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

Optimal. Leaf size=211 \[ -\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)^3}+\frac{d^{3/2} \left (15 a^2 d^2-42 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} (b c-a d)^3}-\frac{15 a^2 d^2-27 a b c d+8 b^2 c^2}{8 a c^3 x (b c-a d)^2}-\frac{d (9 b c-5 a d)}{8 c^2 x \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(8*b^2*c^2 - 27*a*b*c*d + 15*a^2*d^2)/(8*a*c^3*(b*c - a*d)^2*x) - d/(4*c*(b*c -
a*d)*x*(c + d*x^2)^2) - (d*(9*b*c - 5*a*d))/(8*c^2*(b*c - a*d)^2*x*(c + d*x^2))
- (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c - a*d)^3) + (d^(3/2)*(35*
b^2*c^2 - 42*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*(b*c
- a*d)^3)

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Rubi [A]  time = 0.827151, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)^3}+\frac{d^{3/2} \left (15 a^2 d^2-42 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} (b c-a d)^3}-\frac{15 a^2 d^2-27 a b c d+8 b^2 c^2}{8 a c^3 x (b c-a d)^2}-\frac{d (9 b c-5 a d)}{8 c^2 x \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(8*b^2*c^2 - 27*a*b*c*d + 15*a^2*d^2)/(8*a*c^3*(b*c - a*d)^2*x) - d/(4*c*(b*c -
a*d)*x*(c + d*x^2)^2) - (d*(9*b*c - 5*a*d))/(8*c^2*(b*c - a*d)^2*x*(c + d*x^2))
- (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c - a*d)^3) + (d^(3/2)*(35*
b^2*c^2 - 42*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*(b*c
- a*d)^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.677219, size = 172, normalized size = 0.82 \[ \frac{1}{8} \left (\frac{8 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (a d-b c)^3}+\frac{d^{3/2} \left (15 a^2 d^2-42 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^3}+\frac{d^2 x (11 b c-7 a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{2 d^2 x}{c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac{8}{a c^3 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-8/(a*c^3*x) + (2*d^2*x)/(c^2*(b*c - a*d)*(c + d*x^2)^2) + (d^2*(11*b*c - 7*a*d
)*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)) + (8*b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
a^(3/2)*(-(b*c) + a*d)^3) + (d^(3/2)*(35*b^2*c^2 - 42*a*b*c*d + 15*a^2*d^2)*ArcT
an[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^3))/8

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Maple [A]  time = 0.025, size = 335, normalized size = 1.6 \[ -{\frac{1}{a{c}^{3}x}}-{\frac{7\,{d}^{5}{x}^{3}{a}^{2}}{8\,{c}^{3} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,{d}^{4}{x}^{3}ab}{4\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{11\,{d}^{3}{x}^{3}{b}^{2}}{8\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,{d}^{4}x{a}^{2}}{8\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{11\,{d}^{3}xab}{4\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{13\,{d}^{2}x{b}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{15\,{a}^{2}{d}^{4}}{8\,{c}^{3} \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{21\,ab{d}^{3}}{4\,{c}^{2} \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{35\,{d}^{2}{b}^{2}}{8\,c \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{4}}{a \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/c^3/x-7/8*d^5/c^3/(a*d-b*c)^3/(d*x^2+c)^2*x^3*a^2+9/4*d^4/c^2/(a*d-b*c)^3/(
d*x^2+c)^2*x^3*a*b-11/8*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x^3*b^2-9/8*d^4/c^2/(a*d-b
*c)^3/(d*x^2+c)^2*x*a^2+11/4*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x*a*b-13/8*d^2/(a*d-b
*c)^3/(d*x^2+c)^2*x*b^2-15/8*d^4/c^3/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1
/2))*a^2+21/4*d^3/c^2/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b-35/8*d
^2/c/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2+1/a*b^4/(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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36682, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(16*b^3*c^5 - 48*a*b^2*c^4*d + 48*a^2*b*c^3*d^2 - 16*a^3*c^2*d^3 + 2*(8*b
^3*c^3*d^2 - 35*a*b^2*c^2*d^3 + 42*a^2*b*c*d^4 - 15*a^3*d^5)*x^4 + 2*(16*b^3*c^4
*d - 61*a*b^2*c^3*d^2 + 70*a^2*b*c^2*d^3 - 25*a^3*c*d^4)*x^2 + 8*(b^3*c^3*d^2*x^
5 + 2*b^3*c^4*d*x^3 + b^3*c^5*x)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(
b*x^2 + a)) + ((35*a*b^2*c^2*d^3 - 42*a^2*b*c*d^4 + 15*a^3*d^5)*x^5 + 2*(35*a*b^
2*c^3*d^2 - 42*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^3 + (35*a*b^2*c^4*d - 42*a^2*b*c^
3*d^2 + 15*a^3*c^2*d^3)*x)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2
+ c)))/((a*b^3*c^6*d^2 - 3*a^2*b^2*c^5*d^3 + 3*a^3*b*c^4*d^4 - a^4*c^3*d^5)*x^5
+ 2*(a*b^3*c^7*d - 3*a^2*b^2*c^6*d^2 + 3*a^3*b*c^5*d^3 - a^4*c^4*d^4)*x^3 + (a*b
^3*c^8 - 3*a^2*b^2*c^7*d + 3*a^3*b*c^6*d^2 - a^4*c^5*d^3)*x), -1/8*(8*b^3*c^5 -
24*a*b^2*c^4*d + 24*a^2*b*c^3*d^2 - 8*a^3*c^2*d^3 + (8*b^3*c^3*d^2 - 35*a*b^2*c^
2*d^3 + 42*a^2*b*c*d^4 - 15*a^3*d^5)*x^4 + (16*b^3*c^4*d - 61*a*b^2*c^3*d^2 + 70
*a^2*b*c^2*d^3 - 25*a^3*c*d^4)*x^2 - ((35*a*b^2*c^2*d^3 - 42*a^2*b*c*d^4 + 15*a^
3*d^5)*x^5 + 2*(35*a*b^2*c^3*d^2 - 42*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^3 + (35*a*
b^2*c^4*d - 42*a^2*b*c^3*d^2 + 15*a^3*c^2*d^3)*x)*sqrt(d/c)*arctan(d*x/(c*sqrt(d
/c))) + 4*(b^3*c^3*d^2*x^5 + 2*b^3*c^4*d*x^3 + b^3*c^5*x)*sqrt(-b/a)*log((b*x^2
+ 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a*b^3*c^6*d^2 - 3*a^2*b^2*c^5*d^3 + 3*a^
3*b*c^4*d^4 - a^4*c^3*d^5)*x^5 + 2*(a*b^3*c^7*d - 3*a^2*b^2*c^6*d^2 + 3*a^3*b*c^
5*d^3 - a^4*c^4*d^4)*x^3 + (a*b^3*c^8 - 3*a^2*b^2*c^7*d + 3*a^3*b*c^6*d^2 - a^4*
c^5*d^3)*x), -1/16*(16*b^3*c^5 - 48*a*b^2*c^4*d + 48*a^2*b*c^3*d^2 - 16*a^3*c^2*
d^3 + 2*(8*b^3*c^3*d^2 - 35*a*b^2*c^2*d^3 + 42*a^2*b*c*d^4 - 15*a^3*d^5)*x^4 + 2
*(16*b^3*c^4*d - 61*a*b^2*c^3*d^2 + 70*a^2*b*c^2*d^3 - 25*a^3*c*d^4)*x^2 + 16*(b
^3*c^3*d^2*x^5 + 2*b^3*c^4*d*x^3 + b^3*c^5*x)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))
) + ((35*a*b^2*c^2*d^3 - 42*a^2*b*c*d^4 + 15*a^3*d^5)*x^5 + 2*(35*a*b^2*c^3*d^2
- 42*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^3 + (35*a*b^2*c^4*d - 42*a^2*b*c^3*d^2 + 15
*a^3*c^2*d^3)*x)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a
*b^3*c^6*d^2 - 3*a^2*b^2*c^5*d^3 + 3*a^3*b*c^4*d^4 - a^4*c^3*d^5)*x^5 + 2*(a*b^3
*c^7*d - 3*a^2*b^2*c^6*d^2 + 3*a^3*b*c^5*d^3 - a^4*c^4*d^4)*x^3 + (a*b^3*c^8 - 3
*a^2*b^2*c^7*d + 3*a^3*b*c^6*d^2 - a^4*c^5*d^3)*x), -1/8*(8*b^3*c^5 - 24*a*b^2*c
^4*d + 24*a^2*b*c^3*d^2 - 8*a^3*c^2*d^3 + (8*b^3*c^3*d^2 - 35*a*b^2*c^2*d^3 + 42
*a^2*b*c*d^4 - 15*a^3*d^5)*x^4 + (16*b^3*c^4*d - 61*a*b^2*c^3*d^2 + 70*a^2*b*c^2
*d^3 - 25*a^3*c*d^4)*x^2 + 8*(b^3*c^3*d^2*x^5 + 2*b^3*c^4*d*x^3 + b^3*c^5*x)*sqr
t(b/a)*arctan(b*x/(a*sqrt(b/a))) - ((35*a*b^2*c^2*d^3 - 42*a^2*b*c*d^4 + 15*a^3*
d^5)*x^5 + 2*(35*a*b^2*c^3*d^2 - 42*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^3 + (35*a*b^
2*c^4*d - 42*a^2*b*c^3*d^2 + 15*a^3*c^2*d^3)*x)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c
))))/((a*b^3*c^6*d^2 - 3*a^2*b^2*c^5*d^3 + 3*a^3*b*c^4*d^4 - a^4*c^3*d^5)*x^5 +
2*(a*b^3*c^7*d - 3*a^2*b^2*c^6*d^2 + 3*a^3*b*c^5*d^3 - a^4*c^4*d^4)*x^3 + (a*b^3
*c^8 - 3*a^2*b^2*c^7*d + 3*a^3*b*c^6*d^2 - a^4*c^5*d^3)*x)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.253936, size = 319, normalized size = 1.51 \[ -\frac{b^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt{a b}} + \frac{{\left (35 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 15 \, a^{2} d^{4}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} \sqrt{c d}} + \frac{11 \, b c d^{3} x^{3} - 7 \, a d^{4} x^{3} + 13 \, b c^{2} d^{2} x - 9 \, a c d^{3} x}{8 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}{\left (d x^{2} + c\right )}^{2}} - \frac{1}{a c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^4*arctan(b*x/sqrt(a*b))/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d
^3)*sqrt(a*b)) + 1/8*(35*b^2*c^2*d^2 - 42*a*b*c*d^3 + 15*a^2*d^4)*arctan(d*x/sqr
t(c*d))/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*sqrt(c*d)) +
1/8*(11*b*c*d^3*x^3 - 7*a*d^4*x^3 + 13*b*c^2*d^2*x - 9*a*c*d^3*x)/((b^2*c^5 - 2*
a*b*c^4*d + a^2*c^3*d^2)*(d*x^2 + c)^2) - 1/(a*c^3*x)