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

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

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Rubi [A]  time = 0.04, antiderivative size = 68, 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} \begin {gather*} \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} \end {gather*}

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 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^{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*}

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Mathematica [A]  time = 0.03, size = 68, normalized size = 1.00 \begin {gather*} \frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}}-\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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{10}}{b x^2+c x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^10/(b*x^2 + c*x^4),x]

[Out]

IntegrateAlgebraic[x^10/(b*x^2 + c*x^4), x]

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fricas [A]  time = 0.59, size = 148, normalized size = 2.18 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.19, size = 65, normalized size = 0.96 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 60, normalized size = 0.88 \begin {gather*} \frac {x^{7}}{7 c}-\frac {b \,x^{5}}{5 c^{2}}+\frac {b^{2} x^{3}}{3 c^{3}}+\frac {b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{4}}-\frac {b^{3} x}{c^{4}} \end {gather*}

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 [A]  time = 3.01, size = 60, normalized size = 0.88 \begin {gather*} \frac {b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {15 \, c^{3} x^{7} - 21 \, b c^{2} x^{5} + 35 \, b^{2} c x^{3} - 105 \, b^{3} x}{105 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 54, normalized size = 0.79 \begin {gather*} \frac {x^7}{7\,c}-\frac {b\,x^5}{5\,c^2}-\frac {b^3\,x}{c^4}+\frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{c^{9/2}}+\frac {b^2\,x^3}{3\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(b*x^2 + c*x^4),x)

[Out]

x^7/(7*c) - (b*x^5)/(5*c^2) - (b^3*x)/c^4 + (b^(7/2)*atan((c^(1/2)*x)/b^(1/2)))/c^(9/2) + (b^2*x^3)/(3*c^3)

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sympy [A]  time = 0.22, size = 107, normalized size = 1.57 \begin {gather*} - \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 {gather*}

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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