3.173 \(\int \frac{x^5}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=49 \[ -\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0377695, antiderivative size = 49, 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} \[ -\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0183525, size = 39, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+4 b x^2\right )}{\left (a+b x^2\right )^2}+2 \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{a}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 2.11304, size = 74, normalized size = 1.51 \begin{align*} \frac{4 \, a b x^{2} + 3 \, a^{2}}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.15257, size = 143, normalized size = 2.92 \begin{align*} \frac{4 \, a b x^{2} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.468551, size = 53, normalized size = 1.08 \begin{align*} \frac{3 a^{2} + 4 a b x^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**3,x)

[Out]

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

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Giac [A]  time = 2.02583, size = 57, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} - \frac{3 \, b x^{4} + 2 \, a x^{2}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3,x, algorithm="giac")

[Out]

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