3.139 \(\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=168 \[ \frac{3 b c-a d}{a^4 x}-\frac{c}{3 a^3 x^3}+\frac{x \left (\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac{x \left (a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right )}{8 a^4 b \left (a+b x^2\right )} \]

[Out]

-c/(3*a^3*x^3) + (3*b*c - a*d)/(a^4*x) + (((b^2*c)/a^2 - (b*d)/a + e - (a*f)/b)*
x)/(4*a*(a + b*x^2)^2) + ((11*b^3*c - 7*a*b^2*d + 3*a^2*b*e + a^3*f)*x)/(8*a^4*b
*(a + b*x^2)) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/
Sqrt[a]])/(8*a^(9/2)*b^(3/2))

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Rubi [A]  time = 0.556753, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{3 b c-a d}{a^4 x}-\frac{c}{3 a^3 x^3}+\frac{x \left (\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac{x \left (a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right )}{8 a^4 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)^3),x]

[Out]

-c/(3*a^3*x^3) + (3*b*c - a*d)/(a^4*x) + (((b^2*c)/a^2 - (b*d)/a + e - (a*f)/b)*
x)/(4*a*(a + b*x^2)^2) + ((11*b^3*c - 7*a*b^2*d + 3*a^2*b*e + a^3*f)*x)/(8*a^4*b
*(a + b*x^2)) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/
Sqrt[a]])/(8*a^(9/2)*b^(3/2))

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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((f*x**6+e*x**4+d*x**2+c)/x**4/(b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.293769, size = 169, normalized size = 1.01 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac{-3 a^4 f x^4+a^3 b \left (3 x^2 \left (-8 d+5 e x^2+f x^4\right )-8 c\right )+a^2 b^2 x^2 \left (56 c-75 d x^2+9 e x^4\right )+5 a b^3 x^4 \left (35 c-9 d x^2\right )+105 b^4 c x^6}{24 a^4 b x^3 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)^3),x]

[Out]

(-3*a^4*f*x^4 + 105*b^4*c*x^6 + 5*a*b^3*x^4*(35*c - 9*d*x^2) + a^2*b^2*x^2*(56*c
- 75*d*x^2 + 9*e*x^4) + a^3*b*(-8*c + 3*x^2*(-8*d + 5*e*x^2 + f*x^4)))/(24*a^4*
b*x^3*(a + b*x^2)^2) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))

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Maple [A]  time = 0.022, size = 264, normalized size = 1.6 \[ -{\frac{c}{3\,{a}^{3}{x}^{3}}}-{\frac{d}{{a}^{3}x}}+3\,{\frac{bc}{{a}^{4}x}}+{\frac{{x}^{3}f}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,{x}^{3}be}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{x}^{3}{b}^{2}d}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{x}^{3}{b}^{3}c}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{fx}{8\, \left ( b{x}^{2}+a \right ) ^{2}b}}+{\frac{5\,ex}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bxd}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,x{b}^{2}c}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{f}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,e}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,bd}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{b}^{2}c}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^4/(b*x^2+a)^3,x)

[Out]

-1/3*c/a^3/x^3-1/a^3/x*d+3/a^4/x*b*c+1/8/a/(b*x^2+a)^2*x^3*f+3/8/a^2/(b*x^2+a)^2
*x^3*b*e-7/8/a^3/(b*x^2+a)^2*x^3*b^2*d+11/8/a^4/(b*x^2+a)^2*x^3*b^3*c-1/8/(b*x^2
+a)^2*x/b*f+5/8/a/(b*x^2+a)^2*x*e-9/8/a^2/(b*x^2+a)^2*x*b*d+13/8/a^3/(b*x^2+a)^2
*x*b^2*c+1/8/a/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*f+3/8/a^2/(a*b)^(1/2)*arcta
n(x*b/(a*b)^(1/2))*e-15/8/a^3*b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+35/8/a^4*b
^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c

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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((f*x^6 + e*x^4 + d*x^2 + c)/((b*x^2 + a)^3*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240968, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} +{\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \,{\left (35 \, b^{4} c - 15 \, a b^{3} d + 3 \, a^{2} b^{2} e + a^{3} b f\right )} x^{6} - 8 \, a^{3} b c +{\left (175 \, a b^{3} c - 75 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{4} + 8 \,{\left (7 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{2}\right )} \sqrt{-a b}}{48 \,{\left (a^{4} b^{3} x^{7} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} +{\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \,{\left (35 \, b^{4} c - 15 \, a b^{3} d + 3 \, a^{2} b^{2} e + a^{3} b f\right )} x^{6} - 8 \, a^{3} b c +{\left (175 \, a b^{3} c - 75 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{4} + 8 \,{\left (7 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{2}\right )} \sqrt{a b}}{24 \,{\left (a^{4} b^{3} x^{7} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)/((b*x^2 + a)^3*x^4),x, algorithm="fricas")

[Out]

[1/48*(3*((35*b^5*c - 15*a*b^4*d + 3*a^2*b^3*e + a^3*b^2*f)*x^7 + 2*(35*a*b^4*c
- 15*a^2*b^3*d + 3*a^3*b^2*e + a^4*b*f)*x^5 + (35*a^2*b^3*c - 15*a^3*b^2*d + 3*a
^4*b*e + a^5*f)*x^3)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(3*
(35*b^4*c - 15*a*b^3*d + 3*a^2*b^2*e + a^3*b*f)*x^6 - 8*a^3*b*c + (175*a*b^3*c -
75*a^2*b^2*d + 15*a^3*b*e - 3*a^4*f)*x^4 + 8*(7*a^2*b^2*c - 3*a^3*b*d)*x^2)*sqr
t(-a*b))/((a^4*b^3*x^7 + 2*a^5*b^2*x^5 + a^6*b*x^3)*sqrt(-a*b)), 1/24*(3*((35*b^
5*c - 15*a*b^4*d + 3*a^2*b^3*e + a^3*b^2*f)*x^7 + 2*(35*a*b^4*c - 15*a^2*b^3*d +
3*a^3*b^2*e + a^4*b*f)*x^5 + (35*a^2*b^3*c - 15*a^3*b^2*d + 3*a^4*b*e + a^5*f)*
x^3)*arctan(sqrt(a*b)*x/a) + (3*(35*b^4*c - 15*a*b^3*d + 3*a^2*b^2*e + a^3*b*f)*
x^6 - 8*a^3*b*c + (175*a*b^3*c - 75*a^2*b^2*d + 15*a^3*b*e - 3*a^4*f)*x^4 + 8*(7
*a^2*b^2*c - 3*a^3*b*d)*x^2)*sqrt(a*b))/((a^4*b^3*x^7 + 2*a^5*b^2*x^5 + a^6*b*x^
3)*sqrt(a*b))]

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Sympy [A]  time = 99.1656, size = 270, normalized size = 1.61 \[ - \frac{\sqrt{- \frac{1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log{\left (- a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log{\left (a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{16} + \frac{- 8 a^{3} b c + x^{6} \left (3 a^{3} b f + 9 a^{2} b^{2} e - 45 a b^{3} d + 105 b^{4} c\right ) + x^{4} \left (- 3 a^{4} f + 15 a^{3} b e - 75 a^{2} b^{2} d + 175 a b^{3} c\right ) + x^{2} \left (- 24 a^{3} b d + 56 a^{2} b^{2} c\right )}{24 a^{6} b x^{3} + 48 a^{5} b^{2} x^{5} + 24 a^{4} b^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**4/(b*x**2+a)**3,x)

[Out]

-sqrt(-1/(a**9*b**3))*(a**3*f + 3*a**2*b*e - 15*a*b**2*d + 35*b**3*c)*log(-a**5*
b*sqrt(-1/(a**9*b**3)) + x)/16 + sqrt(-1/(a**9*b**3))*(a**3*f + 3*a**2*b*e - 15*
a*b**2*d + 35*b**3*c)*log(a**5*b*sqrt(-1/(a**9*b**3)) + x)/16 + (-8*a**3*b*c + x
**6*(3*a**3*b*f + 9*a**2*b**2*e - 45*a*b**3*d + 105*b**4*c) + x**4*(-3*a**4*f +
15*a**3*b*e - 75*a**2*b**2*d + 175*a*b**3*c) + x**2*(-24*a**3*b*d + 56*a**2*b**2
*c))/(24*a**6*b*x**3 + 48*a**5*b**2*x**5 + 24*a**4*b**3*x**7)

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GIAC/XCAS [A]  time = 0.220755, size = 230, normalized size = 1.37 \[ \frac{{\left (35 \, b^{3} c - 15 \, a b^{2} d + a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{4} b} + \frac{11 \, b^{4} c x^{3} - 7 \, a b^{3} d x^{3} + a^{3} b f x^{3} + 3 \, a^{2} b^{2} x^{3} e + 13 \, a b^{3} c x - 9 \, a^{2} b^{2} d x - a^{4} f x + 5 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{4} b} + \frac{9 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)/((b*x^2 + a)^3*x^4),x, algorithm="giac")

[Out]

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