3.121 \(\int \frac{c+d x^2+e x^4+f x^6}{x^8 (a+b x^2)} \, dx\)

Optimal. Leaf size=137 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{9/2}}-\frac{a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac{b c-a d}{5 a^2 x^5}-\frac{c}{7 a x^7} \]

[Out]

-c/(7*a*x^7) + (b*c - a*d)/(5*a^2*x^5) - (b^2*c - a*b*d + a^2*e)/(3*a^3*x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^
3*f)/(a^4*x) + (Sqrt[b]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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Rubi [A]  time = 0.130352, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1802, 205} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{9/2}}-\frac{a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac{b c-a d}{5 a^2 x^5}-\frac{c}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]

[Out]

-c/(7*a*x^7) + (b*c - a*d)/(5*a^2*x^5) - (b^2*c - a*b*d + a^2*e)/(3*a^3*x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^
3*f)/(a^4*x) + (Sqrt[b]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx &=\int \left (\frac{c}{a x^8}+\frac{-b c+a d}{a^2 x^6}+\frac{b^2 c-a b d+a^2 e}{a^3 x^4}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac{\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{a^4}\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac{\sqrt{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.115209, size = 139, normalized size = 1.01 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{9/2}}+\frac{a^2 (-e)+a b d-b^2 c}{3 a^3 x^3}+\frac{b c-a d}{5 a^2 x^5}-\frac{c}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]

[Out]

-c/(7*a*x^7) + (b*c - a*d)/(5*a^2*x^5) + (-(b^2*c) + a*b*d - a^2*e)/(3*a^3*x^3) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(a^4*x) - (Sqrt[b]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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Maple [A]  time = 0.009, size = 190, normalized size = 1.4 \begin{align*} -{\frac{c}{7\,a{x}^{7}}}-{\frac{d}{5\,a{x}^{5}}}+{\frac{bc}{5\,{x}^{5}{a}^{2}}}-{\frac{e}{3\,a{x}^{3}}}+{\frac{bd}{3\,{x}^{3}{a}^{2}}}-{\frac{{b}^{2}c}{3\,{a}^{3}{x}^{3}}}-{\frac{f}{ax}}+{\frac{be}{{a}^{2}x}}-{\frac{{b}^{2}d}{{a}^{3}x}}+{\frac{{b}^{3}c}{{a}^{4}x}}-{\frac{bf}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}e}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}d}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{4}c}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x)

[Out]

-1/7*c/a/x^7-1/5/a/x^5*d+1/5/a^2/x^5*b*c-1/3/a/x^3*e+1/3/a^2/x^3*b*d-1/3/a^3/x^3*b^2*c-1/a/x*f+1/a^2/x*b*e-1/a
^3/x*b^2*d+1/a^4/x*b^3*c-b/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f+b^2/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
*e-b^3/a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+b^4/a^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51032, size = 624, normalized size = 4.55 \begin{align*} \left [-\frac{105 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 210 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} + 70 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} + 30 \, a^{3} c - 42 \,{\left (a^{2} b c - a^{3} d\right )} x^{2}}{210 \, a^{4} x^{7}}, \frac{105 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 105 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \,{\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/210*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)
) - 210*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6 + 70*(a*b^2*c - a^2*b*d + a^3*e)*x^4 + 30*a^3*c - 42*(a^2*b*c
- a^3*d)*x^2)/(a^4*x^7), 1/105*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7*sqrt(b/a)*arctan(x*sqrt(b/a)) + 10
5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6 - 35*(a*b^2*c - a^2*b*d + a^3*e)*x^4 - 15*a^3*c + 21*(a^2*b*c - a^3*
d)*x^2)/(a^4*x^7)]

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Sympy [B]  time = 13.2633, size = 301, normalized size = 2.2 \begin{align*} \frac{\sqrt{- \frac{b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- \frac{a^{5} \sqrt{- \frac{b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a^{5} \sqrt{- \frac{b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} - \frac{15 a^{3} c + x^{6} \left (105 a^{3} f - 105 a^{2} b e + 105 a b^{2} d - 105 b^{3} c\right ) + x^{4} \left (35 a^{3} e - 35 a^{2} b d + 35 a b^{2} c\right ) + x^{2} \left (21 a^{3} d - 21 a^{2} b c\right )}{105 a^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a),x)

[Out]

sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**5*sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d -
b**3*c)/(a**3*b*f - a**2*b**2*e + a*b**3*d - b**4*c) + x)/2 - sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b*
*3*c)*log(a**5*sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b*f - a**2*b**2*e + a*b**3*d - b**4
*c) + x)/2 - (15*a**3*c + x**6*(105*a**3*f - 105*a**2*b*e + 105*a*b**2*d - 105*b**3*c) + x**4*(35*a**3*e - 35*
a**2*b*d + 35*a*b**2*c) + x**2*(21*a**3*d - 21*a**2*b*c))/(105*a**4*x**7)

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Giac [A]  time = 1.18216, size = 204, normalized size = 1.49 \begin{align*} \frac{{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{105 \, b^{3} c x^{6} - 105 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 105 \, a^{2} b x^{6} e - 35 \, a b^{2} c x^{4} + 35 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 21 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x, algorithm="giac")

[Out]

(b^4*c - a*b^3*d - a^3*b*f + a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(105*b^3*c*x^6 - 105*a*b
^2*d*x^6 - 105*a^3*f*x^6 + 105*a^2*b*x^6*e - 35*a*b^2*c*x^4 + 35*a^2*b*d*x^4 - 35*a^3*x^4*e + 21*a^2*b*c*x^2 -
21*a^3*d*x^2 - 15*a^3*c)/(a^4*x^7)