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

Optimal. Leaf size=31 \[ \frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{3/2}} \]

[Out]

x/c - (Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(3/2)

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Rubi [A]  time = 0.0424931, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(b*x^2 + c*x^4),x]

[Out]

x/c - (Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(3/2)

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Rubi in Sympy [A]  time = 8.56843, size = 26, normalized size = 0.84 \[ - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{3}{2}}} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*atan(sqrt(c)*x/sqrt(b))/c**(3/2) + x/c

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Mathematica [A]  time = 0.0151931, size = 31, normalized size = 1. \[ \frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

x/c - (Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(3/2)

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Maple [A]  time = 0.004, size = 27, normalized size = 0.9 \[{\frac{x}{c}}-{\frac{b}{c}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^4+b*x^2),x)

[Out]

x/c-b/c/(b*c)^(1/2)*arctan(c*x/(b*c)^(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(x^4/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259961, size = 1, normalized size = 0.03 \[ \left [\frac{\sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 2 \, x}{2 \, c}, -\frac{\sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) - x}{c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(-b/c)*log((c*x^2 - 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) + 2*x)/c, -(sqr
t(b/c)*arctan(x/sqrt(b/c)) - x)/c]

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Sympy [A]  time = 1.18445, size = 56, normalized size = 1.81 \[ \frac{\sqrt{- \frac{b}{c^{3}}} \log{\left (- c \sqrt{- \frac{b}{c^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{c^{3}}} \log{\left (c \sqrt{- \frac{b}{c^{3}}} + x \right )}}{2} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-b/c**3)*log(-c*sqrt(-b/c**3) + x)/2 - sqrt(-b/c**3)*log(c*sqrt(-b/c**3) +
x)/2 + x/c

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GIAC/XCAS [A]  time = 0.268052, size = 35, normalized size = 1.13 \[ -\frac{b \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c) + x/c