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

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)/Sqr
t[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 = 0.639138, 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 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{5/2} (b c-a d)}-\frac{x^2 (a d+b c)}{2 b^2 d^2}+\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 d^{5/2} (b c-a d)}+\frac{x^6}{6 b d} \]

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)/Sqr
t[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 in Sympy [A]  time = 73.9624, size = 94, normalized size = 0.84 \[ \frac{a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{5}{2}} \left (a d - b c\right )} - \frac{c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c}} \right )}}{2 d^{\frac{5}{2}} \left (a d - b c\right )} + \frac{x^{6}}{6 b d} - \frac{x^{2} \left (a d + b c\right )}{2 b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(b*x**4+a)/(d*x**4+c),x)

[Out]

a**(5/2)*atan(sqrt(b)*x**2/sqrt(a))/(2*b**(5/2)*(a*d - b*c)) - c**(5/2)*atan(sqr
t(d)*x**2/sqrt(c))/(2*d**(5/2)*(a*d - b*c)) + x**6/(6*b*d) - x**2*(a*d + b*c)/(2
*b**2*d**2)

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Mathematica [A]  time = 0.385159, 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)/Sq
rt[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 = 0.013, size = 105, normalized size = 0.9 \[{\frac{{x}^{6}}{6\,bd}}-{\frac{a{x}^{2}}{2\,{b}^{2}d}}-{\frac{c{x}^{2}}{2\,{d}^{2}b}}-{\frac{{c}^{3}}{2\,{d}^{2} \left ( ad-bc \right ) }\arctan \left ({d{x}^{2}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{a}^{3}}{2\,{b}^{2} \left ( ad-bc \right ) }\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 2.35535, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{6} - 3 \, a^{2} d^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) - 3 \, b^{2} c^{2} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{12 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac{2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{6} - 6 \, a^{2} d^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{b}}}\right ) - 3 \, b^{2} c^{2} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{12 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac{2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{6} + 6 \, b^{2} c^{2} \sqrt{\frac{c}{d}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{c}{d}}}\right ) - 3 \, a^{2} d^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{12 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{6} - 3 \, a^{2} d^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{b}}}\right ) + 3 \, b^{2} c^{2} \sqrt{\frac{c}{d}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{c}{d}}}\right ) - 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}\right ] \]

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*sqr
t(-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(x^2/sqrt(a/b)) - 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(x^2/sqrt(c/d)) - 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(x^2/sqrt(a/b)) + 3*b^2*
c^2*sqrt(c/d)*arctan(x^2/sqrt(c/d)) - 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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.257493, size = 722, normalized size = 6.45 \[ -\frac{{\left (\sqrt{c d} b^{7} c^{2} d^{3} x^{4}{\left | d \right |} + \sqrt{c d} a b^{6} c d^{4} x^{4}{\left | d \right |} + \sqrt{c d} a^{2} b^{5} d^{5} x^{4}{\left | d \right |} + \sqrt{c d} a b^{6} c^{2} d^{3}{\left | d \right |} + \sqrt{c d} a^{2} b^{5} c d^{4}{\left | d \right |}\right )} \arctan \left (\frac{8 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{16 \, b^{4} c d^{3} + 16 \, a b^{3} d^{4} + \sqrt{-1024 \, a b^{7} c d^{7} + 256 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )}^{2}}}{b^{4} d^{4}}}}\right )}{b^{4} c d^{3}{\left | b^{4} c d^{3} - a b^{3} d^{4} \right |} + a b^{3} d^{4}{\left | b^{4} c d^{3} - a b^{3} d^{4} \right |} +{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}^{2}} + \frac{{\left (\sqrt{a b} b^{5} c^{2} d^{5} x^{4}{\left | b \right |} + \sqrt{a b} a b^{4} c d^{6} x^{4}{\left | b \right |} + \sqrt{a b} a^{2} b^{3} d^{7} x^{4}{\left | b \right |} + \sqrt{a b} a b^{4} c^{2} d^{5}{\left | b \right |} + \sqrt{a b} a^{2} b^{3} c d^{6}{\left | b \right |}\right )} \arctan \left (\frac{8 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{16 \, b^{4} c d^{3} + 16 \, a b^{3} d^{4} - \sqrt{-1024 \, a b^{7} c d^{7} + 256 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )}^{2}}}{b^{4} d^{4}}}}\right )}{b^{4} c d^{3}{\left | b^{4} c d^{3} - a b^{3} d^{4} \right |} + a b^{3} d^{4}{\left | b^{4} c d^{3} - a b^{3} d^{4} \right |} -{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}^{2}} + \frac{b^{2} d^{2} x^{6} - 3 \, b^{2} c d x^{2} - 3 \, a b d^{2} x^{2}}{6 \, b^{3} d^{3}} \]

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(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) + sqrt(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(-102
4*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(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) + s
qrt(a*b)*a^2*b^3*d^7*x^4*abs(b) + sqrt(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 - sq
rt(-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*ab
s(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)