3.138 \(\int \frac{c+d x^2+e x^4+f x^6}{x^2 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=153 \[ -\frac{x \left (-a^2 b e+5 a^3 f-3 a b^2 d+7 b^3 c\right )}{8 a^3 b^2 \left (a+b x^2\right )}-\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{4 a \left (a+b x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e-3 a^3 f-3 a b^2 d+15 b^3 c\right )}{8 a^{7/2} b^{5/2}}-\frac{c}{a^3 x} \]

[Out]

-(c/(a^3*x)) - (((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(4*a*(a + b*x^2)^2) - ((7*b^3*c - 3*a*b^2*d - a^2*b*e
+ 5*a^3*f)*x)/(8*a^3*b^2*(a + b*x^2)) - ((15*b^3*c - 3*a*b^2*d - a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a
]])/(8*a^(7/2)*b^(5/2))

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Rubi [A]  time = 0.176184, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1805, 1259, 453, 205} \[ -\frac{x \left (-a^2 b e+5 a^3 f-3 a b^2 d+7 b^3 c\right )}{8 a^3 b^2 \left (a+b x^2\right )}-\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{4 a \left (a+b x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e-3 a^3 f-3 a b^2 d+15 b^3 c\right )}{8 a^{7/2} b^{5/2}}-\frac{c}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^3),x]

[Out]

-(c/(a^3*x)) - (((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(4*a*(a + b*x^2)^2) - ((7*b^3*c - 3*a*b^2*d - a^2*b*e
+ 5*a^3*f)*x)/(8*a^3*b^2*(a + b*x^2)) - ((15*b^3*c - 3*a*b^2*d - a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a
]])/(8*a^(7/2)*b^(5/2))

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 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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^2 \left (a+b x^2\right )^3} \, dx &=-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{-4 c+\left (\frac{3 b c}{a}-3 d-\frac{a e}{b}+\frac{a^2 f}{b^2}\right ) x^2-\frac{4 a f x^4}{b}}{x^2 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{\left (7 b^3 c-3 a b^2 d-a^2 b e+5 a^3 f\right ) x}{8 a^3 b^2 \left (a+b x^2\right )}+\frac{\int \frac{8 a b^2 c-\left (7 b^3 c-3 a b^2 d-a^2 b e-3 a^3 f\right ) x^2}{x^2 \left (a+b x^2\right )} \, dx}{8 a^3 b^2}\\ &=-\frac{c}{a^3 x}-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{\left (7 b^3 c-3 a b^2 d-a^2 b e+5 a^3 f\right ) x}{8 a^3 b^2 \left (a+b x^2\right )}-\frac{\left (15 b^3 c-3 a b^2 d-a^2 b e-3 a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^3 b^2}\\ &=-\frac{c}{a^3 x}-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{\left (7 b^3 c-3 a b^2 d-a^2 b e+5 a^3 f\right ) x}{8 a^3 b^2 \left (a+b x^2\right )}-\frac{\left (15 b^3 c-3 a b^2 d-a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.12391, size = 155, normalized size = 1.01 \[ -\frac{x \left (-a^2 b e+5 a^3 f-3 a b^2 d+7 b^3 c\right )}{8 a^3 b^2 \left (a+b x^2\right )}+\frac{x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{4 a^2 b^2 \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+3 a^3 f+3 a b^2 d-15 b^3 c\right )}{8 a^{7/2} b^{5/2}}-\frac{c}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^3),x]

[Out]

-(c/(a^3*x)) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^2*b^2*(a + b*x^2)^2) - ((7*b^3*c - 3*a*b^2*d -
a^2*b*e + 5*a^3*f)*x)/(8*a^3*b^2*(a + b*x^2)) + ((-15*b^3*c + 3*a*b^2*d + a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x
)/Sqrt[a]])/(8*a^(7/2)*b^(5/2))

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Maple [A]  time = 0.013, size = 237, normalized size = 1.6 \begin{align*} -{\frac{c}{{a}^{3}x}}-{\frac{5\,{x}^{3}f}{8\, \left ( b{x}^{2}+a \right ) ^{2}b}}+{\frac{{x}^{3}e}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,b{x}^{3}d}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{b}^{2}{x}^{3}c}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,axf}{8\, \left ( b{x}^{2}+a \right ) ^{2}{b}^{2}}}-{\frac{ex}{8\, \left ( b{x}^{2}+a \right ) ^{2}b}}+{\frac{5\,dx}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bcx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,f}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,d}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,bc}{8\,{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^2/(b*x^2+a)^3,x)

[Out]

-c/a^3/x-5/8/(b*x^2+a)^2/b*x^3*f+1/8/a/(b*x^2+a)^2*x^3*e+3/8/a^2/(b*x^2+a)^2*b*x^3*d-7/8/a^3/(b*x^2+a)^2*b^2*x
^3*c-3/8*a/(b*x^2+a)^2/b^2*x*f-1/8/(b*x^2+a)^2/b*x*e+5/8/a/(b*x^2+a)^2*x*d-9/8/a^2/(b*x^2+a)^2*b*x*c+3/8/b^2/(
a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f+1/8/a/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e+3/8/a^2/(a*b)^(1/2)*arctan(
b*x/(a*b)^(1/2))*d-15/8/a^3*b/(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^2/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5269, size = 1076, normalized size = 7.03 \begin{align*} \left [-\frac{16 \, a^{3} b^{3} c + 2 \,{\left (15 \, a b^{5} c - 3 \, a^{2} b^{4} d - a^{3} b^{3} e + 5 \, a^{4} b^{2} f\right )} x^{4} + 2 \,{\left (25 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + a^{4} b^{2} e + 3 \, a^{5} b f\right )} x^{2} -{\left ({\left (15 \, b^{5} c - 3 \, a b^{4} d - a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{5} + 2 \,{\left (15 \, a b^{4} c - 3 \, a^{2} b^{3} d - a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{3} +{\left (15 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d - a^{4} b e - 3 \, a^{5} f\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{16 \,{\left (a^{4} b^{5} x^{5} + 2 \, a^{5} b^{4} x^{3} + a^{6} b^{3} x\right )}}, -\frac{8 \, a^{3} b^{3} c +{\left (15 \, a b^{5} c - 3 \, a^{2} b^{4} d - a^{3} b^{3} e + 5 \, a^{4} b^{2} f\right )} x^{4} +{\left (25 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + a^{4} b^{2} e + 3 \, a^{5} b f\right )} x^{2} +{\left ({\left (15 \, b^{5} c - 3 \, a b^{4} d - a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{5} + 2 \,{\left (15 \, a b^{4} c - 3 \, a^{2} b^{3} d - a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{3} +{\left (15 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d - a^{4} b e - 3 \, a^{5} f\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{8 \,{\left (a^{4} b^{5} x^{5} + 2 \, a^{5} b^{4} x^{3} + a^{6} b^{3} x\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^2/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/16*(16*a^3*b^3*c + 2*(15*a*b^5*c - 3*a^2*b^4*d - a^3*b^3*e + 5*a^4*b^2*f)*x^4 + 2*(25*a^2*b^4*c - 5*a^3*b^
3*d + a^4*b^2*e + 3*a^5*b*f)*x^2 - ((15*b^5*c - 3*a*b^4*d - a^2*b^3*e - 3*a^3*b^2*f)*x^5 + 2*(15*a*b^4*c - 3*a
^2*b^3*d - a^3*b^2*e - 3*a^4*b*f)*x^3 + (15*a^2*b^3*c - 3*a^3*b^2*d - a^4*b*e - 3*a^5*f)*x)*sqrt(-a*b)*log((b*
x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^4*b^5*x^5 + 2*a^5*b^4*x^3 + a^6*b^3*x), -1/8*(8*a^3*b^3*c + (15*a*b
^5*c - 3*a^2*b^4*d - a^3*b^3*e + 5*a^4*b^2*f)*x^4 + (25*a^2*b^4*c - 5*a^3*b^3*d + a^4*b^2*e + 3*a^5*b*f)*x^2 +
((15*b^5*c - 3*a*b^4*d - a^2*b^3*e - 3*a^3*b^2*f)*x^5 + 2*(15*a*b^4*c - 3*a^2*b^3*d - a^3*b^2*e - 3*a^4*b*f)*
x^3 + (15*a^2*b^3*c - 3*a^3*b^2*d - a^4*b*e - 3*a^5*f)*x)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^4*b^5*x^5 + 2*a^
5*b^4*x^3 + a^6*b^3*x)]

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Sympy [A]  time = 22.8078, size = 250, normalized size = 1.63 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} b^{5}}} \left (3 a^{3} f + a^{2} b e + 3 a b^{2} d - 15 b^{3} c\right ) \log{\left (- a^{4} b^{2} \sqrt{- \frac{1}{a^{7} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{7} b^{5}}} \left (3 a^{3} f + a^{2} b e + 3 a b^{2} d - 15 b^{3} c\right ) \log{\left (a^{4} b^{2} \sqrt{- \frac{1}{a^{7} b^{5}}} + x \right )}}{16} - \frac{8 a^{2} b^{2} c + x^{4} \left (5 a^{3} b f - a^{2} b^{2} e - 3 a b^{3} d + 15 b^{4} c\right ) + x^{2} \left (3 a^{4} f + a^{3} b e - 5 a^{2} b^{2} d + 25 a b^{3} c\right )}{8 a^{5} b^{2} x + 16 a^{4} b^{3} x^{3} + 8 a^{3} b^{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**2/(b*x**2+a)**3,x)

[Out]

-sqrt(-1/(a**7*b**5))*(3*a**3*f + a**2*b*e + 3*a*b**2*d - 15*b**3*c)*log(-a**4*b**2*sqrt(-1/(a**7*b**5)) + x)/
16 + sqrt(-1/(a**7*b**5))*(3*a**3*f + a**2*b*e + 3*a*b**2*d - 15*b**3*c)*log(a**4*b**2*sqrt(-1/(a**7*b**5)) +
x)/16 - (8*a**2*b**2*c + x**4*(5*a**3*b*f - a**2*b**2*e - 3*a*b**3*d + 15*b**4*c) + x**2*(3*a**4*f + a**3*b*e
- 5*a**2*b**2*d + 25*a*b**3*c))/(8*a**5*b**2*x + 16*a**4*b**3*x**3 + 8*a**3*b**4*x**5)

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Giac [A]  time = 1.17095, size = 207, normalized size = 1.35 \begin{align*} -\frac{c}{a^{3} x} - \frac{{\left (15 \, b^{3} c - 3 \, a b^{2} d - 3 \, a^{3} f - a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3} b^{2}} - \frac{7 \, b^{4} c x^{3} - 3 \, a b^{3} d x^{3} + 5 \, a^{3} b f x^{3} - a^{2} b^{2} x^{3} e + 9 \, a b^{3} c x - 5 \, a^{2} b^{2} d x + 3 \, a^{4} f x + a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{3} b^{2}} \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^2/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-c/(a^3*x) - 1/8*(15*b^3*c - 3*a*b^2*d - 3*a^3*f - a^2*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3*b^2) - 1/8*(7
*b^4*c*x^3 - 3*a*b^3*d*x^3 + 5*a^3*b*f*x^3 - a^2*b^2*x^3*e + 9*a*b^3*c*x - 5*a^2*b^2*d*x + 3*a^4*f*x + a^3*b*x
*e)/((b*x^2 + a)^2*a^3*b^2)