3.126 \(\int \frac{x^9}{a+b x^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{a^2 x^4}{4 b^3}-\frac{a^3 x^2}{2 b^4}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b} \]

[Out]

-(a^3*x^2)/(2*b^4) + (a^2*x^4)/(4*b^3) - (a*x^6)/(6*b^2) + x^8/(8*b) + (a^4*Log[a + b*x^2])/(2*b^5)

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Rubi [A]  time = 0.0443748, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^4}{4 b^3}-\frac{a^3 x^2}{2 b^4}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a + b*x^2),x]

[Out]

-(a^3*x^2)/(2*b^4) + (a^2*x^4)/(4*b^3) - (a*x^6)/(6*b^2) + x^8/(8*b) + (a^4*Log[a + b*x^2])/(2*b^5)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^9}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^4}+\frac{a^2 x}{b^3}-\frac{a x^2}{b^2}+\frac{x^3}{b}+\frac{a^4}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^4}{4 b^3}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0055564, size = 66, normalized size = 1. \[ \frac{a^2 x^4}{4 b^3}-\frac{a^3 x^2}{2 b^4}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x^2),x]

[Out]

-(a^3*x^2)/(2*b^4) + (a^2*x^4)/(4*b^3) - (a*x^6)/(6*b^2) + x^8/(8*b) + (a^4*Log[a + b*x^2])/(2*b^5)

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Maple [A]  time = 0.004, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}{x}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}{x}^{4}}{4\,{b}^{3}}}-{\frac{a{x}^{6}}{6\,{b}^{2}}}+{\frac{{x}^{8}}{8\,b}}+{\frac{{a}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b*x^2+a),x)

[Out]

-1/2*a^3*x^2/b^4+1/4*a^2*x^4/b^3-1/6*a*x^6/b^2+1/8*x^8/b+1/2*a^4*ln(b*x^2+a)/b^5

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Maxima [A]  time = 2.28757, size = 77, normalized size = 1.17 \begin{align*} \frac{a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{5}} + \frac{3 \, b^{3} x^{8} - 4 \, a b^{2} x^{6} + 6 \, a^{2} b x^{4} - 12 \, a^{3} x^{2}}{24 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a),x, algorithm="maxima")

[Out]

1/2*a^4*log(b*x^2 + a)/b^5 + 1/24*(3*b^3*x^8 - 4*a*b^2*x^6 + 6*a^2*b*x^4 - 12*a^3*x^2)/b^4

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Fricas [A]  time = 1.28839, size = 123, normalized size = 1.86 \begin{align*} \frac{3 \, b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 12 \, a^{3} b x^{2} + 12 \, a^{4} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a),x, algorithm="fricas")

[Out]

1/24*(3*b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 12*a^3*b*x^2 + 12*a^4*log(b*x^2 + a))/b^5

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Sympy [A]  time = 0.309818, size = 56, normalized size = 0.85 \begin{align*} \frac{a^{4} \log{\left (a + b x^{2} \right )}}{2 b^{5}} - \frac{a^{3} x^{2}}{2 b^{4}} + \frac{a^{2} x^{4}}{4 b^{3}} - \frac{a x^{6}}{6 b^{2}} + \frac{x^{8}}{8 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x**2+a),x)

[Out]

a**4*log(a + b*x**2)/(2*b**5) - a**3*x**2/(2*b**4) + a**2*x**4/(4*b**3) - a*x**6/(6*b**2) + x**8/(8*b)

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Giac [A]  time = 2.39218, size = 78, normalized size = 1.18 \begin{align*} \frac{a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac{3 \, b^{3} x^{8} - 4 \, a b^{2} x^{6} + 6 \, a^{2} b x^{4} - 12 \, a^{3} x^{2}}{24 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*a^4*log(abs(b*x^2 + a))/b^5 + 1/24*(3*b^3*x^8 - 4*a*b^2*x^6 + 6*a^2*b*x^4 - 12*a^3*x^2)/b^4