3.122 \(\int \frac{c+d x^2+e x^4+f x^6}{x^{10} (a+b x^2)} \, dx\)

Optimal. Leaf size=175 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^5 x}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{11/2}}-\frac{a^2 e-a b d+b^2 c}{5 a^3 x^5}+\frac{b c-a d}{7 a^2 x^7}-\frac{c}{9 a x^9} \]

[Out]

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

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Rubi [A]  time = 0.145924, antiderivative size = 175, 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} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^5 x}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{11/2}}-\frac{a^2 e-a b d+b^2 c}{5 a^3 x^5}+\frac{b c-a d}{7 a^2 x^7}-\frac{c}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)),x]

[Out]

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

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^{10} \left (a+b x^2\right )} \, dx &=\int \left (\frac{c}{a x^{10}}+\frac{-b c+a d}{a^2 x^8}+\frac{b^2 c-a b d+a^2 e}{a^3 x^6}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^4}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^2}+\frac{b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c}{9 a x^9}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac{\left (b^2 \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^5}\\ &=-\frac{c}{9 a x^9}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac{b^{3/2} \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^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.132944, size = 174, normalized size = 0.99 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}+\frac{b \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^5 x}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{11/2}}+\frac{a^2 (-e)+a b d-b^2 c}{5 a^3 x^5}+\frac{b c-a d}{7 a^2 x^7}-\frac{c}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 238, normalized size = 1.4 \begin{align*} -{\frac{c}{9\,a{x}^{9}}}-{\frac{d}{7\,a{x}^{7}}}+{\frac{bc}{7\,{a}^{2}{x}^{7}}}-{\frac{e}{5\,a{x}^{5}}}+{\frac{bd}{5\,{x}^{5}{a}^{2}}}-{\frac{{b}^{2}c}{5\,{a}^{3}{x}^{5}}}-{\frac{f}{3\,a{x}^{3}}}+{\frac{be}{3\,{x}^{3}{a}^{2}}}-{\frac{{b}^{2}d}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}c}{3\,{a}^{4}{x}^{3}}}+{\frac{bf}{{a}^{2}x}}-{\frac{{b}^{2}e}{{a}^{3}x}}+{\frac{{b}^{3}d}{{a}^{4}x}}-{\frac{{b}^{4}c}{{a}^{5}x}}+{\frac{{b}^{2}f}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}e}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{4}d}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{5}c}{{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^10/(b*x^2+a),x)

[Out]

-1/9*c/a/x^9-1/7/a/x^7*d+1/7/a^2/x^7*b*c-1/5/a/x^5*e+1/5/a^2/x^5*b*d-1/5/a^3/x^5*b^2*c-1/3/a/x^3*f+1/3/a^2/x^3
*b*e-1/3/a^3/x^3*b^2*d+1/3/a^4/x^3*b^3*c+1/a^2*b/x*f-1/a^3*b^2/x*e+1/a^4*b^3/x*d-1/a^5*b^4/x*c+b^2/a^2/(a*b)^(
1/2)*arctan(b*x/(a*b)^(1/2))*f-b^3/a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e+b^4/a^4/(a*b)^(1/2)*arctan(b*x/(a
*b)^(1/2))*d-b^5/a^5/(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^10/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50432, size = 788, normalized size = 4.5 \begin{align*} \left [-\frac{315 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{9} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 630 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 210 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 70 \, a^{4} c + 126 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 90 \,{\left (a^{3} b c - a^{4} d\right )} x^{2}}{630 \, a^{5} x^{9}}, -\frac{315 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{9} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 315 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 105 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 35 \, a^{4} c + 63 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 45 \,{\left (a^{3} b c - a^{4} d\right )} x^{2}}{315 \, a^{5} x^{9}}\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^10/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/630*(315*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^9*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2
+ a)) + 630*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^8 - 210*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^6 + 70
*a^4*c + 126*(a^2*b^2*c - a^3*b*d + a^4*e)*x^4 - 90*(a^3*b*c - a^4*d)*x^2)/(a^5*x^9), -1/315*(315*(b^4*c - a*b
^3*d + a^2*b^2*e - a^3*b*f)*x^9*sqrt(b/a)*arctan(x*sqrt(b/a)) + 315*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^
8 - 105*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^6 + 35*a^4*c + 63*(a^2*b^2*c - a^3*b*d + a^4*e)*x^4 - 45*(a^
3*b*c - a^4*d)*x^2)/(a^5*x^9)]

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Sympy [B]  time = 27.4595, size = 354, normalized size = 2.02 \begin{align*} - \frac{\sqrt{- \frac{b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- \frac{a^{6} \sqrt{- \frac{b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a^{6} \sqrt{- \frac{b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac{- 35 a^{4} c + x^{8} \left (315 a^{3} b f - 315 a^{2} b^{2} e + 315 a b^{3} d - 315 b^{4} c\right ) + x^{6} \left (- 105 a^{4} f + 105 a^{3} b e - 105 a^{2} b^{2} d + 105 a b^{3} c\right ) + x^{4} \left (- 63 a^{4} e + 63 a^{3} b d - 63 a^{2} b^{2} c\right ) + x^{2} \left (- 45 a^{4} d + 45 a^{3} b c\right )}{315 a^{5} 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**10/(b*x**2+a),x)

[Out]

-sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**6*sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*
b**2*d - b**3*c)/(a**3*b**2*f - a**2*b**3*e + a*b**4*d - b**5*c) + x)/2 + sqrt(-b**3/a**11)*(a**3*f - a**2*b*e
+ a*b**2*d - b**3*c)*log(a**6*sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b**2*f - a**2*b
**3*e + a*b**4*d - b**5*c) + x)/2 + (-35*a**4*c + x**8*(315*a**3*b*f - 315*a**2*b**2*e + 315*a*b**3*d - 315*b*
*4*c) + x**6*(-105*a**4*f + 105*a**3*b*e - 105*a**2*b**2*d + 105*a*b**3*c) + x**4*(-63*a**4*e + 63*a**3*b*d -
63*a**2*b**2*c) + x**2*(-45*a**4*d + 45*a**3*b*c))/(315*a**5*x**9)

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Giac [A]  time = 1.18385, size = 271, normalized size = 1.55 \begin{align*} -\frac{{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} - \frac{315 \, b^{4} c x^{8} - 315 \, a b^{3} d x^{8} - 315 \, a^{3} b f x^{8} + 315 \, a^{2} b^{2} x^{8} e - 105 \, a b^{3} c x^{6} + 105 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 105 \, a^{3} b x^{6} e + 63 \, a^{2} b^{2} c x^{4} - 63 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 45 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{5} 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^10/(b*x^2+a),x, algorithm="giac")

[Out]

-(b^5*c - a*b^4*d - a^3*b^2*f + a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/315*(315*b^4*c*x^8 - 315*
a*b^3*d*x^8 - 315*a^3*b*f*x^8 + 315*a^2*b^2*x^8*e - 105*a*b^3*c*x^6 + 105*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 105*
a^3*b*x^6*e + 63*a^2*b^2*c*x^4 - 63*a^3*b*d*x^4 + 63*a^4*x^4*e - 45*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^
5*x^9)