3.178 \(\int \frac{x^6}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=42 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.0266642, antiderivative size = 42, 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} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(b*x^2 + c*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0185414, size = 42, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(b*x^2 + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.045, size = 38, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,c}}-{\frac{bx}{{c}^{2}}}+{\frac{{b}^{2}}{{c}^{2}}\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^6/(c*x^4+b*x^2),x)

[Out]

1/3*x^3/c-b*x/c^2+b^2/c^2/(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^6/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54459, size = 217, normalized size = 5.17 \begin{align*} \left [\frac{2 \, c x^{3} + 3 \, b \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 6 \, b x}{6 \, c^{2}}, \frac{c x^{3} + 3 \, b \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) - 3 \, b x}{3 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.385021, size = 80, normalized size = 1.9 \begin{align*} - \frac{b x}{c^{2}} - \frac{\sqrt{- \frac{b^{3}}{c^{5}}} \log{\left (x - \frac{c^{2} \sqrt{- \frac{b^{3}}{c^{5}}}}{b} \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{c^{5}}} \log{\left (x + \frac{c^{2} \sqrt{- \frac{b^{3}}{c^{5}}}}{b} \right )}}{2} + \frac{x^{3}}{3 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(c*x**4+b*x**2),x)

[Out]

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

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Giac [A]  time = 1.21578, size = 54, normalized size = 1.29 \begin{align*} \frac{b^{2} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{2}} + \frac{c^{2} x^{3} - 3 \, b c x}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

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