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

Optimal. Leaf size=74 \[ \frac{a}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{c \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{c \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0653426, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac{a}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{c \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{c \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0331597, size = 74, normalized size = 1. \[ \frac{a}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{c \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{c \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 95, normalized size = 1.3 \begin{align*} -{\frac{c\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{2}}}-{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{2}b \left ( b{x}^{2}+a \right ) }}+{\frac{ac}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981009, size = 142, normalized size = 1.92 \begin{align*} \frac{c \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{c \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{a}{2 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.80083, size = 242, normalized size = 3.27 \begin{align*} \frac{a b c - a^{2} d +{\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) -{\left (b^{2} c x^{2} + a b c\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.24498, size = 253, normalized size = 3.42 \begin{align*} - \frac{a}{2 a^{2} b d - 2 a b^{2} c + x^{2} \left (2 a b^{2} d - 2 b^{3} c\right )} - \frac{c \log{\left (x^{2} + \frac{- \frac{a^{3} c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + a c d + \frac{b^{3} c^{4}}{\left (a d - b c\right )^{2}} + b c^{2}}{2 b c d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{c \log{\left (x^{2} + \frac{\frac{a^{3} c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + a c d - \frac{b^{3} c^{4}}{\left (a d - b c\right )^{2}} + b c^{2}}{2 b c d} \right )}}{2 \left (a d - b c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19154, size = 124, normalized size = 1.68 \begin{align*} -\frac{\frac{b^{2} c \log \left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{a b}{{\left (b^{2} c - a b d\right )}{\left (b x^{2} + a\right )}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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