3.1278 \(\int \frac{x^{19}}{(a+b x^5)^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0453078, antiderivative size = 59, 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} \[ \frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^19/(a + b*x^5)^2,x]

[Out]

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

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^{19}}{\left (a+b x^5\right )^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2}+\frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0156562, size = 49, normalized size = 0.83 \[ \frac{\frac{2 a^3}{a+b x^5}+6 a^2 \log \left (a+b x^5\right )-4 a b x^5+b^2 x^{10}}{10 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^19/(a + b*x^5)^2,x]

[Out]

(-4*a*b*x^5 + b^2*x^10 + (2*a^3)/(a + b*x^5) + 6*a^2*Log[a + b*x^5])/(10*b^4)

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Maple [A]  time = 0.009, size = 52, normalized size = 0.9 \begin{align*} -{\frac{2\,a{x}^{5}}{5\,{b}^{3}}}+{\frac{{x}^{10}}{10\,{b}^{2}}}+{\frac{{a}^{3}}{5\,{b}^{4} \left ( b{x}^{5}+a \right ) }}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(b*x^5+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.04403, size = 73, normalized size = 1.24 \begin{align*} \frac{a^{3}}{5 \,{\left (b^{5} x^{5} + a b^{4}\right )}} + \frac{3 \, a^{2} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac{b x^{10} - 4 \, a x^{5}}{10 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5793, size = 147, normalized size = 2.49 \begin{align*} \frac{b^{3} x^{15} - 3 \, a b^{2} x^{10} - 4 \, a^{2} b x^{5} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{5} + a^{3}\right )} \log \left (b x^{5} + a\right )}{10 \,{\left (b^{5} x^{5} + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.949049, size = 56, normalized size = 0.95 \begin{align*} \frac{a^{3}}{5 a b^{4} + 5 b^{5} x^{5}} + \frac{3 a^{2} \log{\left (a + b x^{5} \right )}}{5 b^{4}} - \frac{2 a x^{5}}{5 b^{3}} + \frac{x^{10}}{10 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19/(b*x**5+a)**2,x)

[Out]

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

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Giac [A]  time = 1.19982, size = 90, normalized size = 1.53 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac{b^{2} x^{10} - 4 \, a b x^{5}}{10 \, b^{4}} - \frac{3 \, a^{2} b x^{5} + 2 \, a^{3}}{5 \,{\left (b x^{5} + a\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a)^2,x, algorithm="giac")

[Out]

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