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

Optimal. Leaf size=82 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{5/2} b^{3/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3}+\frac{f x}{b} \]

[Out]

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

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Rubi [A]  time = 0.087426, antiderivative size = 82, 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} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{5/2} b^{3/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3}+\frac{f x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0798082, size = 83, normalized size = 1.01 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{5/2} b^{3/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3}+\frac{f x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 115, normalized size = 1.4 \begin{align*}{\frac{fx}{b}}-{\frac{c}{3\,a{x}^{3}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}}-{\frac{af}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{e\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{bd}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{{a}^{2}}\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^4/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56508, size = 455, normalized size = 5.55 \begin{align*} \left [\frac{6 \, a^{3} b f x^{4} + 3 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt{-a b} x^{3} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) - 2 \, a^{2} b^{2} c + 6 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2}}{6 \, a^{3} b^{2} x^{3}}, \frac{3 \, a^{3} b f x^{4} + 3 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt{a b} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - a^{2} b^{2} c + 3 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2}}{3 \, a^{3} b^{2} x^{3}}\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^4/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.42794, size = 151, normalized size = 1.84 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{2} + \frac{f x}{b} - \frac{a c + x^{2} \left (3 a d - 3 b c\right )}{3 a^{2} x^{3}} \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**4/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.16674, size = 109, normalized size = 1.33 \begin{align*} \frac{f x}{b} + \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^{2} b} + \frac{3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \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^4/(b*x^2+a),x, algorithm="giac")

[Out]

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