3.176 \(\int \frac{x^8}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=55 \[ \frac{b^2 x}{c^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]

[Out]

(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(7/2)

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Rubi [A]  time = 0.0326585, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 302, 205} \[ \frac{b^2 x}{c^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(b*x^2 + c*x^4),x]

[Out]

(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(7/2)

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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^8}{b x^2+c x^4} \, dx &=\int \frac{x^6}{b+c x^2} \, dx\\ &=\int \left (\frac{b^2}{c^3}-\frac{b x^2}{c^2}+\frac{x^4}{c}-\frac{b^3}{c^3 \left (b+c x^2\right )}\right ) \, dx\\ &=\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c}-\frac{b^3 \int \frac{1}{b+c x^2} \, dx}{c^3}\\ &=\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0265003, size = 55, normalized size = 1. \[ \frac{b^2 x}{c^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(b*x^2 + c*x^4),x]

[Out]

(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(7/2)

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Maple [A]  time = 0.048, size = 49, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,c}}-{\frac{b{x}^{3}}{3\,{c}^{2}}}+{\frac{{b}^{2}x}{{c}^{3}}}-{\frac{{b}^{3}}{{c}^{3}}\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^8/(c*x^4+b*x^2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46424, size = 278, normalized size = 5.05 \begin{align*} \left [\frac{6 \, c^{2} x^{5} - 10 \, b c x^{3} + 15 \, b^{2} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, c^{3}}, \frac{3 \, c^{2} x^{5} - 5 \, b c x^{3} - 15 \, b^{2} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) + 15 \, b^{2} x}{15 \, c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/30*(6*c^2*x^5 - 10*b*c*x^3 + 15*b^2*sqrt(-b/c)*log((c*x^2 - 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) + 30*b^2*x)/
c^3, 1/15*(3*c^2*x^5 - 5*b*c*x^3 - 15*b^2*sqrt(b/c)*arctan(c*x*sqrt(b/c)/b) + 15*b^2*x)/c^3]

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Sympy [A]  time = 0.363509, size = 95, normalized size = 1.73 \begin{align*} \frac{b^{2} x}{c^{3}} - \frac{b x^{3}}{3 c^{2}} + \frac{\sqrt{- \frac{b^{5}}{c^{7}}} \log{\left (x - \frac{c^{3} \sqrt{- \frac{b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{c^{7}}} \log{\left (x + \frac{c^{3} \sqrt{- \frac{b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} + \frac{x^{5}}{5 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(c*x**4+b*x**2),x)

[Out]

b**2*x/c**3 - b*x**3/(3*c**2) + sqrt(-b**5/c**7)*log(x - c**3*sqrt(-b**5/c**7)/b**2)/2 - sqrt(-b**5/c**7)*log(
x + c**3*sqrt(-b**5/c**7)/b**2)/2 + x**5/(5*c)

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Giac [A]  time = 1.27465, size = 74, normalized size = 1.35 \begin{align*} -\frac{b^{3} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{3}} + \frac{3 \, c^{4} x^{5} - 5 \, b c^{3} x^{3} + 15 \, b^{2} c^{2} x}{15 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

-b^3*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^3) + 1/15*(3*c^4*x^5 - 5*b*c^3*x^3 + 15*b^2*c^2*x)/c^5