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

Optimal. Leaf size=70 \[ \frac {1}{2 \left (c+d x^2\right ) (b c-a d)}+\frac {b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 44} \begin {gather*} \frac {1}{2 \left (c+d x^2\right ) (b c-a d)}+\frac {b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

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

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])

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 66, normalized size = 0.94 \begin {gather*} \frac {b \left (c+d x^2\right ) \log \left (a+b x^2\right )-a d-b \left (c+d x^2\right ) \log \left (c+d x^2\right )+b c}{2 \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 103, normalized size = 1.47 \begin {gather*} \frac {b c - a d + {\left (b d x^{2} + b c\right )} \log \left (b x^{2} + a\right ) - {\left (b d x^{2} + b c\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.34, size = 85, normalized size = 1.21 \begin {gather*} \frac {b d \log \left ({\left | b - \frac {b c}{d x^{2} + c} + \frac {a d}{d x^{2} + c} \right |}\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} + \frac {d}{2 \, {\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 90, normalized size = 1.29 \begin {gather*} -\frac {a d}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {b c}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2}}-\frac {b \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 99, normalized size = 1.41 \begin {gather*} \frac {b \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {b \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {1}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.27, size = 160, normalized size = 2.29 \begin {gather*} \frac {-a\,d+c\,\left (b+b\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}\right )+b\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}}{2\,a^2\,c\,d^2+2\,a^2\,d^3\,x^2-4\,a\,b\,c^2\,d-4\,a\,b\,c\,d^2\,x^2+2\,b^2\,c^3+2\,b^2\,c^2\,d\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x^2)*(c + d*x^2)^2),x)

[Out]

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

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sympy [B]  time = 1.89, size = 248, normalized size = 3.54 \begin {gather*} - \frac {b \log {\left (x^{2} + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac {b \log {\left (x^{2} + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac {1}{2 a c d - 2 b c^{2} + x^{2} \left (2 a d^{2} - 2 b c d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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