3.770 \(\int \frac{x^3}{(a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=45 \[ \frac{\log \left (a+b x^4\right )}{4 (b c-a d)}-\frac{\log \left (c+d x^4\right )}{4 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0314069, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {444, 36, 31} \[ \frac{\log \left (a+b x^4\right )}{4 (b c-a d)}-\frac{\log \left (c+d x^4\right )}{4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0190943, size = 31, normalized size = 0.69 \[ \frac{\log \left (a+b x^4\right )-\log \left (c+d x^4\right )}{4 b c-4 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 42, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( d{x}^{4}+c \right ) }{4\,ad-4\,bc}}-{\frac{\ln \left ( b{x}^{4}+a \right ) }{4\,ad-4\,bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^4+a)/(d*x^4+c),x)

[Out]

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

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Maxima [A]  time = 0.927421, size = 55, normalized size = 1.22 \begin{align*} \frac{\log \left (b x^{4} + a\right )}{4 \,{\left (b c - a d\right )}} - \frac{\log \left (d x^{4} + c\right )}{4 \,{\left (b c - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.29518, size = 69, normalized size = 1.53 \begin{align*} \frac{\log \left (b x^{4} + a\right ) - \log \left (d x^{4} + c\right )}{4 \,{\left (b c - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.71828, size = 138, normalized size = 3.07 \begin{align*} \frac{\log{\left (x^{4} + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{4 \left (a d - b c\right )} - \frac{\log{\left (x^{4} + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{4 \left (a d - b c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**4+a)/(d*x**4+c),x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError