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

Optimal. Leaf size=189 \[ \frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^2 b e-3 a^3 f-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7} \]

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(5*a^3*x^5) - (3*b^2*c - 2*a*b*d + a^2*e)/(3*a^4*x^3) + (4*b^3*c - 3*a*b^2*d +
2*a^2*b*e - a^3*f)/(a^5*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(2*a^5*(a + b*x^2)) + (Sqrt[b]*(9*b^3*c
- 7*a*b^2*d + 5*a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))

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Rubi [A]  time = 0.293351, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1805, 1802, 205} \[ \frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^2 b e-3 a^3 f-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(5*a^3*x^5) - (3*b^2*c - 2*a*b*d + a^2*e)/(3*a^4*x^3) + (4*b^3*c - 3*a*b^2*d +
2*a^2*b*e - a^3*f)/(a^5*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(2*a^5*(a + b*x^2)) + (Sqrt[b]*(9*b^3*c
- 7*a*b^2*d + 5*a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

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 )^2} \, dx &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac{\int \frac{-2 c+2 \left (\frac{b c}{a}-d\right ) x^2-\frac{2 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac{2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )} \, dx}{2 a}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 c}{a x^8}-\frac{2 (-2 b c+a d)}{a^2 x^6}-\frac{2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^4}-\frac{2 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^2}+\frac{b \left (-9 b^3 c+7 a b^2 d-5 a^2 b e+3 a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac{\left (b \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^5}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac{\sqrt{b} \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.101657, size = 190, normalized size = 1.01 \[ -\frac{b x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-5 a^2 b e+3 a^3 f+7 a b^2 d-9 b^3 c\right )}{2 a^{11/2}}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(5*a^3*x^5) + (-3*b^2*c + 2*a*b*d - a^2*e)/(3*a^4*x^3) + (4*b^3*c - 3*a*b^2*d +
2*a^2*b*e - a^3*f)/(a^5*x) - (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^5*(a + b*x^2)) - (Sqrt[b]*(-9*
b^3*c + 7*a*b^2*d - 5*a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))

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Maple [A]  time = 0.016, size = 268, normalized size = 1.4 \begin{align*} -{\frac{c}{7\,{a}^{2}{x}^{7}}}-{\frac{d}{5\,{x}^{5}{a}^{2}}}+{\frac{2\,bc}{5\,{a}^{3}{x}^{5}}}-{\frac{e}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bd}{3\,{a}^{3}{x}^{3}}}-{\frac{{b}^{2}c}{{a}^{4}{x}^{3}}}-{\frac{f}{{a}^{2}x}}+2\,{\frac{be}{{a}^{3}x}}-3\,{\frac{{b}^{2}d}{{a}^{4}x}}+4\,{\frac{{b}^{3}c}{{a}^{5}x}}-{\frac{bxf}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}xe}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xd}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xc}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bf}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}e}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{3}d}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{b}^{4}c}{2\,{a}^{5}}\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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37974, size = 1057, normalized size = 5.59 \begin{align*} \left [\frac{210 \,{\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 140 \,{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 60 \, a^{4} c - 28 \,{\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 12 \,{\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} - 105 \,{\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} +{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{420 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac{105 \,{\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 70 \,{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 30 \, a^{4} c - 14 \,{\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 6 \,{\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} + 105 \,{\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} +{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{210 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\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)^2,x, algorithm="fricas")

[Out]

[1/420*(210*(9*b^4*c - 7*a*b^3*d + 5*a^2*b^2*e - 3*a^3*b*f)*x^8 + 140*(9*a*b^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3
*a^4*f)*x^6 - 60*a^4*c - 28*(9*a^2*b^2*c - 7*a^3*b*d + 5*a^4*e)*x^4 + 12*(9*a^3*b*c - 7*a^4*d)*x^2 - 105*((9*b
^4*c - 7*a*b^3*d + 5*a^2*b^2*e - 3*a^3*b*f)*x^9 + (9*a*b^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)*x^7)*sqrt(-b
/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b*x^9 + a^6*x^7), 1/210*(105*(9*b^4*c - 7*a*b^3*d +
5*a^2*b^2*e - 3*a^3*b*f)*x^8 + 70*(9*a*b^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)*x^6 - 30*a^4*c - 14*(9*a^2*b
^2*c - 7*a^3*b*d + 5*a^4*e)*x^4 + 6*(9*a^3*b*c - 7*a^4*d)*x^2 + 105*((9*b^4*c - 7*a*b^3*d + 5*a^2*b^2*e - 3*a^
3*b*f)*x^9 + (9*a*b^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)*x^7)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^5*b*x^9 +
a^6*x^7)]

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Sympy [B]  time = 76.7361, size = 394, normalized size = 2.08 \begin{align*} \frac{\sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log{\left (- \frac{a^{6} \sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log{\left (\frac{a^{6} \sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac{30 a^{4} c + x^{8} \left (315 a^{3} b f - 525 a^{2} b^{2} e + 735 a b^{3} d - 945 b^{4} c\right ) + x^{6} \left (210 a^{4} f - 350 a^{3} b e + 490 a^{2} b^{2} d - 630 a b^{3} c\right ) + x^{4} \left (70 a^{4} e - 98 a^{3} b d + 126 a^{2} b^{2} c\right ) + x^{2} \left (42 a^{4} d - 54 a^{3} b c\right )}{210 a^{6} x^{7} + 210 a^{5} b x^{9}} \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)**2,x)

[Out]

sqrt(-b/a**11)*(3*a**3*f - 5*a**2*b*e + 7*a*b**2*d - 9*b**3*c)*log(-a**6*sqrt(-b/a**11)*(3*a**3*f - 5*a**2*b*e
+ 7*a*b**2*d - 9*b**3*c)/(3*a**3*b*f - 5*a**2*b**2*e + 7*a*b**3*d - 9*b**4*c) + x)/4 - sqrt(-b/a**11)*(3*a**3
*f - 5*a**2*b*e + 7*a*b**2*d - 9*b**3*c)*log(a**6*sqrt(-b/a**11)*(3*a**3*f - 5*a**2*b*e + 7*a*b**2*d - 9*b**3*
c)/(3*a**3*b*f - 5*a**2*b**2*e + 7*a*b**3*d - 9*b**4*c) + x)/4 - (30*a**4*c + x**8*(315*a**3*b*f - 525*a**2*b*
*2*e + 735*a*b**3*d - 945*b**4*c) + x**6*(210*a**4*f - 350*a**3*b*e + 490*a**2*b**2*d - 630*a*b**3*c) + x**4*(
70*a**4*e - 98*a**3*b*d + 126*a**2*b**2*c) + x**2*(42*a**4*d - 54*a**3*b*c))/(210*a**6*x**7 + 210*a**5*b*x**9)

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Giac [A]  time = 1.19663, size = 271, normalized size = 1.43 \begin{align*} \frac{{\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{5}} + \frac{b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{2 \,{\left (b x^{2} + a\right )} a^{5}} + \frac{420 \, b^{3} c x^{6} - 315 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 210 \, a^{2} b x^{6} e - 105 \, a b^{2} c x^{4} + 70 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 42 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{5} 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)^2,x, algorithm="giac")

[Out]

1/2*(9*b^4*c - 7*a*b^3*d - 3*a^3*b*f + 5*a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/2*(b^4*c*x - a*b
^3*d*x - a^3*b*f*x + a^2*b^2*x*e)/((b*x^2 + a)*a^5) + 1/105*(420*b^3*c*x^6 - 315*a*b^2*d*x^6 - 105*a^3*f*x^6 +
210*a^2*b*x^6*e - 105*a*b^2*c*x^4 + 70*a^2*b*d*x^4 - 35*a^3*x^4*e + 42*a^2*b*c*x^2 - 21*a^3*d*x^2 - 15*a^3*c)
/(a^5*x^7)