3.40 \(\int \frac{1}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=161 \[ \frac{b x (3 b c-7 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^3}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^3}+\frac{b x}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]

[Out]

(b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(8*a^2*(b*c - a*d)
^2*(a + b*x^2)) + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/
(Sqrt[c]*(b*c - a*d)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.44586, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{b x (3 b c-7 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^3}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^3}+\frac{b x}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

(b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(8*a^2*(b*c - a*d)
^2*(a + b*x^2)) + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/
(Sqrt[c]*(b*c - a*d)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 98.3856, size = 146, normalized size = 0.91 \[ \frac{d^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{c} \left (a d - b c\right )^{3}} - \frac{b x}{4 a \left (a + b x^{2}\right )^{2} \left (a d - b c\right )} - \frac{b x \left (7 a d - 3 b c\right )}{8 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )^{2}} - \frac{\sqrt{b} \left (15 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**3/(d*x**2+c),x)

[Out]

d**(5/2)*atan(sqrt(d)*x/sqrt(c))/(sqrt(c)*(a*d - b*c)**3) - b*x/(4*a*(a + b*x**2
)**2*(a*d - b*c)) - b*x*(7*a*d - 3*b*c)/(8*a**2*(a + b*x**2)*(a*d - b*c)**2) - s
qrt(b)*(15*a**2*d**2 - 10*a*b*c*d + 3*b**2*c**2)*atan(sqrt(b)*x/sqrt(a))/(8*a**(
5/2)*(a*d - b*c)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.54819, size = 158, normalized size = 0.98 \[ \frac{1}{8} \left (\frac{b x (3 b c-7 a d)}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (a d-b c)^3}-\frac{8 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^3}-\frac{2 b x}{a \left (a+b x^2\right )^2 (a d-b c)}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

((-2*b*x)/(a*(-(b*c) + a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(a^2*(b*c - a
*d)^2*(a + b*x^2)) - (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) - (8*d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt
[c]])/(Sqrt[c]*(b*c - a*d)^3))/8

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 309, normalized size = 1.9 \[{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{7\,{b}^{2}{x}^{3}{d}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{b}^{3}{x}^{3}cd}{4\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{3\,{b}^{4}{x}^{3}{c}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}-{\frac{9\,abx{d}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{b}^{2}xcd}{4\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{b}^{3}x{c}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{15\,{d}^{2}b}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}cd}{4\, \left ( ad-bc \right ) ^{3}a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,{b}^{3}{c}^{2}}{8\, \left ( ad-bc \right ) ^{3}{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^3/(d*x^2+c),x)

[Out]

d^3/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))-7/8*b^2/(a*d-b*c)^3/(b*x^2+a
)^2*x^3*d^2+5/4*b^3/(a*d-b*c)^3/(b*x^2+a)^2/a*x^3*c*d-3/8*b^4/(a*d-b*c)^3/(b*x^2
+a)^2/a^2*x^3*c^2-9/8*b/(a*d-b*c)^3/(b*x^2+a)^2*a*x*d^2+7/4*b^2/(a*d-b*c)^3/(b*x
^2+a)^2*x*c*d-5/8*b^3/(a*d-b*c)^3/(b*x^2+a)^2/a*x*c^2-15/8*b/(a*d-b*c)^3/(a*b)^(
1/2)*arctan(x*b/(a*b)^(1/2))*d^2+5/4*b^2/(a*d-b*c)^3/a/(a*b)^(1/2)*arctan(x*b/(a
*b)^(1/2))*c*d-3/8*b^3/(a*d-b*c)^3/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^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(1/((b*x^2 + a)^3*(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.983517, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^3*(d*x^2 + c)),x, algorithm="fricas")

[Out]

[1/16*(2*(3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 - (3*a^2*b^2*c^2 - 10*a^
3*b*c*d + 15*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*
b^3*c^2 - 10*a^2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt
(-b/a) - a)/(b*x^2 + a)) - 8*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2 + a^4*d^2)*sqrt(
-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^3*c^2 - 14*a^2*
b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d
^3 + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^4 + 2*(a^
3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x^2), 1/16*(2*(3*b^4*
c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 - 16*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2
+ a^4*d^2)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) - (3*a^2*b^2*c^2 - 10*a^3*b*c*d +
15*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 -
10*a^2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) -
a)/(b*x^2 + a)) + 2*(5*a*b^3*c^2 - 14*a^2*b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3
- 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3 + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d +
3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^4 + 2*(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^
2*c*d^2 - a^6*b*d^3)*x^2), 1/8*((3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 +
(3*a^2*b^2*c^2 - 10*a^3*b*c*d + 15*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2
*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(b/a)*a
rctan(b*x/(a*sqrt(b/a))) - 4*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2 + a^4*d^2)*sqrt(
-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + (5*a*b^3*c^2 - 14*a^2*b^
2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3
+ (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^4 + 2*(a^3*
b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x^2), 1/8*((3*b^4*c^2 -
10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 + (3*a^2*b^2*c^2 - 10*a^3*b*c*d + 15*a^4*d^2
+ (3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*
c*d + 15*a^3*b*d^2)*x^2)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 8*(a^2*b^2*d^2*x^
4 + 2*a^3*b*d^2*x^2 + a^4*d^2)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) + (5*a*b^3*c^
2 - 14*a^2*b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*
d^2 - a^7*d^3 + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*
x^4 + 2*(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x^2)]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230471, size = 294, normalized size = 1.83 \[ -\frac{d^{3} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{c d}} + \frac{{\left (3 \, b^{3} c^{2} - 10 \, a b^{2} c d + 15 \, a^{2} b d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt{a b}} + \frac{3 \, b^{3} c x^{3} - 7 \, a b^{2} d x^{3} + 5 \, a b^{2} c x - 9 \, a^{2} b d x}{8 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}{\left (b x^{2} + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^3*(d*x^2 + c)),x, algorithm="giac")

[Out]

-d^3*arctan(d*x/sqrt(c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
sqrt(c*d)) + 1/8*(3*b^3*c^2 - 10*a*b^2*c*d + 15*a^2*b*d^2)*arctan(b*x/sqrt(a*b))
/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*sqrt(a*b)) + 1/8*(3*
b^3*c*x^3 - 7*a*b^2*d*x^3 + 5*a*b^2*c*x - 9*a^2*b*d*x)/((a^2*b^2*c^2 - 2*a^3*b*c
*d + a^4*d^2)*(b*x^2 + a)^2)