3.773 \(\int \frac{x^{13}}{(a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=112 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{5/2} (b c-a d)}-\frac{x^2 (a d+b c)}{2 b^2 d^2}+\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 d^{5/2} (b c-a d)}+\frac{x^6}{6 b d} \]

[Out]

-((b*c + a*d)*x^2)/(2*b^2*d^2) + x^6/(6*b*d) - (a^(5/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*b^(5/2)*(b*c - a*d))
+ (c^(5/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*d^(5/2)*(b*c - a*d))

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Rubi [A]  time = 0.273042, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {465, 479, 582, 522, 205} \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{5/2} (b c-a d)}-\frac{x^2 (a d+b c)}{2 b^2 d^2}+\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 d^{5/2} (b c-a d)}+\frac{x^6}{6 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^13/((a + b*x^4)*(c + d*x^4)),x]

[Out]

-((b*c + a*d)*x^2)/(2*b^2*d^2) + x^6/(6*b*d) - (a^(5/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*b^(5/2)*(b*c - a*d))
+ (c^(5/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*d^(5/2)*(b*c - a*d))

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(2*n
- 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q) + 1)), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 582

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q
+ 1) + 1)), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{13}}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^6}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=\frac{x^6}{6 b d}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 a c+3 (b c+a d) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 b d}\\ &=-\frac{(b c+a d) x^2}{2 b^2 d^2}+\frac{x^6}{6 b d}+\frac{\operatorname{Subst}\left (\int \frac{3 a c (b c+a d)+3 \left (b^2 c^2+a d (b c+a d)\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 b^2 d^2}\\ &=-\frac{(b c+a d) x^2}{2 b^2 d^2}+\frac{x^6}{6 b d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 b^2 (b c-a d)}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{c+d x^2} \, dx,x,x^2\right )}{2 d^2 (b c-a d)}\\ &=-\frac{(b c+a d) x^2}{2 b^2 d^2}+\frac{x^6}{6 b d}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{5/2} (b c-a d)}+\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 d^{5/2} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.158419, size = 104, normalized size = 0.93 \[ \frac{1}{6} \left (\frac{3 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{b^{5/2} (a d-b c)}+\frac{x^2 \left (-3 a d-3 b c+b d x^4\right )}{b^2 d^2}+\frac{3 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{d^{5/2} (b c-a d)}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^13/((a + b*x^4)*(c + d*x^4)),x]

[Out]

((x^2*(-3*b*c - 3*a*d + b*d*x^4))/(b^2*d^2) + (3*a^(5/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(b^(5/2)*(-(b*c) + a*d
)) + (3*c^(5/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(d^(5/2)*(b*c - a*d)))/6

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Maple [A]  time = 0.011, size = 105, normalized size = 0.9 \begin{align*}{\frac{{x}^{6}}{6\,bd}}-{\frac{{x}^{2}a}{2\,{b}^{2}d}}-{\frac{{x}^{2}c}{2\,b{d}^{2}}}-{\frac{{c}^{3}}{2\,{d}^{2} \left ( ad-bc \right ) }\arctan \left ({{x}^{2}d{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{a}^{3}}{2\,{b}^{2} \left ( ad-bc \right ) }\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(b*x^4+a)/(d*x^4+c),x)

[Out]

1/6*x^6/b/d-1/2/b^2/d*x^2*a-1/2/b/d^2*x^2*c-1/2*c^3/d^2/(a*d-b*c)/(c*d)^(1/2)*arctan(x^2*d/(c*d)^(1/2))+1/2*a^
3/b^2/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 10.3222, size = 1125, normalized size = 10.04 \begin{align*} \left [\frac{2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{6} - 3 \, a^{2} d^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) - 3 \, b^{2} c^{2} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{12 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac{2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{6} - 6 \, a^{2} d^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{2} \sqrt{\frac{a}{b}}}{a}\right ) - 3 \, b^{2} c^{2} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{12 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac{2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{6} + 6 \, b^{2} c^{2} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x^{2} \sqrt{\frac{c}{d}}}{c}\right ) - 3 \, a^{2} d^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{12 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{6} - 3 \, a^{2} d^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{2} \sqrt{\frac{a}{b}}}{a}\right ) + 3 \, b^{2} c^{2} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x^{2} \sqrt{\frac{c}{d}}}{c}\right ) - 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

[1/12*(2*(b^2*c*d - a*b*d^2)*x^6 - 3*a^2*d^2*sqrt(-a/b)*log((b*x^4 + 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a)) - 3*
b^2*c^2*sqrt(-c/d)*log((d*x^4 - 2*d*x^2*sqrt(-c/d) - c)/(d*x^4 + c)) - 6*(b^2*c^2 - a^2*d^2)*x^2)/(b^3*c*d^2 -
a*b^2*d^3), 1/12*(2*(b^2*c*d - a*b*d^2)*x^6 - 6*a^2*d^2*sqrt(a/b)*arctan(b*x^2*sqrt(a/b)/a) - 3*b^2*c^2*sqrt(
-c/d)*log((d*x^4 - 2*d*x^2*sqrt(-c/d) - c)/(d*x^4 + c)) - 6*(b^2*c^2 - a^2*d^2)*x^2)/(b^3*c*d^2 - a*b^2*d^3),
1/12*(2*(b^2*c*d - a*b*d^2)*x^6 + 6*b^2*c^2*sqrt(c/d)*arctan(d*x^2*sqrt(c/d)/c) - 3*a^2*d^2*sqrt(-a/b)*log((b*
x^4 + 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a)) - 6*(b^2*c^2 - a^2*d^2)*x^2)/(b^3*c*d^2 - a*b^2*d^3), 1/6*((b^2*c*d
- a*b*d^2)*x^6 - 3*a^2*d^2*sqrt(a/b)*arctan(b*x^2*sqrt(a/b)/a) + 3*b^2*c^2*sqrt(c/d)*arctan(d*x^2*sqrt(c/d)/c
) - 3*(b^2*c^2 - a^2*d^2)*x^2)/(b^3*c*d^2 - a*b^2*d^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(b*x**4+a)/(d*x**4+c),x)

[Out]

Timed out

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Giac [B]  time = 1.24234, size = 722, normalized size = 6.45 \begin{align*} -\frac{{\left (\sqrt{a b} b^{5} c^{2} d^{5} x^{4}{\left | b \right |} + \sqrt{a b} a b^{4} c d^{6} x^{4}{\left | b \right |} + \sqrt{a b} a^{2} b^{3} d^{7} x^{4}{\left | b \right |} + \sqrt{a b} a b^{4} c^{2} d^{5}{\left | b \right |} + \sqrt{a b} a^{2} b^{3} c d^{6}{\left | b \right |}\right )} \arctan \left (\frac{8 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{16 \, b^{4} c d^{3} + 16 \, a b^{3} d^{4} + \sqrt{-1024 \, a b^{7} c d^{7} + 256 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )}^{2}}}{b^{4} d^{4}}}}\right )}{b^{4} c d^{3}{\left | -b^{4} c d^{3} + a b^{3} d^{4} \right |} + a b^{3} d^{4}{\left | -b^{4} c d^{3} + a b^{3} d^{4} \right |} +{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}^{2}} + \frac{{\left (\sqrt{c d} b^{7} c^{2} d^{3} x^{4}{\left | d \right |} + \sqrt{c d} a b^{6} c d^{4} x^{4}{\left | d \right |} + \sqrt{c d} a^{2} b^{5} d^{5} x^{4}{\left | d \right |} + \sqrt{c d} a b^{6} c^{2} d^{3}{\left | d \right |} + \sqrt{c d} a^{2} b^{5} c d^{4}{\left | d \right |}\right )} \arctan \left (\frac{8 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{16 \, b^{4} c d^{3} + 16 \, a b^{3} d^{4} - \sqrt{-1024 \, a b^{7} c d^{7} + 256 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )}^{2}}}{b^{4} d^{4}}}}\right )}{b^{4} c d^{3}{\left | -b^{4} c d^{3} + a b^{3} d^{4} \right |} + a b^{3} d^{4}{\left | -b^{4} c d^{3} + a b^{3} d^{4} \right |} -{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}^{2}} + \frac{b^{2} d^{2} x^{6} - 3 \, b^{2} c d x^{2} - 3 \, a b d^{2} x^{2}}{6 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

-(sqrt(a*b)*b^5*c^2*d^5*x^4*abs(b) + sqrt(a*b)*a*b^4*c*d^6*x^4*abs(b) + sqrt(a*b)*a^2*b^3*d^7*x^4*abs(b) + sqr
t(a*b)*a*b^4*c^2*d^5*abs(b) + sqrt(a*b)*a^2*b^3*c*d^6*abs(b))*arctan(8*sqrt(1/2)*x^2/sqrt((16*b^4*c*d^3 + 16*a
*b^3*d^4 + sqrt(-1024*a*b^7*c*d^7 + 256*(b^4*c*d^3 + a*b^3*d^4)^2))/(b^4*d^4)))/(b^4*c*d^3*abs(-b^4*c*d^3 + a*
b^3*d^4) + a*b^3*d^4*abs(-b^4*c*d^3 + a*b^3*d^4) + (b^4*c*d^3 - a*b^3*d^4)^2) + (sqrt(c*d)*b^7*c^2*d^3*x^4*abs
(d) + sqrt(c*d)*a*b^6*c*d^4*x^4*abs(d) + sqrt(c*d)*a^2*b^5*d^5*x^4*abs(d) + sqrt(c*d)*a*b^6*c^2*d^3*abs(d) + s
qrt(c*d)*a^2*b^5*c*d^4*abs(d))*arctan(8*sqrt(1/2)*x^2/sqrt((16*b^4*c*d^3 + 16*a*b^3*d^4 - sqrt(-1024*a*b^7*c*d
^7 + 256*(b^4*c*d^3 + a*b^3*d^4)^2))/(b^4*d^4)))/(b^4*c*d^3*abs(-b^4*c*d^3 + a*b^3*d^4) + a*b^3*d^4*abs(-b^4*c
*d^3 + a*b^3*d^4) - (b^4*c*d^3 - a*b^3*d^4)^2) + 1/6*(b^2*d^2*x^6 - 3*b^2*c*d*x^2 - 3*a*b*d^2*x^2)/(b^3*d^3)