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

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

[Out]

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

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Rubi [A]  time = 0.0642443, 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} \[ -\frac{c}{2 d \left (c+d x^2\right ) (b c-a d)}-\frac{a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0341205, size = 74, normalized size = 1. \[ \frac{c}{2 d \left (c+d x^2\right ) (a d-b c)}-\frac{a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

c/(2*d*(-(b*c) + a*d)*(c + d*x^2)) - (a*Log[a + b*x^2])/(2*(b*c - a*d)^2) + (a*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{a\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}+{\frac{ac}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{b{c}^{2}}{2\, \left ( ad-bc \right ) ^{2}d \left ( d{x}^{2}+c \right ) }}-{\frac{a\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05796, size = 142, normalized size = 1.92 \begin{align*} -\frac{a \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{a \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{c}{2 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54888, size = 243, normalized size = 3.28 \begin{align*} -\frac{b c^{2} - a c d +{\left (a d^{2} x^{2} + a c d\right )} \log \left (b x^{2} + a\right ) -{\left (a d^{2} x^{2} + a c d\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.37832, size = 253, normalized size = 3.42 \begin{align*} \frac{a \log{\left (x^{2} + \frac{- \frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{a \log{\left (x^{2} + \frac{\frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{c}{2 a c d^{2} - 2 b c^{2} d + x^{2} \left (2 a d^{3} - 2 b c d^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17782, size = 123, normalized size = 1.66 \begin{align*} -\frac{\frac{a d^{2} \log \left ({\left | b - \frac{b c}{d x^{2} + c} + \frac{a d}{d x^{2} + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{c d}{{\left (b c d - a d^{2}\right )}{\left (d x^{2} + c\right )}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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