3.239 \(\int \frac{1}{x^6 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=134 \[ -\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5} \]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^
3*c^3*x) - (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)) + (d^(7/2
)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d))

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Rubi [A]  time = 0.65961, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^
3*c^3*x) - (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)) + (d^(7/2
)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d))

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Rubi in Sympy [A]  time = 136.503, size = 112, normalized size = 0.84 \[ - \frac{d^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{7}{2}} \left (a d - b c\right )} - \frac{1}{5 a c x^{5}} + \frac{a d + b c}{3 a^{2} c^{2} x^{3}} + \frac{a b c d - \left (a d + b c\right )^{2}}{a^{3} c^{3} x} + \frac{b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{7}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**2+a)/(d*x**2+c),x)

[Out]

-d**(7/2)*atan(sqrt(d)*x/sqrt(c))/(c**(7/2)*(a*d - b*c)) - 1/(5*a*c*x**5) + (a*d
+ b*c)/(3*a**2*c**2*x**3) + (a*b*c*d - (a*d + b*c)**2)/(a**3*c**3*x) + b**(7/2)
*atan(sqrt(b)*x/sqrt(a))/(a**(7/2)*(a*d - b*c))

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Mathematica [A]  time = 0.241922, size = 135, normalized size = 1.01 \[ \frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (a d-b c)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{-a^2 d^2-a b c d-b^2 c^2}{a^3 c^3 x}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) + (-(b^2*c^2) - a*b*c*d - a^2*d^2)/
(a^3*c^3*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(-(b*c) + a*d)) + (
d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d))

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Maple [A]  time = 0.016, size = 141, normalized size = 1.1 \[ -{\frac{1}{5\,ac{x}^{5}}}+{\frac{d}{3\,{x}^{3}a{c}^{2}}}+{\frac{b}{3\,{a}^{2}c{x}^{3}}}-{\frac{{d}^{2}}{a{c}^{3}x}}-{\frac{bd}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}cx}}-{\frac{{d}^{4}}{{c}^{3} \left ( ad-bc \right ) }\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{4}}{{a}^{3} \left ( ad-bc \right ) }\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^2+a)/(d*x^2+c),x)

[Out]

-1/5/a/c/x^5+1/3/x^3/a/c^2*d+1/3/x^3/a^2/c*b-1/a/c^3/x*d^2-1/a^2/c^2/x*b*d-1/a^3
/c/x*b^2-1/c^3*d^4/(a*d-b*c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))+1/a^3*b^4/(a*d-
b*c)/(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)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313368, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \, b^{3} c^{3} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 15 \, a^{3} d^{3} x^{5} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 6 \, a^{2} b c^{3} - 6 \, a^{3} c^{2} d + 30 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} - 10 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{30 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}, \frac{30 \, a^{3} d^{3} x^{5} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) - 15 \, b^{3} c^{3} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d - 30 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} + 10 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{30 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}, -\frac{30 \, b^{3} c^{3} x^{5} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 15 \, a^{3} d^{3} x^{5} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 6 \, a^{2} b c^{3} - 6 \, a^{3} c^{2} d + 30 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} - 10 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{30 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}, -\frac{15 \, b^{3} c^{3} x^{5} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - 15 \, a^{3} d^{3} x^{5} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) + 3 \, a^{2} b c^{3} - 3 \, a^{3} c^{2} d + 15 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} - 5 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{15 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^6),x, algorithm="fricas")

[Out]

[-1/30*(15*b^3*c^3*x^5*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)
) + 15*a^3*d^3*x^5*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) +
6*a^2*b*c^3 - 6*a^3*c^2*d + 30*(b^3*c^3 - a^3*d^3)*x^4 - 10*(a*b^2*c^3 - a^3*c*d
^2)*x^2)/((a^3*b*c^4 - a^4*c^3*d)*x^5), 1/30*(30*a^3*d^3*x^5*sqrt(d/c)*arctan(d*
x/(c*sqrt(d/c))) - 15*b^3*c^3*x^5*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/
(b*x^2 + a)) - 6*a^2*b*c^3 + 6*a^3*c^2*d - 30*(b^3*c^3 - a^3*d^3)*x^4 + 10*(a*b^
2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*d)*x^5), -1/30*(30*b^3*c^3*x^5*sqr
t(b/a)*arctan(b*x/(a*sqrt(b/a))) + 15*a^3*d^3*x^5*sqrt(-d/c)*log((d*x^2 - 2*c*x*
sqrt(-d/c) - c)/(d*x^2 + c)) + 6*a^2*b*c^3 - 6*a^3*c^2*d + 30*(b^3*c^3 - a^3*d^3
)*x^4 - 10*(a*b^2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*d)*x^5), -1/15*(15
*b^3*c^3*x^5*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 15*a^3*d^3*x^5*sqrt(d/c)*arct
an(d*x/(c*sqrt(d/c))) + 3*a^2*b*c^3 - 3*a^3*c^2*d + 15*(b^3*c^3 - a^3*d^3)*x^4 -
5*(a*b^2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*d)*x^5)]

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Sympy [A]  time = 63.8952, size = 1504, normalized size = 11.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**2+a)/(d*x**2+c),x)

[Out]

-sqrt(-b**7/a**7)*log(x + (-a**13*c**7*d**6*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 +
2*a**12*b*c**8*d**5*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 - a**11*b**2*c**9*d**4*(
-b**7/a**7)**(3/2)/(a*d - b*c)**3 - a**11*d**11*sqrt(-b**7/a**7)/(a*d - b*c) - a
**9*b**4*c**11*d**2*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 + 2*a**8*b**5*c**12*d*(-b
**7/a**7)**(3/2)/(a*d - b*c)**3 - a**7*b**6*c**13*(-b**7/a**7)**(3/2)/(a*d - b*c
)**3 - b**11*c**11*sqrt(-b**7/a**7)/(a*d - b*c))/(a**6*b**4*d**10 + a**5*b**5*c*
d**9 + a**4*b**6*c**2*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d**6 + a*b**9*
c**5*d**5 + b**10*c**6*d**4))/(2*(a*d - b*c)) + sqrt(-b**7/a**7)*log(x + (a**13*
c**7*d**6*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 - 2*a**12*b*c**8*d**5*(-b**7/a**7)*
*(3/2)/(a*d - b*c)**3 + a**11*b**2*c**9*d**4*(-b**7/a**7)**(3/2)/(a*d - b*c)**3
+ a**11*d**11*sqrt(-b**7/a**7)/(a*d - b*c) + a**9*b**4*c**11*d**2*(-b**7/a**7)**
(3/2)/(a*d - b*c)**3 - 2*a**8*b**5*c**12*d*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 +
a**7*b**6*c**13*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 + b**11*c**11*sqrt(-b**7/a**7
)/(a*d - b*c))/(a**6*b**4*d**10 + a**5*b**5*c*d**9 + a**4*b**6*c**2*d**8 + a**3*
b**7*c**3*d**7 + a**2*b**8*c**4*d**6 + a*b**9*c**5*d**5 + b**10*c**6*d**4))/(2*(
a*d - b*c)) - sqrt(-d**7/c**7)*log(x + (-a**13*c**7*d**6*(-d**7/c**7)**(3/2)/(a*
d - b*c)**3 + 2*a**12*b*c**8*d**5*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 - a**11*b**
2*c**9*d**4*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 - a**11*d**11*sqrt(-d**7/c**7)/(a
*d - b*c) - a**9*b**4*c**11*d**2*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + 2*a**8*b**
5*c**12*d*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 - a**7*b**6*c**13*(-d**7/c**7)**(3/
2)/(a*d - b*c)**3 - b**11*c**11*sqrt(-d**7/c**7)/(a*d - b*c))/(a**6*b**4*d**10 +
a**5*b**5*c*d**9 + a**4*b**6*c**2*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d
**6 + a*b**9*c**5*d**5 + b**10*c**6*d**4))/(2*(a*d - b*c)) + sqrt(-d**7/c**7)*lo
g(x + (a**13*c**7*d**6*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 - 2*a**12*b*c**8*d**5*
(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + a**11*b**2*c**9*d**4*(-d**7/c**7)**(3/2)/(a
*d - b*c)**3 + a**11*d**11*sqrt(-d**7/c**7)/(a*d - b*c) + a**9*b**4*c**11*d**2*(
-d**7/c**7)**(3/2)/(a*d - b*c)**3 - 2*a**8*b**5*c**12*d*(-d**7/c**7)**(3/2)/(a*d
- b*c)**3 + a**7*b**6*c**13*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + b**11*c**11*sq
rt(-d**7/c**7)/(a*d - b*c))/(a**6*b**4*d**10 + a**5*b**5*c*d**9 + a**4*b**6*c**2
*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d**6 + a*b**9*c**5*d**5 + b**10*c**
6*d**4))/(2*(a*d - b*c)) - (3*a**2*c**2 + x**4*(15*a**2*d**2 + 15*a*b*c*d + 15*b
**2*c**2) + x**2*(-5*a**2*c*d - 5*a*b*c**2))/(15*a**3*c**3*x**5)

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GIAC/XCAS [A]  time = 0.313644, size = 890, normalized size = 6.64 \[ \frac{{\left (\sqrt{c d} a^{3} b^{4} c^{6}{\left | d \right |} + \sqrt{c d} a^{6} b c^{3} d^{3}{\left | d \right |} - \sqrt{c d} b^{3} c^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | d \right |} - \sqrt{c d} a b^{2} c d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | d \right |} - \sqrt{c d} a^{2} b d^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{3} b c^{4} + a^{4} c^{3} d + \sqrt{-4 \, a^{7} b c^{7} d +{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )}^{2}}}{a^{3} b c^{3} d}}}\right )}{a^{3} b c^{4} d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} + a^{4} c^{3} d^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} +{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )}^{2} d} - \frac{{\left (\sqrt{a b} a^{3} b^{3} c^{6} d{\left | b \right |} + \sqrt{a b} a^{6} c^{3} d^{4}{\left | b \right |} + \sqrt{a b} b^{2} c^{2} d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | b \right |} + \sqrt{a b} a b c d^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | b \right |} + \sqrt{a b} a^{2} d^{3}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{3} b c^{4} + a^{4} c^{3} d - \sqrt{-4 \, a^{7} b c^{7} d +{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )}^{2}}}{a^{3} b c^{3} d}}}\right )}{a^{3} b^{2} c^{4}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} + a^{4} b c^{3} d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} -{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )}^{2} b} - \frac{15 \, b^{2} c^{2} x^{4} + 15 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} - 5 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, a^{3} c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^6),x, algorithm="giac")

[Out]

(sqrt(c*d)*a^3*b^4*c^6*abs(d) + sqrt(c*d)*a^6*b*c^3*d^3*abs(d) - sqrt(c*d)*b^3*c
^2*abs(a^3*b*c^4 - a^4*c^3*d)*abs(d) - sqrt(c*d)*a*b^2*c*d*abs(a^3*b*c^4 - a^4*c
^3*d)*abs(d) - sqrt(c*d)*a^2*b*d^2*abs(a^3*b*c^4 - a^4*c^3*d)*abs(d))*arctan(2*s
qrt(1/2)*x/sqrt((a^3*b*c^4 + a^4*c^3*d + sqrt(-4*a^7*b*c^7*d + (a^3*b*c^4 + a^4*
c^3*d)^2))/(a^3*b*c^3*d)))/(a^3*b*c^4*d*abs(a^3*b*c^4 - a^4*c^3*d) + a^4*c^3*d^2
*abs(a^3*b*c^4 - a^4*c^3*d) + (a^3*b*c^4 - a^4*c^3*d)^2*d) - (sqrt(a*b)*a^3*b^3*
c^6*d*abs(b) + sqrt(a*b)*a^6*c^3*d^4*abs(b) + sqrt(a*b)*b^2*c^2*d*abs(a^3*b*c^4
- a^4*c^3*d)*abs(b) + sqrt(a*b)*a*b*c*d^2*abs(a^3*b*c^4 - a^4*c^3*d)*abs(b) + sq
rt(a*b)*a^2*d^3*abs(a^3*b*c^4 - a^4*c^3*d)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((a^
3*b*c^4 + a^4*c^3*d - sqrt(-4*a^7*b*c^7*d + (a^3*b*c^4 + a^4*c^3*d)^2))/(a^3*b*c
^3*d)))/(a^3*b^2*c^4*abs(a^3*b*c^4 - a^4*c^3*d) + a^4*b*c^3*d*abs(a^3*b*c^4 - a^
4*c^3*d) - (a^3*b*c^4 - a^4*c^3*d)^2*b) - 1/15*(15*b^2*c^2*x^4 + 15*a*b*c*d*x^4
+ 15*a^2*d^2*x^4 - 5*a*b*c^2*x^2 - 5*a^2*c*d*x^2 + 3*a^2*c^2)/(a^3*c^3*x^5)