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

Optimal. Leaf size=160 \[ -\frac{\sqrt{d} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^3}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^3}-\frac{d x (7 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.459235, antiderivative size = 160, 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 \[ -\frac{\sqrt{d} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^3}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^3}-\frac{d x (7 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 99.1836, size = 146, normalized size = 0.91 \[ \frac{d x}{4 c \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{d x \left (3 a d - 7 b c\right )}{8 c^{2} \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{\sqrt{d} \left (3 a^{2} d^{2} - 10 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}} \left (a d - b c\right )^{3}} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.507882, size = 158, normalized size = 0.99 \[ \frac{1}{8} \left (-\frac{\sqrt{d} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^3}-\frac{8 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (a d-b c)^3}+\frac{d x (3 a d-7 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{2 d x}{c \left (c+d x^2\right )^2 (b c-a d)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 310, normalized size = 1.9 \[{\frac{3\,{d}^{4}{x}^{3}{a}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}{c}^{2}}}-{\frac{5\,{d}^{3}{x}^{3}ab}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{7\,{d}^{2}{b}^{2}{x}^{3}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,{d}^{3}x{a}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}c}}-{\frac{7\,{d}^{2}xab}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,dx{b}^{2}c}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{3}{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{5\,a{d}^{2}b}{4\, \left ( ad-bc \right ) ^{3}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{15\,d{b}^{2}}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}}{ \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01554, 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)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

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

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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)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.262079, size = 293, normalized size = 1.83 \[ \frac{b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b}} - \frac{{\left (15 \, b^{2} c^{2} d - 10 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \sqrt{c d}} - \frac{7 \, b c d^{2} x^{3} - 3 \, a d^{3} x^{3} + 9 \, b c^{2} d x - 5 \, a c d^{2} x}{8 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}{\left (d x^{2} + c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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