3.767 \(\int \frac {x^{15}}{(a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=90 \[ -\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}-\frac {x^4 (a d+b c)}{4 b^2 d^2}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}+\frac {x^8}{8 b d} \]

[Out]

-1/4*(a*d+b*c)*x^4/b^2/d^2+1/8*x^8/b/d-1/4*a^3*ln(b*x^4+a)/b^3/(-a*d+b*c)+1/4*c^3*ln(d*x^4+c)/d^3/(-a*d+b*c)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, 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, 72} \[ -\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}-\frac {x^4 (a d+b c)}{4 b^2 d^2}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}+\frac {x^8}{8 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^15/((a + b*x^4)*(c + d*x^4)),x]

[Out]

-((b*c + a*d)*x^4)/(4*b^2*d^2) + x^8/(8*b*d) - (a^3*Log[a + b*x^4])/(4*b^3*(b*c - a*d)) + (c^3*Log[c + d*x^4])
/(4*d^3*(b*c - a*d))

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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]]

Rubi steps

\begin {align*} \int \frac {x^{15}}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x) (c+d x)} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {-b c-a d}{b^2 d^2}+\frac {x}{b d}-\frac {a^3}{b^2 (b c-a d) (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx,x,x^4\right )\\ &=-\frac {(b c+a d) x^4}{4 b^2 d^2}+\frac {x^8}{8 b d}-\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 92, normalized size = 1.02 \[ -\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}+\frac {x^4 (-a d-b c)}{4 b^2 d^2}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}+\frac {x^8}{8 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15/((a + b*x^4)*(c + d*x^4)),x]

[Out]

((-(b*c) - a*d)*x^4)/(4*b^2*d^2) + x^8/(8*b*d) - (a^3*Log[a + b*x^4])/(4*b^3*(b*c - a*d)) + (c^3*Log[c + d*x^4
])/(4*d^3*(b*c - a*d))

________________________________________________________________________________________

fricas [A]  time = 7.13, size = 100, normalized size = 1.11 \[ \frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{8} - 2 \, a^{3} d^{3} \log \left (b x^{4} + a\right ) + 2 \, b^{3} c^{3} \log \left (d x^{4} + c\right ) - 2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{4}}{8 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

1/8*((b^3*c*d^2 - a*b^2*d^3)*x^8 - 2*a^3*d^3*log(b*x^4 + a) + 2*b^3*c^3*log(d*x^4 + c) - 2*(b^3*c^2*d - a^2*b*
d^3)*x^4)/(b^4*c*d^3 - a*b^3*d^4)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 88, normalized size = 0.98 \[ -\frac {a^{3} \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )}} + \frac {c^{3} \log \left ({\left | d x^{4} + c \right |}\right )}{4 \, {\left (b c d^{3} - a d^{4}\right )}} + \frac {b d x^{8} - 2 \, b c x^{4} - 2 \, a d x^{4}}{8 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

-1/4*a^3*log(abs(b*x^4 + a))/(b^4*c - a*b^3*d) + 1/4*c^3*log(abs(d*x^4 + c))/(b*c*d^3 - a*d^4) + 1/8*(b*d*x^8
- 2*b*c*x^4 - 2*a*d*x^4)/(b^2*d^2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 89, normalized size = 0.99 \[ \frac {x^{8}}{8 b d}-\frac {a \,x^{4}}{4 b^{2} d}-\frac {c \,x^{4}}{4 b \,d^{2}}+\frac {a^{3} \ln \left (b \,x^{4}+a \right )}{4 \left (a d -b c \right ) b^{3}}-\frac {c^{3} \ln \left (d \,x^{4}+c \right )}{4 \left (a d -b c \right ) d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(b*x^4+a)/(d*x^4+c),x)

[Out]

1/8*x^8/b/d-1/4/b^2/d*x^4*a-1/4/b/d^2*x^4*c-1/4*c^3/d^3/(a*d-b*c)*ln(d*x^4+c)+1/4*a^3/b^3/(a*d-b*c)*ln(b*x^4+a
)

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 84, normalized size = 0.93 \[ -\frac {a^{3} \log \left (b x^{4} + a\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )}} + \frac {c^{3} \log \left (d x^{4} + c\right )}{4 \, {\left (b c d^{3} - a d^{4}\right )}} + \frac {b d x^{8} - 2 \, {\left (b c + a d\right )} x^{4}}{8 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")

[Out]

-1/4*a^3*log(b*x^4 + a)/(b^4*c - a*b^3*d) + 1/4*c^3*log(d*x^4 + c)/(b*c*d^3 - a*d^4) + 1/8*(b*d*x^8 - 2*(b*c +
a*d)*x^4)/(b^2*d^2)

________________________________________________________________________________________

mupad [B]  time = 5.88, size = 88, normalized size = 0.98 \[ \frac {x^8}{8\,b\,d}-\frac {c^3\,\ln \left (d\,x^4+c\right )}{4\,\left (a\,d^4-b\,c\,d^3\right )}-\frac {a^3\,\ln \left (b\,x^4+a\right )}{4\,\left (b^4\,c-a\,b^3\,d\right )}-\frac {x^4\,\left (a\,d+b\,c\right )}{4\,b^2\,d^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/((a + b*x^4)*(c + d*x^4)),x)

[Out]

x^8/(8*b*d) - (c^3*log(c + d*x^4))/(4*(a*d^4 - b*c*d^3)) - (a^3*log(a + b*x^4))/(4*(b^4*c - a*b^3*d)) - (x^4*(
a*d + b*c))/(4*b^2*d^2)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15/(b*x**4+a)/(d*x**4+c),x)

[Out]

Timed out

________________________________________________________________________________________