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

Optimal. Leaf size=152 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 (-f)-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.213107, antiderivative size = 152, 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} \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 (-f)-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0824897, size = 151, normalized size = 0.99 \[ \frac{x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 a^4 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^2 b e+a^3 f+5 a b^2 d-7 b^3 c\right )}{2 a^{9/2} \sqrt{b}}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{a^4 x}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.014, size = 219, normalized size = 1.4 \begin{align*} -{\frac{c}{5\,{x}^{5}{a}^{2}}}-{\frac{d}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bc}{3\,{a}^{3}{x}^{3}}}-{\frac{e}{{a}^{2}x}}+2\,{\frac{bd}{{a}^{3}x}}-3\,{\frac{{b}^{2}c}{{a}^{4}x}}+{\frac{fx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bxe}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{x{b}^{2}d}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{x{b}^{3}c}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{f}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,be}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}d}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{3}c}{2\,{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^6/(b*x^2+a)^2,x)

[Out]

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

________________________________________________________________________________________

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^6/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.39414, size = 926, normalized size = 6.09 \begin{align*} \left [-\frac{30 \,{\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 12 \, a^{4} b c + 20 \,{\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 4 \,{\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} - 15 \,{\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{60 \,{\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}, -\frac{15 \,{\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 6 \, a^{4} b c + 10 \,{\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 2 \,{\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} + 15 \,{\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{30 \,{\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\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^6/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[-1/60*(30*(7*a*b^4*c - 5*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^6 + 12*a^4*b*c + 20*(7*a^2*b^3*c - 5*a^3*b^2*d
+ 3*a^4*b*e)*x^4 - 4*(7*a^3*b^2*c - 5*a^4*b*d)*x^2 - 15*((7*b^4*c - 5*a*b^3*d + 3*a^2*b^2*e - a^3*b*f)*x^7 + (
7*a*b^3*c - 5*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^5)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a
^5*b^2*x^7 + a^6*b*x^5), -1/30*(15*(7*a*b^4*c - 5*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^6 + 6*a^4*b*c + 10*(7*a
^2*b^3*c - 5*a^3*b^2*d + 3*a^4*b*e)*x^4 - 2*(7*a^3*b^2*c - 5*a^4*b*d)*x^2 + 15*((7*b^4*c - 5*a*b^3*d + 3*a^2*b
^2*e - a^3*b*f)*x^7 + (7*a*b^3*c - 5*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^5)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5
*b^2*x^7 + a^6*b*x^5)]

________________________________________________________________________________________

Sympy [A]  time = 30.5347, size = 226, normalized size = 1.49 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log{\left (- a^{5} \sqrt{- \frac{1}{a^{9} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log{\left (a^{5} \sqrt{- \frac{1}{a^{9} b}} + x \right )}}{4} + \frac{- 6 a^{3} c + x^{6} \left (15 a^{3} f - 45 a^{2} b e + 75 a b^{2} d - 105 b^{3} c\right ) + x^{4} \left (- 30 a^{3} e + 50 a^{2} b d - 70 a b^{2} c\right ) + x^{2} \left (- 10 a^{3} d + 14 a^{2} b c\right )}{30 a^{5} x^{5} + 30 a^{4} b 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**6/(b*x**2+a)**2,x)

[Out]

-sqrt(-1/(a**9*b))*(a**3*f - 3*a**2*b*e + 5*a*b**2*d - 7*b**3*c)*log(-a**5*sqrt(-1/(a**9*b)) + x)/4 + sqrt(-1/
(a**9*b))*(a**3*f - 3*a**2*b*e + 5*a*b**2*d - 7*b**3*c)*log(a**5*sqrt(-1/(a**9*b)) + x)/4 + (-6*a**3*c + x**6*
(15*a**3*f - 45*a**2*b*e + 75*a*b**2*d - 105*b**3*c) + x**4*(-30*a**3*e + 50*a**2*b*d - 70*a*b**2*c) + x**2*(-
10*a**3*d + 14*a**2*b*c))/(30*a**5*x**5 + 30*a**4*b*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.1733, size = 204, normalized size = 1.34 \begin{align*} -\frac{{\left (7 \, b^{3} c - 5 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \,{\left (b x^{2} + a\right )} a^{4}} - \frac{45 \, b^{2} c x^{4} - 30 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{4} x^{5}} \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^6/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(7*b^3*c - 5*a*b^2*d - a^3*f + 3*a^2*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/2*(b^3*c*x - a*b^2*d*
x - a^3*f*x + a^2*b*x*e)/((b*x^2 + a)*a^4) - 1/15*(45*b^2*c*x^4 - 30*a*b*d*x^4 + 15*a^2*x^4*e - 10*a*b*c*x^2 +
5*a^2*d*x^2 + 3*a^2*c)/(a^4*x^5)