3.123 \(\int \frac{c+d x^2+e x^4+f x^6}{x^{12} (a+b x^2)} \, dx\)

Optimal. Leaf size=211 \[ -\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac{b^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{13/2}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{11 a x^{11}} \]

[Out]

-c/(11*a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(5*a^4*x^5) - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(13/2)

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Rubi [A]  time = 0.175188, antiderivative size = 211, 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} \[ -\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac{b^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{13/2}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^12*(a + b*x^2)),x]

[Out]

-c/(11*a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(5*a^4*x^5) - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(13/2)

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^{12} \left (a+b x^2\right )} \, dx &=\int \left (\frac{c}{a x^{12}}+\frac{-b c+a d}{a^2 x^{10}}+\frac{b^2 c-a b d+a^2 e}{a^3 x^8}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^6}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^4}+\frac{b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 x^2}-\frac{b^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c}{11 a x^{11}}+\frac{b c-a d}{9 a^2 x^9}-\frac{b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac{\left (b^3 \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^6}\\ &=-\frac{c}{11 a x^{11}}+\frac{b c-a d}{9 a^2 x^9}-\frac{b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac{b^{5/2} \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^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.160742, size = 211, normalized size = 1. \[ \frac{b \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac{b^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{13/2}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^12*(a + b*x^2)),x]

[Out]

-c/(11*a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(5*a^4*x^5) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(13/2)

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Maple [A]  time = 0.011, size = 286, normalized size = 1.4 \begin{align*} -{\frac{c}{11\,a{x}^{11}}}-{\frac{d}{9\,a{x}^{9}}}+{\frac{bc}{9\,{a}^{2}{x}^{9}}}-{\frac{e}{7\,a{x}^{7}}}+{\frac{bd}{7\,{a}^{2}{x}^{7}}}-{\frac{{b}^{2}c}{7\,{a}^{3}{x}^{7}}}-{\frac{f}{5\,a{x}^{5}}}+{\frac{be}{5\,{x}^{5}{a}^{2}}}-{\frac{{b}^{2}d}{5\,{a}^{3}{x}^{5}}}+{\frac{{b}^{3}c}{5\,{a}^{4}{x}^{5}}}-{\frac{{b}^{2}f}{{a}^{3}x}}+{\frac{{b}^{3}e}{{a}^{4}x}}-{\frac{{b}^{4}d}{{a}^{5}x}}+{\frac{{b}^{5}c}{{a}^{6}x}}+{\frac{bf}{3\,{x}^{3}{a}^{2}}}-{\frac{{b}^{2}e}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}d}{3\,{a}^{4}{x}^{3}}}-{\frac{{b}^{4}c}{3\,{a}^{5}{x}^{3}}}-{\frac{{b}^{3}f}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{4}e}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{5}d}{{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{6}c}{{a}^{6}}\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^12/(b*x^2+a),x)

[Out]

-1/11*c/a/x^11-1/9/a/x^9*d+1/9/a^2/x^9*b*c-1/7/a/x^7*e+1/7/a^2/x^7*b*d-1/7/a^3/x^7*b^2*c-1/5/a/x^5*f+1/5/a^2/x
^5*b*e-1/5/a^3/x^5*b^2*d+1/5/a^4/x^5*b^3*c-1/a^3*b^2/x*f+1/a^4*b^3/x*e-1/a^5*b^4/x*d+1/a^6*b^5/x*c+1/3/a^2*b/x
^3*f-1/3/a^3*b^2/x^3*e+1/3/a^4*b^3/x^3*d-1/3/a^5*b^4/x^3*c-b^3/a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f+b^4/a
^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e-b^5/a^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+b^6/a^6/(a*b)^(1/2)*arc
tan(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^12/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53972, size = 976, normalized size = 4.63 \begin{align*} \left [-\frac{3465 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{11} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 6930 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} + 2310 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} - 1386 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} + 630 \, a^{5} c + 990 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} - 770 \,{\left (a^{4} b c - a^{5} d\right )} x^{2}}{6930 \, a^{6} x^{11}}, \frac{3465 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{11} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 3465 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} - 1155 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} + 693 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} - 315 \, a^{5} c - 495 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} + 385 \,{\left (a^{4} b c - a^{5} d\right )} x^{2}}{3465 \, a^{6} x^{11}}\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^12/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/6930*(3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^11*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b
*x^2 + a)) - 6930*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^10 + 2310*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4
*b*f)*x^8 - 1386*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^6 + 630*a^5*c + 990*(a^3*b^2*c - a^4*b*d + a^5*e)
*x^4 - 770*(a^4*b*c - a^5*d)*x^2)/(a^6*x^11), 1/3465*(3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^11*sqrt
(b/a)*arctan(x*sqrt(b/a)) + 3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^10 - 1155*(a*b^4*c - a^2*b^3*d +
a^3*b^2*e - a^4*b*f)*x^8 + 693*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^6 - 315*a^5*c - 495*(a^3*b^2*c - a^
4*b*d + a^5*e)*x^4 + 385*(a^4*b*c - a^5*d)*x^2)/(a^6*x^11)]

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Sympy [A]  time = 51.001, size = 398, normalized size = 1.89 \begin{align*} \frac{\sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- \frac{a^{7} \sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a^{7} \sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} - \frac{315 a^{5} c + x^{10} \left (3465 a^{3} b^{2} f - 3465 a^{2} b^{3} e + 3465 a b^{4} d - 3465 b^{5} c\right ) + x^{8} \left (- 1155 a^{4} b f + 1155 a^{3} b^{2} e - 1155 a^{2} b^{3} d + 1155 a b^{4} c\right ) + x^{6} \left (693 a^{5} f - 693 a^{4} b e + 693 a^{3} b^{2} d - 693 a^{2} b^{3} c\right ) + x^{4} \left (495 a^{5} e - 495 a^{4} b d + 495 a^{3} b^{2} c\right ) + x^{2} \left (385 a^{5} d - 385 a^{4} b c\right )}{3465 a^{6} x^{11}} \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**12/(b*x**2+a),x)

[Out]

sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**7*sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b
**2*d - b**3*c)/(a**3*b**3*f - a**2*b**4*e + a*b**5*d - b**6*c) + x)/2 - sqrt(-b**5/a**13)*(a**3*f - a**2*b*e
+ a*b**2*d - b**3*c)*log(a**7*sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b**3*f - a**2*b*
*4*e + a*b**5*d - b**6*c) + x)/2 - (315*a**5*c + x**10*(3465*a**3*b**2*f - 3465*a**2*b**3*e + 3465*a*b**4*d -
3465*b**5*c) + x**8*(-1155*a**4*b*f + 1155*a**3*b**2*e - 1155*a**2*b**3*d + 1155*a*b**4*c) + x**6*(693*a**5*f
- 693*a**4*b*e + 693*a**3*b**2*d - 693*a**2*b**3*c) + x**4*(495*a**5*e - 495*a**4*b*d + 495*a**3*b**2*c) + x**
2*(385*a**5*d - 385*a**4*b*c))/(3465*a**6*x**11)

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Giac [A]  time = 1.17518, size = 336, normalized size = 1.59 \begin{align*} \frac{{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{6}} + \frac{3465 \, b^{5} c x^{10} - 3465 \, a b^{4} d x^{10} - 3465 \, a^{3} b^{2} f x^{10} + 3465 \, a^{2} b^{3} x^{10} e - 1155 \, a b^{4} c x^{8} + 1155 \, a^{2} b^{3} d x^{8} + 1155 \, a^{4} b f x^{8} - 1155 \, a^{3} b^{2} x^{8} e + 693 \, a^{2} b^{3} c x^{6} - 693 \, a^{3} b^{2} d x^{6} - 693 \, a^{5} f x^{6} + 693 \, a^{4} b x^{6} e - 495 \, a^{3} b^{2} c x^{4} + 495 \, a^{4} b d x^{4} - 495 \, a^{5} x^{4} e + 385 \, a^{4} b c x^{2} - 385 \, a^{5} d x^{2} - 315 \, a^{5} c}{3465 \, a^{6} x^{11}} \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^12/(b*x^2+a),x, algorithm="giac")

[Out]

(b^6*c - a*b^5*d - a^3*b^3*f + a^2*b^4*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) + 1/3465*(3465*b^5*c*x^10 - 34
65*a*b^4*d*x^10 - 3465*a^3*b^2*f*x^10 + 3465*a^2*b^3*x^10*e - 1155*a*b^4*c*x^8 + 1155*a^2*b^3*d*x^8 + 1155*a^4
*b*f*x^8 - 1155*a^3*b^2*x^8*e + 693*a^2*b^3*c*x^6 - 693*a^3*b^2*d*x^6 - 693*a^5*f*x^6 + 693*a^4*b*x^6*e - 495*
a^3*b^2*c*x^4 + 495*a^4*b*d*x^4 - 495*a^5*x^4*e + 385*a^4*b*c*x^2 - 385*a^5*d*x^2 - 315*a^5*c)/(a^6*x^11)