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

Optimal. Leaf size=104 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{7/2} \sqrt{b}}-\frac{a^2 e-a b d+b^2 c}{a^3 x}+\frac{b c-a d}{3 a^2 x^3}-\frac{c}{5 a x^5} \]

[Out]

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

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Rubi [A]  time = 0.102393, antiderivative size = 104, 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} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{7/2} \sqrt{b}}-\frac{a^2 e-a b d+b^2 c}{a^3 x}+\frac{b c-a d}{3 a^2 x^3}-\frac{c}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0826154, size = 103, normalized size = 0.99 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{7/2} \sqrt{b}}+\frac{a^2 (-e)+a b d-b^2 c}{a^3 x}+\frac{b c-a d}{3 a^2 x^3}-\frac{c}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 142, normalized size = 1.4 \begin{align*} -{\frac{c}{5\,a{x}^{5}}}-{\frac{d}{3\,a{x}^{3}}}+{\frac{bc}{3\,{x}^{3}{a}^{2}}}-{\frac{e}{ax}}+{\frac{bd}{{a}^{2}x}}-{\frac{{b}^{2}c}{{a}^{3}x}}+{f\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{be}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}d}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}c}{{a}^{3}}\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),x)

[Out]

-1/5*c/a/x^5-1/3/a/x^3*d+1/3/a^2/x^3*b*c-1/a/x*e+1/a^2/x*b*d-1/a^3/x*b^2*c+1/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2
))*f-1/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*b*e+1/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*b^2*d-1/a^3/(a*b)^(
1/2)*arctan(b*x/(a*b)^(1/2))*b^3*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^6/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50064, size = 522, normalized size = 5.02 \begin{align*} \left [\frac{15 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt{-a b} x^{5} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) - 6 \, a^{3} b c - 30 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e\right )} x^{4} + 10 \,{\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2}}{30 \, a^{4} b x^{5}}, -\frac{15 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt{a b} x^{5} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \, a^{3} b c + 15 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e\right )} x^{4} - 5 \,{\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2}}{15 \, a^{4} b x^{5}}\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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.3041, size = 167, normalized size = 1.61 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- a^{4} \sqrt{- \frac{1}{a^{7} b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{4} \sqrt{- \frac{1}{a^{7} b}} + x \right )}}{2} - \frac{3 a^{2} c + x^{4} \left (15 a^{2} e - 15 a b d + 15 b^{2} c\right ) + x^{2} \left (5 a^{2} d - 5 a b c\right )}{15 a^{3} 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),x)

[Out]

-sqrt(-1/(a**7*b))*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**4*sqrt(-1/(a**7*b)) + x)/2 + sqrt(-1/(a**7*
b))*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**4*sqrt(-1/(a**7*b)) + x)/2 - (3*a**2*c + x**4*(15*a**2*e -
15*a*b*d + 15*b**2*c) + x**2*(5*a**2*d - 5*a*b*c))/(15*a**3*x**5)

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Giac [A]  time = 1.15126, size = 142, normalized size = 1.37 \begin{align*} -\frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{15 \, b^{2} c x^{4} - 15 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 5 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{3} 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),x, algorithm="giac")

[Out]

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