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 = 0.0181689, 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, Rules used = {1584, 321, 205} \[ \frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{3/2}} \]

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)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^4}{b x^2+c x^4} \, dx &=\int \frac{x^2}{b+c x^2} \, dx\\ &=\frac{x}{c}-\frac{b \int \frac{1}{b+c x^2} \, dx}{c}\\ &=\frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.008255, 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 = 0.043, size = 27, normalized size = 0.9 \begin{align*}{\frac{x}{c}}-{\frac{b}{c}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

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(x*c/(b*c)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 1.51991, size = 165, normalized size = 5.32 \begin{align*} \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{c x \sqrt{\frac{b}{c}}}{b}\right ) - x}{c}\right ] \end{align*}

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, -(sqrt(b/c)*arctan(c*x*sqrt(b/c)/b)
- x)/c]

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Sympy [B]  time = 0.312609, size = 56, normalized size = 1.81 \begin{align*} \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} \end{align*}

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 [A]  time = 1.24588, size = 35, normalized size = 1.13 \begin{align*} -\frac{b \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c} + \frac{x}{c} \end{align*}

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