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

Optimal. Leaf size=33 \[ \frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 33, 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} \[ \frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^2,x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 27, normalized size = 0.82 \[ \frac {\frac {a}{a+b x^2}+\log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 35, normalized size = 1.06 \[ \frac {{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.64, size = 48, normalized size = 1.45 \[ -\frac {\frac {\log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x^{2} + a\right )} b}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 30, normalized size = 0.91 \[ \frac {a}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 32, normalized size = 0.97 \[ \frac {a}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 29, normalized size = 0.88 \[ \frac {\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {a}{2\,b^2\,\left (b\,x^2+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 29, normalized size = 0.88 \[ \frac {a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(2*a*b**2 + 2*b**3*x**2) + log(a + b*x**2)/(2*b**2)

________________________________________________________________________________________