3.289 \(\int \frac{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\sqrt{a} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.23461, antiderivative size = 109, 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 \[ -\frac{\sqrt{a} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^2)^2*(c + d*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 36.8098, size = 94, normalized size = 0.86 \[ \frac{\sqrt{a} \left (a d - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{d} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

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

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Mathematica [A]  time = 0.249216, size = 95, normalized size = 0.87 \[ \frac{\frac{\sqrt{a} (a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}+\frac{a x (b c-a d)}{b \left (a+b x^2\right )}+\frac{2 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d}}}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 144, normalized size = 1.3 \[{\frac{{c}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{x{a}^{2}d}{2\, \left ( ad-bc \right ) ^{2}b \left ( b{x}^{2}+a \right ) }}+{\frac{acx}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{2}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,ac}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^2/(d*x^2+c),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.311494, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 2 \,{\left (b^{2} c x^{2} + a b c\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} + 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (a b c - a^{2} d\right )} x}{4 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac{{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) -{\left (b^{2} c x^{2} + a b c\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} + 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) -{\left (a b c - a^{2} d\right )} x}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, \frac{4 \,{\left (b^{2} c x^{2} + a b c\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) -{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 2 \,{\left (a b c - a^{2} d\right )} x}{4 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac{{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 2 \,{\left (b^{2} c x^{2} + a b c\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) -{\left (a b c - a^{2} d\right )} x}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 42.8901, size = 1850, normalized size = 16.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

-a*x/(2*a**2*b*d - 2*a*b**2*c + x**2*(2*a*b**2*d - 2*b**3*c)) - sqrt(-a/b**3)*(a
*d - 3*b*c)*log(x + (-a**5*b**3*d**6*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d -
b*c)**6) + 9*a**4*b**4*c*d**5*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)*
*6) - a**4*d**4*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 13*a**3*b**5*c*
*2*d**4*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 + 9*a**3*b*c*d**3*sqrt(
-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) + 17*a**2*b**6*c**3*d**3*(-a/b**3)**(3
/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-a/b**3)*(a*d
- 3*b*c)/(2*(a*d - b*c)**2) - 21*a*b**7*c**4*d**2*(-a/b**3)**(3/2)*(a*d - 3*b*c)
**3/(2*(a*d - b*c)**6) + 27*a*b**3*c**3*d*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d -
b*c)**2) + 5*b**8*c**5*d*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) +
4*b**4*c**4*sqrt(-a/b**3)*(a*d - 3*b*c)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**
2*d + 12*b**2*c**3))/(4*(a*d - b*c)**2) + sqrt(-a/b**3)*(a*d - 3*b*c)*log(x + (a
**5*b**3*d**6*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) - 9*a**4*b**4
*c*d**5*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) + a**4*d**4*sqrt(-a
/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) + 13*a**3*b**5*c**2*d**4*(-a/b**3)**(3/2
)*(a*d - 3*b*c)**3/(a*d - b*c)**6 - 9*a**3*b*c*d**3*sqrt(-a/b**3)*(a*d - 3*b*c)/
(2*(a*d - b*c)**2) - 17*a**2*b**6*c**3*d**3*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(a
*d - b*c)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)
**2) + 21*a*b**7*c**4*d**2*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6)
- 27*a*b**3*c**3*d*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 5*b**8*c**5*
d*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) - 4*b**4*c**4*sqrt(-a/b**
3)*(a*d - 3*b*c)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**2*d + 12*b**2*c**3))/(4
*(a*d - b*c)**2) - sqrt(-c**3/d)*log(x + (-4*a**5*b**3*d**6*(-c**3/d)**(3/2)/(a*
d - b*c)**6 + 36*a**4*b**4*c*d**5*(-c**3/d)**(3/2)/(a*d - b*c)**6 - a**4*d**4*sq
rt(-c**3/d)/(a*d - b*c)**2 - 104*a**3*b**5*c**2*d**4*(-c**3/d)**(3/2)/(a*d - b*c
)**6 + 9*a**3*b*c*d**3*sqrt(-c**3/d)/(a*d - b*c)**2 + 136*a**2*b**6*c**3*d**3*(-
c**3/d)**(3/2)/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-c**3/d)/(a*d - b*c)
**2 - 84*a*b**7*c**4*d**2*(-c**3/d)**(3/2)/(a*d - b*c)**6 + 27*a*b**3*c**3*d*sqr
t(-c**3/d)/(a*d - b*c)**2 + 20*b**8*c**5*d*(-c**3/d)**(3/2)/(a*d - b*c)**6 + 8*b
**4*c**4*sqrt(-c**3/d)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**2*d + 12*b**2*c**
3))/(2*(a*d - b*c)**2) + sqrt(-c**3/d)*log(x + (4*a**5*b**3*d**6*(-c**3/d)**(3/2
)/(a*d - b*c)**6 - 36*a**4*b**4*c*d**5*(-c**3/d)**(3/2)/(a*d - b*c)**6 + a**4*d*
*4*sqrt(-c**3/d)/(a*d - b*c)**2 + 104*a**3*b**5*c**2*d**4*(-c**3/d)**(3/2)/(a*d
- b*c)**6 - 9*a**3*b*c*d**3*sqrt(-c**3/d)/(a*d - b*c)**2 - 136*a**2*b**6*c**3*d*
*3*(-c**3/d)**(3/2)/(a*d - b*c)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-c**3/d)/(a*d -
b*c)**2 + 84*a*b**7*c**4*d**2*(-c**3/d)**(3/2)/(a*d - b*c)**6 - 27*a*b**3*c**3*
d*sqrt(-c**3/d)/(a*d - b*c)**2 - 20*b**8*c**5*d*(-c**3/d)**(3/2)/(a*d - b*c)**6
- 8*b**4*c**4*sqrt(-c**3/d)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**2*d + 12*b**
2*c**3))/(2*(a*d - b*c)**2)

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GIAC/XCAS [A]  time = 0.241564, size = 165, normalized size = 1.51 \[ \frac{c^{2} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d}} - \frac{{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{a b}} + \frac{a x}{2 \,{\left (b^{2} c - a b d\right )}{\left (b x^{2} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="giac")

[Out]

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