3.252 \(\int \frac{x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=157 \[ \frac{a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{3/2} (b c-a d)^3}+\frac{x (b c-5 a d)}{8 d \left (c+d x^2\right ) (b c-a d)^2}-\frac{c x}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(c*x)/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(8*d*(b*c - a*d)^2*(c
+ d*x^2)) + (a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2
*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*d^(3/2)*(b
*c - a*d)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.434987, antiderivative size = 157, 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 \[ \frac{a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{3/2} (b c-a d)^3}+\frac{x (b c-5 a d)}{8 d \left (c+d x^2\right ) (b c-a d)^2}-\frac{c x}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-(c*x)/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(8*d*(b*c - a*d)^2*(c
+ d*x^2)) + (a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2
*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*d^(3/2)*(b
*c - a*d)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 73.4853, size = 139, normalized size = 0.89 \[ - \frac{a^{\frac{3}{2}} \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\left (a d - b c\right )^{3}} + \frac{c x}{4 d \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} - \frac{x \left (5 a d - b c\right )}{8 d \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{\left (3 a^{2} d^{2} + 6 a b c d - b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{c} d^{\frac{3}{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

-a**(3/2)*sqrt(b)*atan(sqrt(b)*x/sqrt(a))/(a*d - b*c)**3 + c*x/(4*d*(c + d*x**2)
**2*(a*d - b*c)) - x*(5*a*d - b*c)/(8*d*(c + d*x**2)*(a*d - b*c)**2) + (3*a**2*d
**2 + 6*a*b*c*d - b**2*c**2)*atan(sqrt(d)*x/sqrt(c))/(8*sqrt(c)*d**(3/2)*(a*d -
b*c)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.424337, size = 154, normalized size = 0.98 \[ \frac{1}{8} \left (\frac{8 a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2} (b c-a d)^3}+\frac{x (b c-5 a d)}{d \left (c+d x^2\right ) (b c-a d)^2}+\frac{2 c x}{d \left (c+d x^2\right )^2 (a d-b c)}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((2*c*x)/(d*(-(b*c) + a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(d*(b*c - a*d)^2*(
c + d*x^2)) + (8*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((
b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(3/2)*(
b*c - a*d)^3))/8

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 299, normalized size = 1.9 \[ -{\frac{5\,{x}^{3}{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}abcd}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{x}^{3}{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,x{a}^{2}cd}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{xab{c}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{x{b}^{2}{c}^{3}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}d}}+{\frac{3\,{a}^{2}d}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,acb}{4\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{3}d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b{a}^{2}}{ \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

-5/8/(a*d-b*c)^3/(d*x^2+c)^2*x^3*a^2*d^2+3/4/(a*d-b*c)^3/(d*x^2+c)^2*x^3*a*b*c*d
-1/8/(a*d-b*c)^3/(d*x^2+c)^2*x^3*b^2*c^2-3/8/(a*d-b*c)^3/(d*x^2+c)^2*x*a^2*c*d+1
/4/(a*d-b*c)^3/(d*x^2+c)^2*x*a*b*c^2+1/8/(a*d-b*c)^3/(d*x^2+c)^2*c^3/d*x*b^2+3/8
/(a*d-b*c)^3*d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2+3/4/(a*d-b*c)^3/(c*d)^(1/
2)*arctan(x*d/(c*d)^(1/2))*c*a*b-1/8/(a*d-b*c)^3/d/(c*d)^(1/2)*arctan(x*d/(c*d)^
(1/2))*b^2*c^2-b/(a*d-b*c)^3*a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))

_______________________________________________________________________________________

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)*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.596803, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

[-1/16*(8*(a*d^3*x^4 + 2*a*c*d^2*x^2 + a*c^2*d)*sqrt(-a*b)*sqrt(-c*d)*log((b*x^2
- 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (
b^2*c^2*d^2 - 6*a*b*c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^
2*c*d^3)*x^2)*log((2*c*d*x + (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) - 2*((b^2*c^2*
d - 6*a*b*c*d^2 + 5*a^2*d^3)*x^3 - (b^2*c^3 + 2*a*b*c^2*d - 3*a^2*c*d^2)*x)*sqrt
(-c*d))/((b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4 + (b^3*c^3
*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6)*x^4 + 2*(b^3*c^4*d^2 - 3*a*b^2
*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^2)*sqrt(-c*d)), -1/8*(4*(a*d^3*x^4 + 2
*a*c*d^2*x^2 + a*c^2*d)*sqrt(-a*b)*sqrt(c*d)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b
*x^2 + a)) - (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*c*d^3
- 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*arctan(sqrt
(c*d)*x/c) - ((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3)*x^3 - (b^2*c^3 + 2*a*b*c^2*d
- 3*a^2*c*d^2)*x)*sqrt(c*d))/((b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 -
a^3*c^2*d^4 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6)*x^4 + 2*
(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^2)*sqrt(c*d)), 1
/16*(16*(a*d^3*x^4 + 2*a*c*d^2*x^2 + a*c^2*d)*sqrt(a*b)*sqrt(-c*d)*arctan(b*x/sq
rt(a*b)) + (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*c*d^3 -
3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*log((2*c*d*x
+ (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) + 2*((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3
)*x^3 - (b^2*c^3 + 2*a*b*c^2*d - 3*a^2*c*d^2)*x)*sqrt(-c*d))/((b^3*c^5*d - 3*a*b
^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*
a^2*b*c*d^5 - a^3*d^6)*x^4 + 2*(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4
- a^3*c*d^5)*x^2)*sqrt(-c*d)), 1/8*(8*(a*d^3*x^4 + 2*a*c*d^2*x^2 + a*c^2*d)*sqrt
(a*b)*sqrt(c*d)*arctan(b*x/sqrt(a*b)) + (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 +
(b^2*c^2*d^2 - 6*a*b*c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*
a^2*c*d^3)*x^2)*arctan(sqrt(c*d)*x/c) + ((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3)*x
^3 - (b^2*c^3 + 2*a*b*c^2*d - 3*a^2*c*d^2)*x)*sqrt(c*d))/((b^3*c^5*d - 3*a*b^2*c
^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*
b*c*d^5 - a^3*d^6)*x^4 + 2*(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^
3*c*d^5)*x^2)*sqrt(c*d))]

_______________________________________________________________________________________

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**4/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.258904, size = 275, normalized size = 1.75 \[ \frac{a^{2} b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b}} + \frac{{\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt{c d}} + \frac{b c d x^{3} - 5 \, a d^{2} x^{3} - b c^{2} x - 3 \, a c d x}{8 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

a^2*b*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)
*sqrt(a*b)) + 1/8*(b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*arctan(d*x/sqrt(c*d))/((b^3*
c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*sqrt(c*d)) + 1/8*(b*c*d*x^3 -
5*a*d^2*x^3 - b*c^2*x - 3*a*c*d*x)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(d*x^2
+ c)^2)