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]

-((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))

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Rubi [A]  time = 0.0955465, antiderivative size = 90, 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, 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 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 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]

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 = 0.0469224, 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))

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Maple [A]  time = 0.007, size = 89, normalized size = 1. \begin{align*}{\frac{{x}^{8}}{8\,bd}}-{\frac{{x}^{4}a}{4\,{b}^{2}d}}-{\frac{{x}^{4}c}{4\,b{d}^{2}}}-{\frac{{c}^{3}\ln \left ( d{x}^{4}+c \right ) }{4\,{d}^{3} \left ( ad-bc \right ) }}+{\frac{{a}^{3}\ln \left ( b{x}^{4}+a \right ) }{4\,{b}^{3} \left ( ad-bc \right ) }} \end{align*}

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
)

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Maxima [A]  time = 0.935028, size = 113, normalized size = 1.26 \begin{align*} -\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}} \end{align*}

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)

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Fricas [A]  time = 28.199, size = 196, normalized size = 2.18 \begin{align*} \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 )}} \end{align*}

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)

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Sympy [B]  time = 9.56232, size = 230, normalized size = 2.56 \begin{align*} \frac{a^{3} \log{\left (x^{4} + \frac{\frac{a^{5} d^{4}}{b \left (a d - b c\right )} - \frac{2 a^{4} c d^{3}}{a d - b c} + \frac{a^{3} b c^{2} d^{2}}{a d - b c} + a^{3} c d^{2} + a b^{2} c^{3}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{4 b^{3} \left (a d - b c\right )} - \frac{c^{3} \log{\left (x^{4} + \frac{a^{3} c d^{2} - \frac{a^{2} b^{2} c^{3} d}{a d - b c} + \frac{2 a b^{3} c^{4}}{a d - b c} + a b^{2} c^{3} - \frac{b^{4} c^{5}}{d \left (a d - b c\right )}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{4 d^{3} \left (a d - b c\right )} + \frac{x^{8}}{8 b d} - \frac{x^{4} \left (a d + b c\right )}{4 b^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x**4 + (a**5*d**4/(b*(a*d - b*c)) - 2*a**4*c*d**3/(a*d - b*c) + a**3*b*c**2*d**2/(a*d - b*c) + a**3*c
*d**2 + a*b**2*c**3)/(a**3*d**3 + b**3*c**3))/(4*b**3*(a*d - b*c)) - c**3*log(x**4 + (a**3*c*d**2 - a**2*b**2*
c**3*d/(a*d - b*c) + 2*a*b**3*c**4/(a*d - b*c) + a*b**2*c**3 - b**4*c**5/(d*(a*d - b*c)))/(a**3*d**3 + b**3*c*
*3))/(4*d**3*(a*d - b*c)) + x**8/(8*b*d) - x**4*(a*d + b*c)/(4*b**2*d**2)

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

[Out]

Exception raised: NotImplementedError