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

Optimal. Leaf size=67 \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]

[Out]

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

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Rubi [A]  time = 0.0681163, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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

Mathematica [A]  time = 0.0240263, size = 75, normalized size = 1.12 \[ \frac{-a^2 d^2-2 a b d \left (c+2 d x^2\right )+b^2 c \left (3 c+4 d x^2\right )+2 b^2 \left (c+d x^2\right )^2 \log \left (c+d x^2\right )}{4 d^3 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 105, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\,{d}^{3}}}-{\frac{{a}^{2}}{4\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{abc}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab}{{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}c}{{d}^{3} \left ( d{x}^{2}+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01857, size = 117, normalized size = 1.75 \begin{align*} \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} + \frac{b^{2} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45565, size = 216, normalized size = 3.22 \begin{align*} \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2 + 4*(b^2*c*d - a*b*d^2)*x^2 + 2*(b^2*d^2*x^4 + 2*b^2*c*d*x^2 + b^2*c^2)*l
og(d*x^2 + c))/(d^5*x^4 + 2*c*d^4*x^2 + c^2*d^3)

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Sympy [A]  time = 1.66126, size = 87, normalized size = 1.3 \begin{align*} \frac{b^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x^{2} \left (4 a b d^{2} - 4 b^{2} c d\right )}{4 c^{2} d^{3} + 8 c d^{4} x^{2} + 4 d^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18528, size = 103, normalized size = 1.54 \begin{align*} \frac{b^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x^{2} + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{4 \,{\left (d x^{2} + c\right )}^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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