3.1277 \(\int \frac{x^{24}}{(a+b x^5)^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{3 a^2 x^5}{5 b^4}-\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2} \]

[Out]

(3*a^2*x^5)/(5*b^4) - (a*x^10)/(5*b^3) + x^15/(15*b^2) - a^4/(5*b^5*(a + b*x^5)) - (4*a^3*Log[a + b*x^5])/(5*b
^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0583653, antiderivative size = 72, 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} \[ \frac{3 a^2 x^5}{5 b^4}-\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^24/(a + b*x^5)^2,x]

[Out]

(3*a^2*x^5)/(5*b^4) - (a*x^10)/(5*b^3) + x^15/(15*b^2) - a^4/(5*b^5*(a + b*x^5)) - (4*a^3*Log[a + b*x^5])/(5*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^{24}}{\left (a+b x^5\right )^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=\frac{3 a^2 x^5}{5 b^4}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2}-\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0233569, size = 60, normalized size = 0.83 \[ \frac{9 a^2 b x^5-\frac{3 a^4}{a+b x^5}-12 a^3 \log \left (a+b x^5\right )-3 a b^2 x^{10}+b^3 x^{15}}{15 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^24/(a + b*x^5)^2,x]

[Out]

(9*a^2*b*x^5 - 3*a*b^2*x^10 + b^3*x^15 - (3*a^4)/(a + b*x^5) - 12*a^3*Log[a + b*x^5])/(15*b^5)

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 63, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{5}{a}^{2}}{5\,{b}^{4}}}-{\frac{a{x}^{10}}{5\,{b}^{3}}}+{\frac{{x}^{15}}{15\,{b}^{2}}}-{\frac{{a}^{4}}{5\,{b}^{5} \left ( b{x}^{5}+a \right ) }}-{\frac{4\,{a}^{3}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24/(b*x^5+a)^2,x)

[Out]

3/5*a^2*x^5/b^4-1/5*a*x^10/b^3+1/15*x^15/b^2-1/5*a^4/b^5/(b*x^5+a)-4/5*a^3*ln(b*x^5+a)/b^5

________________________________________________________________________________________

Maxima [A]  time = 1.00645, size = 88, normalized size = 1.22 \begin{align*} -\frac{a^{4}}{5 \,{\left (b^{6} x^{5} + a b^{5}\right )}} - \frac{4 \, a^{3} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac{b^{2} x^{15} - 3 \, a b x^{10} + 9 \, a^{2} x^{5}}{15 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^5+a)^2,x, algorithm="maxima")

[Out]

-1/5*a^4/(b^6*x^5 + a*b^5) - 4/5*a^3*log(b*x^5 + a)/b^5 + 1/15*(b^2*x^15 - 3*a*b*x^10 + 9*a^2*x^5)/b^4

________________________________________________________________________________________

Fricas [A]  time = 1.72371, size = 171, normalized size = 2.38 \begin{align*} \frac{b^{4} x^{20} - 2 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} + 9 \, a^{3} b x^{5} - 3 \, a^{4} - 12 \,{\left (a^{3} b x^{5} + a^{4}\right )} \log \left (b x^{5} + a\right )}{15 \,{\left (b^{6} x^{5} + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^5+a)^2,x, algorithm="fricas")

[Out]

1/15*(b^4*x^20 - 2*a*b^3*x^15 + 6*a^2*b^2*x^10 + 9*a^3*b*x^5 - 3*a^4 - 12*(a^3*b*x^5 + a^4)*log(b*x^5 + a))/(b
^6*x^5 + a*b^5)

________________________________________________________________________________________

Sympy [A]  time = 1.01986, size = 68, normalized size = 0.94 \begin{align*} - \frac{a^{4}}{5 a b^{5} + 5 b^{6} x^{5}} - \frac{4 a^{3} \log{\left (a + b x^{5} \right )}}{5 b^{5}} + \frac{3 a^{2} x^{5}}{5 b^{4}} - \frac{a x^{10}}{5 b^{3}} + \frac{x^{15}}{15 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**24/(b*x**5+a)**2,x)

[Out]

-a**4/(5*a*b**5 + 5*b**6*x**5) - 4*a**3*log(a + b*x**5)/(5*b**5) + 3*a**2*x**5/(5*b**4) - a*x**10/(5*b**3) + x
**15/(15*b**2)

________________________________________________________________________________________

Giac [A]  time = 1.19263, size = 108, normalized size = 1.5 \begin{align*} -\frac{4 \, a^{3} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac{b^{4} x^{15} - 3 \, a b^{3} x^{10} + 9 \, a^{2} b^{2} x^{5}}{15 \, b^{6}} + \frac{4 \, a^{3} b x^{5} + 3 \, a^{4}}{5 \,{\left (b x^{5} + a\right )} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^5+a)^2,x, algorithm="giac")

[Out]

-4/5*a^3*log(abs(b*x^5 + a))/b^5 + 1/15*(b^4*x^15 - 3*a*b^3*x^10 + 9*a^2*b^2*x^5)/b^6 + 1/5*(4*a^3*b*x^5 + 3*a
^4)/((b*x^5 + a)*b^5)