3.174 \(\int \frac{x^{10}}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=68 \[ \frac{b^2 x^3}{3 c^3}-\frac{b^3 x}{c^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c} \]

[Out]

-((b^3*x)/c^4) + (b^2*x^3)/(3*c^3) - (b*x^5)/(5*c^2) + x^7/(7*c) + (b^(7/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/
2)

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Rubi [A]  time = 0.0382052, antiderivative size = 68, 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} \[ \frac{b^2 x^3}{3 c^3}-\frac{b^3 x}{c^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(b*x^2 + c*x^4),x]

[Out]

-((b^3*x)/c^4) + (b^2*x^3)/(3*c^3) - (b*x^5)/(5*c^2) + x^7/(7*c) + (b^(7/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/
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^{10}}{b x^2+c x^4} \, dx &=\int \frac{x^8}{b+c x^2} \, dx\\ &=\int \left (-\frac{b^3}{c^4}+\frac{b^2 x^2}{c^3}-\frac{b x^4}{c^2}+\frac{x^6}{c}+\frac{b^4}{c^4 \left (b+c x^2\right )}\right ) \, dx\\ &=-\frac{b^3 x}{c^4}+\frac{b^2 x^3}{3 c^3}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c}+\frac{b^4 \int \frac{1}{b+c x^2} \, dx}{c^4}\\ &=-\frac{b^3 x}{c^4}+\frac{b^2 x^3}{3 c^3}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0264946, size = 68, normalized size = 1. \[ \frac{b^2 x^3}{3 c^3}-\frac{b^3 x}{c^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(b*x^2 + c*x^4),x]

[Out]

-((b^3*x)/c^4) + (b^2*x^3)/(3*c^3) - (b*x^5)/(5*c^2) + x^7/(7*c) + (b^(7/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/
2)

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Maple [A]  time = 0.046, size = 60, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,c}}-{\frac{b{x}^{5}}{5\,{c}^{2}}}+{\frac{{b}^{2}{x}^{3}}{3\,{c}^{3}}}-{\frac{{b}^{3}x}{{c}^{4}}}+{\frac{{b}^{4}}{{c}^{4}}\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^10/(c*x^4+b*x^2),x)

[Out]

1/7*x^7/c-1/5*b*x^5/c^2+1/3*b^2*x^3/c^3-b^3*x/c^4+b^4/c^4/(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^10/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45701, size = 336, normalized size = 4.94 \begin{align*} \left [\frac{30 \, c^{3} x^{7} - 42 \, b c^{2} x^{5} + 70 \, b^{2} c x^{3} + 105 \, b^{3} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 210 \, b^{3} x}{210 \, c^{4}}, \frac{15 \, c^{3} x^{7} - 21 \, b c^{2} x^{5} + 35 \, b^{2} c x^{3} + 105 \, b^{3} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) - 105 \, b^{3} x}{105 \, c^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/210*(30*c^3*x^7 - 42*b*c^2*x^5 + 70*b^2*c*x^3 + 105*b^3*sqrt(-b/c)*log((c*x^2 + 2*c*x*sqrt(-b/c) - b)/(c*x^
2 + b)) - 210*b^3*x)/c^4, 1/105*(15*c^3*x^7 - 21*b*c^2*x^5 + 35*b^2*c*x^3 + 105*b^3*sqrt(b/c)*arctan(c*x*sqrt(
b/c)/b) - 105*b^3*x)/c^4]

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Sympy [A]  time = 0.38582, size = 107, normalized size = 1.57 \begin{align*} - \frac{b^{3} x}{c^{4}} + \frac{b^{2} x^{3}}{3 c^{3}} - \frac{b x^{5}}{5 c^{2}} - \frac{\sqrt{- \frac{b^{7}}{c^{9}}} \log{\left (x - \frac{c^{4} \sqrt{- \frac{b^{7}}{c^{9}}}}{b^{3}} \right )}}{2} + \frac{\sqrt{- \frac{b^{7}}{c^{9}}} \log{\left (x + \frac{c^{4} \sqrt{- \frac{b^{7}}{c^{9}}}}{b^{3}} \right )}}{2} + \frac{x^{7}}{7 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(c*x**4+b*x**2),x)

[Out]

-b**3*x/c**4 + b**2*x**3/(3*c**3) - b*x**5/(5*c**2) - sqrt(-b**7/c**9)*log(x - c**4*sqrt(-b**7/c**9)/b**3)/2 +
sqrt(-b**7/c**9)*log(x + c**4*sqrt(-b**7/c**9)/b**3)/2 + x**7/(7*c)

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Giac [A]  time = 1.22828, size = 88, normalized size = 1.29 \begin{align*} \frac{b^{4} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{4}} + \frac{15 \, c^{6} x^{7} - 21 \, b c^{5} x^{5} + 35 \, b^{2} c^{4} x^{3} - 105 \, b^{3} c^{3} x}{105 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

b^4*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^4) + 1/105*(15*c^6*x^7 - 21*b*c^5*x^5 + 35*b^2*c^4*x^3 - 105*b^3*c^3*x)
/c^7