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+1/3*x^3/c+b^(3/2)*arctan(x*c^(1/2)/b^(1/2))/c^(5/2)

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Rubi [A]  time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 42, normalized size = 1.00 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{5/2}}-\frac {b x}{c^2}+\frac {x^3}{3 c} \]

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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fricas [A]  time = 0.84, size = 99, normalized size = 2.36 \[ \left [\frac {2 \, c x^{3} + 3 \, b \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} + 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) - 6 \, b x}{6 \, c^{2}}, \frac {c x^{3} + 3 \, b \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 3 \, b x}{3 \, c^{2}}\right ] \]

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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giac [A]  time = 0.17, size = 40, normalized size = 0.95 \[ \frac {b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {c^{2} x^{3} - 3 \, b c x}{3 \, c^{3}} \]

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

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maple [A]  time = 0.00, size = 38, normalized size = 0.90 \[ \frac {x^{3}}{3 c}+\frac {b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{2}}-\frac {b x}{c^{2}} \]

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

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maxima [A]  time = 2.95, size = 37, normalized size = 0.88 \[ \frac {b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {c x^{3} - 3 \, b x}{3 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 32, normalized size = 0.76 \[ \frac {x^3}{3\,c}+\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{c^{5/2}}-\frac {b\,x}{c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^2 + c*x^4),x)

[Out]

x^3/(3*c) + (b^(3/2)*atan((c^(1/2)*x)/b^(1/2)))/c^(5/2) - (b*x)/c^2

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sympy [B]  time = 0.19, size = 80, normalized size = 1.90 \[ - \frac {b x}{c^{2}} - \frac {\sqrt {- \frac {b^{3}}{c^{5}}} \log {\left (x - \frac {c^{2} \sqrt {- \frac {b^{3}}{c^{5}}}}{b} \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{c^{5}}} \log {\left (x + \frac {c^{2} \sqrt {- \frac {b^{3}}{c^{5}}}}{b} \right )}}{2} + \frac {x^{3}}{3 c} \]

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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