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

Optimal. Leaf size=27 \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^3,x]

[Out]

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

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])

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]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 27, normalized size = 1.00 \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^3,x]

[Out]

-1/2*a^2/x^2 + (b^2*x^2)/2 + 2*a*b*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 27, normalized size = 1.00 \[ \frac {b^{2} x^{4} + 4 \, a b x^{2} \log \relax (x) - a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.14, size = 32, normalized size = 1.19 \[ \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 24, normalized size = 0.89 \[ \frac {b^{2} x^{2}}{2}+2 a b \ln \relax (x )-\frac {a^{2}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.39, size = 24, normalized size = 0.89 \[ \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.94, size = 23, normalized size = 0.85 \[ \frac {b^2\,x^2}{2}-\frac {a^2}{2\,x^2}+2\,a\,b\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^3,x)

[Out]

(b^2*x^2)/2 - a^2/(2*x^2) + 2*a*b*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 24, normalized size = 0.89 \[ - \frac {a^{2}}{2 x^{2}} + 2 a b \log {\relax (x )} + \frac {b^{2} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/(2*x**2) + 2*a*b*log(x) + b**2*x**2/2

________________________________________________________________________________________