3.605 \(\int \frac{x^9}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=92 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)}-\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)}+\frac{x^2}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.306129, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)}-\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)}+\frac{x^2}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^9/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 44.9054, size = 75, normalized size = 0.82 \[ - \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}} \left (a d - b c\right )} + \frac{c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c}} \right )}}{2 d^{\frac{3}{2}} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)/(d*x**4+c),x)

[Out]

-a**(3/2)*atan(sqrt(b)*x**2/sqrt(a))/(2*b**(3/2)*(a*d - b*c)) + c**(3/2)*atan(sq
rt(d)*x**2/sqrt(c))/(2*d**(3/2)*(a*d - b*c)) + x**2/(2*b*d)

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Mathematica [A]  time = 0.258417, size = 82, normalized size = 0.89 \[ \frac{\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{b^{3/2}}+x^2 \left (\frac{c}{d}-\frac{a}{b}\right )-\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{d^{3/2}}}{2 b c-2 a d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 81, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,bd}}+{\frac{{c}^{2}}{ \left ( 2\,ad-2\,bc \right ) d}\arctan \left ({d{x}^{2}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{ \left ( 2\,ad-2\,bc \right ) b}\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)/(d*x^4+c),x)

[Out]

1/2*x^2/b/d+1/2*c^2/(a*d-b*c)/d/(c*d)^(1/2)*arctan(x^2*d/(c*d)^(1/2))-1/2*a^2/(a
*d-b*c)/b/(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^9/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.556566, size = 1, normalized size = 0.01 \[ \left [-\frac{a d \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} - 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) + b c \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} + 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right ) - 2 \,{\left (b c - a d\right )} x^{2}}{4 \,{\left (b^{2} c d - a b d^{2}\right )}}, \frac{2 \, a d \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{b}}}\right ) - b c \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} + 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right ) + 2 \,{\left (b c - a d\right )} x^{2}}{4 \,{\left (b^{2} c d - a b d^{2}\right )}}, -\frac{2 \, b c \sqrt{\frac{c}{d}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{c}{d}}}\right ) + a d \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} - 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) - 2 \,{\left (b c - a d\right )} x^{2}}{4 \,{\left (b^{2} c d - a b d^{2}\right )}}, \frac{a d \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{b}}}\right ) - b c \sqrt{\frac{c}{d}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{c}{d}}}\right ) +{\left (b c - a d\right )} x^{2}}{2 \,{\left (b^{2} c d - a b d^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 17.7964, size = 932, normalized size = 10.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)/(d*x**4+c),x)

[Out]

-sqrt(-a**3/b**3)*log(x**2 + (-a**4*d**4*sqrt(-a**3/b**3)/(a*d - b*c) - a**3*b**
3*d**6*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 + a**2*b**4*c*d**5*(-a**3/b**3)**(3/2)
/(a*d - b*c)**3 + a*b**5*c**2*d**4*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - b**6*c**
3*d**3*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - b**4*c**4*sqrt(-a**3/b**3)/(a*d - b*
c))/(a**3*c*d**2 + a**2*b*c**2*d + a*b**2*c**3))/(4*(a*d - b*c)) + sqrt(-a**3/b*
*3)*log(x**2 + (a**4*d**4*sqrt(-a**3/b**3)/(a*d - b*c) + a**3*b**3*d**6*(-a**3/b
**3)**(3/2)/(a*d - b*c)**3 - a**2*b**4*c*d**5*(-a**3/b**3)**(3/2)/(a*d - b*c)**3
- a*b**5*c**2*d**4*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 + b**6*c**3*d**3*(-a**3/b
**3)**(3/2)/(a*d - b*c)**3 + b**4*c**4*sqrt(-a**3/b**3)/(a*d - b*c))/(a**3*c*d**
2 + a**2*b*c**2*d + a*b**2*c**3))/(4*(a*d - b*c)) - sqrt(-c**3/d**3)*log(x**2 +
(-a**4*d**4*sqrt(-c**3/d**3)/(a*d - b*c) - a**3*b**3*d**6*(-c**3/d**3)**(3/2)/(a
*d - b*c)**3 + a**2*b**4*c*d**5*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + a*b**5*c**2
*d**4*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - b**6*c**3*d**3*(-c**3/d**3)**(3/2)/(a
*d - b*c)**3 - b**4*c**4*sqrt(-c**3/d**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c**
2*d + a*b**2*c**3))/(4*(a*d - b*c)) + sqrt(-c**3/d**3)*log(x**2 + (a**4*d**4*sqr
t(-c**3/d**3)/(a*d - b*c) + a**3*b**3*d**6*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 -
a**2*b**4*c*d**5*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - a*b**5*c**2*d**4*(-c**3/d*
*3)**(3/2)/(a*d - b*c)**3 + b**6*c**3*d**3*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 +
b**4*c**4*sqrt(-c**3/d**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c**2*d + a*b**2*c*
*3))/(4*(a*d - b*c)) + x**2/(2*b*d)

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GIAC/XCAS [A]  time = 0.250056, size = 482, normalized size = 5.24 \[ \frac{{\left (\sqrt{c d} b^{3} c x^{4}{\left | d \right |} + \sqrt{c d} a b^{2} d x^{4}{\left | d \right |} + \sqrt{c d} a b^{2} c{\left | d \right |}\right )} \arctan \left (\frac{4 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{4 \, b^{2} c d + 4 \, a b d^{2} + \sqrt{-64 \, a b^{3} c d^{3} + 16 \,{\left (b^{2} c d + a b d^{2}\right )}^{2}}}{b^{2} d^{2}}}}\right )}{b^{2} c d{\left | b^{2} c d - a b d^{2} \right |} + a b d^{2}{\left | b^{2} c d - a b d^{2} \right |} +{\left (b^{2} c d - a b d^{2}\right )}^{2}} - \frac{{\left (\sqrt{a b} b c d^{2} x^{4}{\left | b \right |} + \sqrt{a b} a d^{3} x^{4}{\left | b \right |} + \sqrt{a b} a c d^{2}{\left | b \right |}\right )} \arctan \left (\frac{4 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{4 \, b^{2} c d + 4 \, a b d^{2} - \sqrt{-64 \, a b^{3} c d^{3} + 16 \,{\left (b^{2} c d + a b d^{2}\right )}^{2}}}{b^{2} d^{2}}}}\right )}{b^{2} c d{\left | b^{2} c d - a b d^{2} \right |} + a b d^{2}{\left | b^{2} c d - a b d^{2} \right |} -{\left (b^{2} c d - a b d^{2}\right )}^{2}} + \frac{x^{2}}{2 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="giac")

[Out]

(sqrt(c*d)*b^3*c*x^4*abs(d) + sqrt(c*d)*a*b^2*d*x^4*abs(d) + sqrt(c*d)*a*b^2*c*a
bs(d))*arctan(4*sqrt(1/2)*x^2/sqrt((4*b^2*c*d + 4*a*b*d^2 + sqrt(-64*a*b^3*c*d^3
+ 16*(b^2*c*d + a*b*d^2)^2))/(b^2*d^2)))/(b^2*c*d*abs(b^2*c*d - a*b*d^2) + a*b*
d^2*abs(b^2*c*d - a*b*d^2) + (b^2*c*d - a*b*d^2)^2) - (sqrt(a*b)*b*c*d^2*x^4*abs
(b) + sqrt(a*b)*a*d^3*x^4*abs(b) + sqrt(a*b)*a*c*d^2*abs(b))*arctan(4*sqrt(1/2)*
x^2/sqrt((4*b^2*c*d + 4*a*b*d^2 - sqrt(-64*a*b^3*c*d^3 + 16*(b^2*c*d + a*b*d^2)^
2))/(b^2*d^2)))/(b^2*c*d*abs(b^2*c*d - a*b*d^2) + a*b*d^2*abs(b^2*c*d - a*b*d^2)
- (b^2*c*d - a*b*d^2)^2) + 1/2*x^2/(b*d)