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

Optimal. Leaf size=40 \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{4 x^4}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0186338, antiderivative size = 40, 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} \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{4 x^4}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0039348, size = 40, normalized size = 1. \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{4 x^4}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^5,x]

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+{\frac{{b}^{3}{x}^{2}}{2}}+3\,a{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.36099, size = 50, normalized size = 1.25 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28901, size = 85, normalized size = 2.12 \begin{align*} \frac{2 \, b^{3} x^{6} + 12 \, a b^{2} x^{4} \log \left (x\right ) - 6 \, a^{2} b x^{2} - a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b^3*x^6 + 12*a*b^2*x^4*log(x) - 6*a^2*b*x^2 - a^3)/x^4

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Sympy [A]  time = 0.339119, size = 36, normalized size = 0.9 \begin{align*} 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{2}}{2} - \frac{a^{3} + 6 a^{2} b x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.3507, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{9 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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