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

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

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Rubi [A]  time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 {gather*}

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 31

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

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 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^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*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.69 \begin {gather*} \frac {\log \left (a+b x^4\right )-\log \left (c+d x^4\right )}{4 b c-4 a d} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 31, normalized size = 0.69 \begin {gather*} \frac {\log \left (b x^{4} + a\right ) - \log \left (d x^{4} + c\right )}{4 \, {\left (b c - a d\right )}} \end {gather*}

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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giac [A]  time = 0.34, size = 51, normalized size = 1.13 \begin {gather*} \frac {b \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (b^{2} c - a b d\right )}} - \frac {d \log \left ({\left | d x^{4} + c \right |}\right )}{4 \, {\left (b c d - a d^{2}\right )}} \end {gather*}

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]

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

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maple [A]  time = 0.05, size = 42, normalized size = 0.93 \begin {gather*} -\frac {\ln \left (b \,x^{4}+a \right )}{4 \left (a d -b c \right )}+\frac {\ln \left (d \,x^{4}+c \right )}{4 a d -4 b c} \end {gather*}

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 = 0.52, size = 41, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 4.99, size = 1012, normalized size = 22.49 \begin {gather*} -\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x^4\,\left (96\,c\,b^5\,d^4+96\,a\,b^4\,d^5\right )+\frac {\frac {x^4\,\left (512\,a^3\,b^4\,d^7+1536\,a^2\,b^5\,c\,d^6+1536\,a\,b^6\,c^2\,d^5+512\,b^7\,c^3\,d^4\right )+1024\,a\,b^6\,c^3\,d^4+1024\,a^3\,b^4\,c\,d^6+2048\,a^2\,b^5\,c^2\,d^5}{4\,a\,d-4\,b\,c}+x^4\,\left (384\,a^2\,b^4\,d^6+768\,a\,b^5\,c\,d^5+384\,b^6\,c^2\,d^4\right )+512\,a\,b^5\,c^2\,d^4+512\,a^2\,b^4\,c\,d^5}{4\,a\,d-4\,b\,c}+64\,a\,b^4\,c\,d^4}{4\,a\,d-4\,b\,c}+8\,b^4\,d^4\,x^4\right )\,1{}\mathrm {i}}{4\,a\,d-4\,b\,c}-\frac {\left (\frac {x^4\,\left (96\,c\,b^5\,d^4+96\,a\,b^4\,d^5\right )-\frac {x^4\,\left (384\,a^2\,b^4\,d^6+768\,a\,b^5\,c\,d^5+384\,b^6\,c^2\,d^4\right )-\frac {x^4\,\left (512\,a^3\,b^4\,d^7+1536\,a^2\,b^5\,c\,d^6+1536\,a\,b^6\,c^2\,d^5+512\,b^7\,c^3\,d^4\right )+1024\,a\,b^6\,c^3\,d^4+1024\,a^3\,b^4\,c\,d^6+2048\,a^2\,b^5\,c^2\,d^5}{4\,a\,d-4\,b\,c}+512\,a\,b^5\,c^2\,d^4+512\,a^2\,b^4\,c\,d^5}{4\,a\,d-4\,b\,c}+64\,a\,b^4\,c\,d^4}{4\,a\,d-4\,b\,c}-8\,b^4\,d^4\,x^4\right )\,1{}\mathrm {i}}{4\,a\,d-4\,b\,c}}{\frac {\frac {x^4\,\left (96\,c\,b^5\,d^4+96\,a\,b^4\,d^5\right )+\frac {\frac {x^4\,\left (512\,a^3\,b^4\,d^7+1536\,a^2\,b^5\,c\,d^6+1536\,a\,b^6\,c^2\,d^5+512\,b^7\,c^3\,d^4\right )+1024\,a\,b^6\,c^3\,d^4+1024\,a^3\,b^4\,c\,d^6+2048\,a^2\,b^5\,c^2\,d^5}{4\,a\,d-4\,b\,c}+x^4\,\left (384\,a^2\,b^4\,d^6+768\,a\,b^5\,c\,d^5+384\,b^6\,c^2\,d^4\right )+512\,a\,b^5\,c^2\,d^4+512\,a^2\,b^4\,c\,d^5}{4\,a\,d-4\,b\,c}+64\,a\,b^4\,c\,d^4}{4\,a\,d-4\,b\,c}+8\,b^4\,d^4\,x^4}{4\,a\,d-4\,b\,c}+\frac {\frac {x^4\,\left (96\,c\,b^5\,d^4+96\,a\,b^4\,d^5\right )-\frac {x^4\,\left (384\,a^2\,b^4\,d^6+768\,a\,b^5\,c\,d^5+384\,b^6\,c^2\,d^4\right )-\frac {x^4\,\left (512\,a^3\,b^4\,d^7+1536\,a^2\,b^5\,c\,d^6+1536\,a\,b^6\,c^2\,d^5+512\,b^7\,c^3\,d^4\right )+1024\,a\,b^6\,c^3\,d^4+1024\,a^3\,b^4\,c\,d^6+2048\,a^2\,b^5\,c^2\,d^5}{4\,a\,d-4\,b\,c}+512\,a\,b^5\,c^2\,d^4+512\,a^2\,b^4\,c\,d^5}{4\,a\,d-4\,b\,c}+64\,a\,b^4\,c\,d^4}{4\,a\,d-4\,b\,c}-8\,b^4\,d^4\,x^4}{4\,a\,d-4\,b\,c}}\right )\,2{}\mathrm {i}}{4\,a\,d-4\,b\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x^4)*(c + d*x^4)),x)

[Out]

-(atan(((((x^4*(96*a*b^4*d^5 + 96*b^5*c*d^4) + ((x^4*(512*a^3*b^4*d^7 + 512*b^7*c^3*d^4 + 1536*a*b^6*c^2*d^5 +
1536*a^2*b^5*c*d^6) + 1024*a*b^6*c^3*d^4 + 1024*a^3*b^4*c*d^6 + 2048*a^2*b^5*c^2*d^5)/(4*a*d - 4*b*c) + x^4*(
384*a^2*b^4*d^6 + 384*b^6*c^2*d^4 + 768*a*b^5*c*d^5) + 512*a*b^5*c^2*d^4 + 512*a^2*b^4*c*d^5)/(4*a*d - 4*b*c)
+ 64*a*b^4*c*d^4)/(4*a*d - 4*b*c) + 8*b^4*d^4*x^4)*1i)/(4*a*d - 4*b*c) - (((x^4*(96*a*b^4*d^5 + 96*b^5*c*d^4)
- (x^4*(384*a^2*b^4*d^6 + 384*b^6*c^2*d^4 + 768*a*b^5*c*d^5) - (x^4*(512*a^3*b^4*d^7 + 512*b^7*c^3*d^4 + 1536*
a*b^6*c^2*d^5 + 1536*a^2*b^5*c*d^6) + 1024*a*b^6*c^3*d^4 + 1024*a^3*b^4*c*d^6 + 2048*a^2*b^5*c^2*d^5)/(4*a*d -
4*b*c) + 512*a*b^5*c^2*d^4 + 512*a^2*b^4*c*d^5)/(4*a*d - 4*b*c) + 64*a*b^4*c*d^4)/(4*a*d - 4*b*c) - 8*b^4*d^4
*x^4)*1i)/(4*a*d - 4*b*c))/(((x^4*(96*a*b^4*d^5 + 96*b^5*c*d^4) + ((x^4*(512*a^3*b^4*d^7 + 512*b^7*c^3*d^4 + 1
536*a*b^6*c^2*d^5 + 1536*a^2*b^5*c*d^6) + 1024*a*b^6*c^3*d^4 + 1024*a^3*b^4*c*d^6 + 2048*a^2*b^5*c^2*d^5)/(4*a
*d - 4*b*c) + x^4*(384*a^2*b^4*d^6 + 384*b^6*c^2*d^4 + 768*a*b^5*c*d^5) + 512*a*b^5*c^2*d^4 + 512*a^2*b^4*c*d^
5)/(4*a*d - 4*b*c) + 64*a*b^4*c*d^4)/(4*a*d - 4*b*c) + 8*b^4*d^4*x^4)/(4*a*d - 4*b*c) + ((x^4*(96*a*b^4*d^5 +
96*b^5*c*d^4) - (x^4*(384*a^2*b^4*d^6 + 384*b^6*c^2*d^4 + 768*a*b^5*c*d^5) - (x^4*(512*a^3*b^4*d^7 + 512*b^7*c
^3*d^4 + 1536*a*b^6*c^2*d^5 + 1536*a^2*b^5*c*d^6) + 1024*a*b^6*c^3*d^4 + 1024*a^3*b^4*c*d^6 + 2048*a^2*b^5*c^2
*d^5)/(4*a*d - 4*b*c) + 512*a*b^5*c^2*d^4 + 512*a^2*b^4*c*d^5)/(4*a*d - 4*b*c) + 64*a*b^4*c*d^4)/(4*a*d - 4*b*
c) - 8*b^4*d^4*x^4)/(4*a*d - 4*b*c)))*2i)/(4*a*d - 4*b*c)

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sympy [B]  time = 1.91, size = 138, normalized size = 3.07 \begin {gather*} \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 {gather*}

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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