3.1267 \(\int \frac{x^{24}}{a+b x^5} \, dx\)

Optimal. Leaf size=66 \[ \frac{a^2 x^{10}}{10 b^3}-\frac{a^3 x^5}{5 b^4}+\frac{a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b} \]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^4*Log[a + b*x^5])/(5*b^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0455001, 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} \[ \frac{a^2 x^{10}}{10 b^3}-\frac{a^3 x^5}{5 b^4}+\frac{a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b} \]

Antiderivative was successfully verified.

[In]

Int[x^24/(a + b*x^5),x]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^4*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}}{a+b x^5} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^4}{a+b x} \, dx,x,x^5\right )\\ &=\frac{1}{5} \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^5\right )\\ &=-\frac{a^3 x^5}{5 b^4}+\frac{a^2 x^{10}}{10 b^3}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b}+\frac{a^4 \log \left (a+b x^5\right )}{5 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0068657, size = 66, normalized size = 1. \[ \frac{a^2 x^{10}}{10 b^3}-\frac{a^3 x^5}{5 b^4}+\frac{a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^24/(a + b*x^5),x]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^4*Log[a + b*x^5])/(5*b^5)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}{x}^{5}}{5\,{b}^{4}}}+{\frac{{a}^{2}{x}^{10}}{10\,{b}^{3}}}-{\frac{a{x}^{15}}{15\,{b}^{2}}}+{\frac{{x}^{20}}{20\,b}}+{\frac{{a}^{4}\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),x)

[Out]

-1/5*a^3*x^5/b^4+1/10*a^2*x^10/b^3-1/15*a*x^15/b^2+1/20*x^20/b+1/5*a^4*ln(b*x^5+a)/b^5

________________________________________________________________________________________

Maxima [A]  time = 1.46723, size = 77, normalized size = 1.17 \begin{align*} \frac{a^{4} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac{3 \, b^{3} x^{20} - 4 \, a b^{2} x^{15} + 6 \, a^{2} b x^{10} - 12 \, a^{3} x^{5}}{60 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^4*log(b*x^5 + a)/b^5 + 1/60*(3*b^3*x^20 - 4*a*b^2*x^15 + 6*a^2*b*x^10 - 12*a^3*x^5)/b^4

________________________________________________________________________________________

Fricas [A]  time = 1.51399, size = 127, normalized size = 1.92 \begin{align*} \frac{3 \, b^{4} x^{20} - 4 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} - 12 \, a^{3} b x^{5} + 12 \, a^{4} \log \left (b x^{5} + a\right )}{60 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(3*b^4*x^20 - 4*a*b^3*x^15 + 6*a^2*b^2*x^10 - 12*a^3*b*x^5 + 12*a^4*log(b*x^5 + a))/b^5

________________________________________________________________________________________

Sympy [A]  time = 0.527185, size = 56, normalized size = 0.85 \begin{align*} \frac{a^{4} \log{\left (a + b x^{5} \right )}}{5 b^{5}} - \frac{a^{3} x^{5}}{5 b^{4}} + \frac{a^{2} x^{10}}{10 b^{3}} - \frac{a x^{15}}{15 b^{2}} + \frac{x^{20}}{20 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*log(a + b*x**5)/(5*b**5) - a**3*x**5/(5*b**4) + a**2*x**10/(10*b**3) - a*x**15/(15*b**2) + x**20/(20*b)

________________________________________________________________________________________

Giac [A]  time = 1.14214, size = 78, normalized size = 1.18 \begin{align*} \frac{a^{4} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac{3 \, b^{3} x^{20} - 4 \, a b^{2} x^{15} + 6 \, a^{2} b x^{10} - 12 \, a^{3} x^{5}}{60 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^4*log(abs(b*x^5 + a))/b^5 + 1/60*(3*b^3*x^20 - 4*a*b^2*x^15 + 6*a^2*b*x^10 - 12*a^3*x^5)/b^4