3.606 \(\int \frac{x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=79 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.17717, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.1415, size = 66, normalized size = 0.84 \[ \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b} \left (a d - b c\right )} - \frac{\sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c}} \right )}}{2 \sqrt{d} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**4+a)/(d*x**4+c),x)

[Out]

sqrt(a)*atan(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)*(a*d - b*c)) - sqrt(c)*atan(sqrt(d
)*x**2/sqrt(c))/(2*sqrt(d)*(a*d - b*c))

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Mathematica [A]  time = 0.0639652, size = 66, normalized size = 0.84 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{b}}}{2 b c-2 a d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 60, normalized size = 0.8 \[ -{\frac{c}{2\,ad-2\,bc}\arctan \left ({d{x}^{2}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{a}{2\,ad-2\,bc}\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)/(d*x^4+c),x)

[Out]

-1/2*c/(a*d-b*c)/(c*d)^(1/2)*arctan(x^2*d/(c*d)^(1/2))+1/2*a/(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^5/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271902, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{b}}}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{c}{d}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{c}{d}}}\right ) - \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{b}}}\right ) - \sqrt{\frac{c}{d}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{c}{d}}}\right )}{2 \,{\left (b c - a d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 52.6507, size = 576, normalized size = 7.29 \[ \frac{\sqrt{- \frac{a}{b}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} - \frac{\sqrt{- \frac{a}{b}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} + \frac{\sqrt{- \frac{c}{d}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} - \frac{\sqrt{- \frac{c}{d}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**4+a)/(d*x**4+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.234224, size = 271, normalized size = 3.43 \[ -\frac{\sqrt{c d} b x^{4}{\left | d \right |} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, b c + 2 \, a d + \sqrt{-16 \, a b c d + 4 \,{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b c d{\left | b c - a d \right |} + a d^{2}{\left | b c - a d \right |} +{\left (b c - a d\right )}^{2} d} + \frac{\sqrt{a b} d x^{4}{\left | b \right |} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, b c + 2 \, a d - \sqrt{-16 \, a b c d + 4 \,{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2} c{\left | b c - a d \right |} + a b d{\left | b c - a d \right |} -{\left (b c - a d\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="giac")

[Out]

-sqrt(c*d)*b*x^4*abs(d)*arctan(2*x^2/sqrt((2*b*c + 2*a*d + sqrt(-16*a*b*c*d + 4*
(b*c + a*d)^2))/(b*d)))/(b*c*d*abs(b*c - a*d) + a*d^2*abs(b*c - a*d) + (b*c - a*
d)^2*d) + sqrt(a*b)*d*x^4*abs(b)*arctan(2*x^2/sqrt((2*b*c + 2*a*d - sqrt(-16*a*b
*c*d + 4*(b*c + a*d)^2))/(b*d)))/(b^2*c*abs(b*c - a*d) + a*b*d*abs(b*c - a*d) -
(b*c - a*d)^2*b)