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

Optimal. Leaf size=126 \[ -\frac{b^2}{2 \left (a+b x^2\right ) (b c-a d)^3}-\frac{3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac{3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4}-\frac{b d}{\left (c+d x^2\right ) (b c-a d)^3}-\frac{d}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.109223, antiderivative size = 126, 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, 44} \[ -\frac{b^2}{2 \left (a+b x^2\right ) (b c-a d)^3}-\frac{3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac{3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4}-\frac{b d}{\left (c+d x^2\right ) (b c-a d)^3}-\frac{d}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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{1}{(a+b x)^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^3}{(b c-a d)^3 (a+b x)^2}-\frac{3 b^3 d}{(b c-a d)^4 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)^3}+\frac{2 b d^2}{(b c-a d)^3 (c+d x)^2}+\frac{3 b^2 d^2}{(b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{2 (b c-a d)^3 \left (a+b x^2\right )}-\frac{d}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{b d}{(b c-a d)^3 \left (c+d x^2\right )}-\frac{3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac{3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.133882, size = 107, normalized size = 0.85 \[ -\frac{\frac{2 b^2 (b c-a d)}{a+b x^2}+6 b^2 d \log \left (a+b x^2\right )+\frac{4 b d (b c-a d)}{c+d x^2}+\frac{d (b c-a d)^2}{\left (c+d x^2\right )^2}-6 b^2 d \log \left (c+d x^2\right )}{4 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*b^2*(b*c - a*d))/(a + b*x^2) + (d*(b*c - a*d)^2)/(c + d*x^2)^2 + (4*b*d*(b*c - a*d))/(c + d*x^2) + 6*b^2*
d*Log[a + b*x^2] - 6*b^2*d*Log[c + d*x^2])/(4*(b*c - a*d)^4)

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Maple [A]  time = 0.016, size = 234, normalized size = 1.9 \begin{align*}{\frac{3\,d\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\, \left ( ad-bc \right ) ^{4}}}-{\frac{{a}^{2}{d}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{abc{d}^{2}}{2\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}d}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{ab{d}^{2}}{ \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}cd}{ \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\, \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{2}ad}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}c}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \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]

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

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Maxima [B]  time = 1.08609, size = 532, normalized size = 4.22 \begin{align*} -\frac{3 \, b^{2} d \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac{3 \, b^{2} d \log \left (d x^{2} + c\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac{6 \, b^{2} d^{2} x^{4} + 2 \, b^{2} c^{2} + 5 \, a b c d - a^{2} d^{2} + 3 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}}{4 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\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="maxima")

[Out]

-3/2*b^2*d*log(b*x^2 + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 3/2*b^2*d*
log(d*x^2 + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 1/4*(6*b^2*d^2*x^4 +
2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x^2)/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2
- a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3
*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c
^2*d^3 - 2*a^4*c*d^4)*x^2)

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Fricas [B]  time = 1.37767, size = 1007, normalized size = 7.99 \begin{align*} -\frac{2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 3 \,{\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} d^{3} x^{6} + a b^{2} c^{2} d +{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{4} +{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 6 \,{\left (b^{3} d^{3} x^{6} + a b^{2} c^{2} d +{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{4} +{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} +{\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\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*(2*b^3*c^3 + 3*a*b^2*c^2*d - 6*a^2*b*c*d^2 + a^3*d^3 + 6*(b^3*c*d^2 - a*b^2*d^3)*x^4 + 3*(3*b^3*c^2*d - 2
*a*b^2*c*d^2 - a^2*b*d^3)*x^2 + 6*(b^3*d^3*x^6 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^4 + (b^3*c^2*d + 2*
a*b^2*c*d^2)*x^2)*log(b*x^2 + a) - 6*(b^3*d^3*x^6 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^4 + (b^3*c^2*d +
2*a*b^2*c*d^2)*x^2)*log(d*x^2 + c))/(a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*
c^2*d^4 + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d^6)*x^6 + (2*b^5*c^5*d
- 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^4 + (b^5*c^6 - 2*a*b^4
*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x^2)

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Sympy [B]  time = 15.0777, size = 643, normalized size = 5.1 \begin{align*} \frac{3 b^{2} d \log{\left (x^{2} + \frac{- \frac{3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac{15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} - \frac{30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac{30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac{15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} + \frac{3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{2 \left (a d - b c\right )^{4}} - \frac{3 b^{2} d \log{\left (x^{2} + \frac{\frac{3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac{15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} + \frac{30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac{30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac{15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} - \frac{3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{2 \left (a d - b c\right )^{4}} + \frac{- a^{2} d^{2} + 5 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{4} + x^{2} \left (3 a b d^{2} + 9 b^{2} c d\right )}{4 a^{4} c^{2} d^{3} - 12 a^{3} b c^{3} d^{2} + 12 a^{2} b^{2} c^{4} d - 4 a b^{3} c^{5} + x^{6} \left (4 a^{3} b d^{5} - 12 a^{2} b^{2} c d^{4} + 12 a b^{3} c^{2} d^{3} - 4 b^{4} c^{3} d^{2}\right ) + x^{4} \left (4 a^{4} d^{5} - 4 a^{3} b c d^{4} - 12 a^{2} b^{2} c^{2} d^{3} + 20 a b^{3} c^{3} d^{2} - 8 b^{4} c^{4} d\right ) + x^{2} \left (8 a^{4} c d^{4} - 20 a^{3} b c^{2} d^{3} + 12 a^{2} b^{2} c^{3} d^{2} + 4 a b^{3} c^{4} d - 4 b^{4} c^{5}\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)

[Out]

3*b**2*d*log(x**2 + (-3*a**5*b**2*d**6/(a*d - b*c)**4 + 15*a**4*b**3*c*d**5/(a*d - b*c)**4 - 30*a**3*b**4*c**2
*d**4/(a*d - b*c)**4 + 30*a**2*b**5*c**3*d**3/(a*d - b*c)**4 - 15*a*b**6*c**4*d**2/(a*d - b*c)**4 + 3*a*b**2*d
**2 + 3*b**7*c**5*d/(a*d - b*c)**4 + 3*b**3*c*d)/(6*b**3*d**2))/(2*(a*d - b*c)**4) - 3*b**2*d*log(x**2 + (3*a*
*5*b**2*d**6/(a*d - b*c)**4 - 15*a**4*b**3*c*d**5/(a*d - b*c)**4 + 30*a**3*b**4*c**2*d**4/(a*d - b*c)**4 - 30*
a**2*b**5*c**3*d**3/(a*d - b*c)**4 + 15*a*b**6*c**4*d**2/(a*d - b*c)**4 + 3*a*b**2*d**2 - 3*b**7*c**5*d/(a*d -
b*c)**4 + 3*b**3*c*d)/(6*b**3*d**2))/(2*(a*d - b*c)**4) + (-a**2*d**2 + 5*a*b*c*d + 2*b**2*c**2 + 6*b**2*d**2
*x**4 + x**2*(3*a*b*d**2 + 9*b**2*c*d))/(4*a**4*c**2*d**3 - 12*a**3*b*c**3*d**2 + 12*a**2*b**2*c**4*d - 4*a*b*
*3*c**5 + x**6*(4*a**3*b*d**5 - 12*a**2*b**2*c*d**4 + 12*a*b**3*c**2*d**3 - 4*b**4*c**3*d**2) + x**4*(4*a**4*d
**5 - 4*a**3*b*c*d**4 - 12*a**2*b**2*c**2*d**3 + 20*a*b**3*c**3*d**2 - 8*b**4*c**4*d) + x**2*(8*a**4*c*d**4 -
20*a**3*b*c**2*d**3 + 12*a**2*b**2*c**3*d**2 + 4*a*b**3*c**4*d - 4*b**4*c**5))

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Giac [A]  time = 1.17023, size = 309, normalized size = 2.45 \begin{align*} \frac{3 \, b^{3} d \log \left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{2 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac{b^{5}}{2 \,{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} + \frac{5 \, b^{2} d^{3} + \frac{6 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}{{\left (b x^{2} + a\right )} b}}{4 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d\right )}^{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]

3/2*b^3*d*log(abs(b*c/(b*x^2 + a) - a*d/(b*x^2 + a) + d))/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3
*b^2*c*d^3 + a^4*b*d^4) - 1/2*b^5/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*(b*x^2 + a)) + 1/
4*(5*b^2*d^3 + 6*(b^4*c*d^2 - a*b^3*d^3)/((b*x^2 + a)*b))/((b*c - a*d)^4*(b*c/(b*x^2 + a) - a*d/(b*x^2 + a) +
d)^2)