3.169 \(\int \frac{x^{13}}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{3 a^2 x^4}{2 b^5}-\frac{5 a^3 x^2}{b^6}+\frac{3 a^5}{b^7 \left (a+b x^2\right )}-\frac{a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac{15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3} \]

[Out]

(-5*a^3*x^2)/b^6 + (3*a^2*x^4)/(2*b^5) - (a*x^6)/(2*b^4) + x^8/(8*b^3) - a^6/(4*b^7*(a + b*x^2)^2) + (3*a^5)/(
b^7*(a + b*x^2)) + (15*a^4*Log[a + b*x^2])/(2*b^7)

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Rubi [A]  time = 0.0821957, antiderivative size = 100, 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} \[ \frac{3 a^2 x^4}{2 b^5}-\frac{5 a^3 x^2}{b^6}+\frac{3 a^5}{b^7 \left (a+b x^2\right )}-\frac{a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac{15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^13/(a + b*x^2)^3,x]

[Out]

(-5*a^3*x^2)/b^6 + (3*a^2*x^4)/(2*b^5) - (a*x^6)/(2*b^4) + x^8/(8*b^3) - a^6/(4*b^7*(a + b*x^2)^2) + (3*a^5)/(
b^7*(a + b*x^2)) + (15*a^4*Log[a + b*x^2])/(2*b^7)

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^{13}}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^6}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{10 a^3}{b^6}+\frac{6 a^2 x}{b^5}-\frac{3 a x^2}{b^4}+\frac{x^3}{b^3}+\frac{a^6}{b^6 (a+b x)^3}-\frac{6 a^5}{b^6 (a+b x)^2}+\frac{15 a^4}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{5 a^3 x^2}{b^6}+\frac{3 a^2 x^4}{2 b^5}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3}-\frac{a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac{3 a^5}{b^7 \left (a+b x^2\right )}+\frac{15 a^4 \log \left (a+b x^2\right )}{2 b^7}\\ \end{align*}

Mathematica [A]  time = 0.0272045, size = 85, normalized size = 0.85 \[ \frac{12 a^2 b^2 x^4-40 a^3 b x^2+\frac{24 a^5}{a+b x^2}-\frac{2 a^6}{\left (a+b x^2\right )^2}+60 a^4 \log \left (a+b x^2\right )-4 a b^3 x^6+b^4 x^8}{8 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13/(a + b*x^2)^3,x]

[Out]

(-40*a^3*b*x^2 + 12*a^2*b^2*x^4 - 4*a*b^3*x^6 + b^4*x^8 - (2*a^6)/(a + b*x^2)^2 + (24*a^5)/(a + b*x^2) + 60*a^
4*Log[a + b*x^2])/(8*b^7)

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Maple [A]  time = 0.009, size = 91, normalized size = 0.9 \begin{align*} -5\,{\frac{{a}^{3}{x}^{2}}{{b}^{6}}}+{\frac{3\,{a}^{2}{x}^{4}}{2\,{b}^{5}}}-{\frac{a{x}^{6}}{2\,{b}^{4}}}+{\frac{{x}^{8}}{8\,{b}^{3}}}-{\frac{{a}^{6}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{a}^{5}}{{b}^{7} \left ( b{x}^{2}+a \right ) }}+{\frac{15\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(b*x^2+a)^3,x)

[Out]

-5*a^3*x^2/b^6+3/2*a^2*x^4/b^5-1/2*a*x^6/b^4+1/8*x^8/b^3-1/4*a^6/b^7/(b*x^2+a)^2+3*a^5/b^7/(b*x^2+a)+15/2*a^4*
ln(b*x^2+a)/b^7

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Maxima [A]  time = 2.31735, size = 134, normalized size = 1.34 \begin{align*} \frac{12 \, a^{5} b x^{2} + 11 \, a^{6}}{4 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac{15 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{7}} + \frac{b^{3} x^{8} - 4 \, a b^{2} x^{6} + 12 \, a^{2} b x^{4} - 40 \, a^{3} x^{2}}{8 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/4*(12*a^5*b*x^2 + 11*a^6)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7) + 15/2*a^4*log(b*x^2 + a)/b^7 + 1/8*(b^3*x^8 - 4
*a*b^2*x^6 + 12*a^2*b*x^4 - 40*a^3*x^2)/b^6

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Fricas [A]  time = 1.20368, size = 261, normalized size = 2.61 \begin{align*} \frac{b^{6} x^{12} - 2 \, a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} - 20 \, a^{3} b^{3} x^{6} - 68 \, a^{4} b^{2} x^{4} - 16 \, a^{5} b x^{2} + 22 \, a^{6} + 60 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{8 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/8*(b^6*x^12 - 2*a*b^5*x^10 + 5*a^2*b^4*x^8 - 20*a^3*b^3*x^6 - 68*a^4*b^2*x^4 - 16*a^5*b*x^2 + 22*a^6 + 60*(a
^4*b^2*x^4 + 2*a^5*b*x^2 + a^6)*log(b*x^2 + a))/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7)

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Sympy [A]  time = 0.597757, size = 104, normalized size = 1.04 \begin{align*} \frac{15 a^{4} \log{\left (a + b x^{2} \right )}}{2 b^{7}} - \frac{5 a^{3} x^{2}}{b^{6}} + \frac{3 a^{2} x^{4}}{2 b^{5}} - \frac{a x^{6}}{2 b^{4}} + \frac{11 a^{6} + 12 a^{5} b x^{2}}{4 a^{2} b^{7} + 8 a b^{8} x^{2} + 4 b^{9} x^{4}} + \frac{x^{8}}{8 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(b*x**2+a)**3,x)

[Out]

15*a**4*log(a + b*x**2)/(2*b**7) - 5*a**3*x**2/b**6 + 3*a**2*x**4/(2*b**5) - a*x**6/(2*b**4) + (11*a**6 + 12*a
**5*b*x**2)/(4*a**2*b**7 + 8*a*b**8*x**2 + 4*b**9*x**4) + x**8/(8*b**3)

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Giac [A]  time = 1.81621, size = 138, normalized size = 1.38 \begin{align*} \frac{15 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{7}} - \frac{45 \, a^{4} b^{2} x^{4} + 78 \, a^{5} b x^{2} + 34 \, a^{6}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{7}} + \frac{b^{9} x^{8} - 4 \, a b^{8} x^{6} + 12 \, a^{2} b^{7} x^{4} - 40 \, a^{3} b^{6} x^{2}}{8 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^3,x, algorithm="giac")

[Out]

15/2*a^4*log(abs(b*x^2 + a))/b^7 - 1/4*(45*a^4*b^2*x^4 + 78*a^5*b*x^2 + 34*a^6)/((b*x^2 + a)^2*b^7) + 1/8*(b^9
*x^8 - 4*a*b^8*x^6 + 12*a^2*b^7*x^4 - 40*a^3*b^6*x^2)/b^12