt(x) + sqrt(-531441*b^7*c^5*
sqrt(-c^5/b^13) + 531441*c^8*x))) + 45*(b^3*c*x^4 + b^4*x^2)*sqrt(x)*(-c^5/b^13)
^(1/4)*log(729*b^10*(-c^5/b^13)^(3/4) + 729*c^4*sqrt(x)) - 45*(b^3*c*x^4 + b^4*x
^2)*sqrt(x)*(-c^5/b^13)^(1/4)*log(-729*b^10*(-c^5/b^13)^(3/4) + 729*c^4*sqrt(x))
- 16*b^2)/((b^3*c*x^4 + b^4*x^2)*sqrt(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(x**(1/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280156, size = 297, normalized size = 1.22 \[ \frac{c^{2} x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b^{3}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4} c} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4} c} - \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4} c} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4} c} + \frac{2 \,{\left (10 \, c x^{2} - b\right )}}{5 \, b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c^2*x^(3/2)/((c*x^2 + b)*b^3) + 9/8*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^4*c) + 9/8*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^4*c) -
9/16*sqrt(2)*(b*c^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4
*c) + 9/16*sqrt(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c)
)/(b^4*c) + 2/5*(10*c*x^2 - b)/(b^3*x^(5/2))