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

Optimal. Leaf size=168 \[ \frac{x \left (3 a^2 b e+a^3 f-7 a b^2 d+11 b^3 c\right )}{8 a^4 b \left (a+b x^2\right )}+\frac{x \left (\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 f-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac{3 b c-a d}{a^4 x}-\frac{c}{3 a^3 x^3} \]

[Out]

-c/(3*a^3*x^3) + (3*b*c - a*d)/(a^4*x) + (((b^2*c)/a^2 - (b*d)/a + e - (a*f)/b)*x)/(4*a*(a + b*x^2)^2) + ((11*
b^3*c - 7*a*b^2*d + 3*a^2*b*e + a^3*f)*x)/(8*a^4*b*(a + b*x^2)) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))

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Rubi [A]  time = 0.242445, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1805, 1259, 1261, 205} \[ \frac{x \left (3 a^2 b e+a^3 f-7 a b^2 d+11 b^3 c\right )}{8 a^4 b \left (a+b x^2\right )}+\frac{x \left (\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 f-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac{3 b c-a d}{a^4 x}-\frac{c}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(3*a^3*x^3) + (3*b*c - a*d)/(a^4*x) + (((b^2*c)/a^2 - (b*d)/a + e - (a*f)/b)*x)/(4*a*(a + b*x^2)^2) + ((11*
b^3*c - 7*a*b^2*d + 3*a^2*b*e + a^3*f)*x)/(8*a^4*b*(a + b*x^2)) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))

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 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -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 )^3} \, dx &=\frac{\left (\frac{b^2 c}{a^2}-\frac{b d}{a}+e-\frac{a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{-4 c+4 \left (\frac{b c}{a}-d\right ) x^2+\left (-\frac{3 b^2 c}{a^2}+\frac{3 b d}{a}-3 e-\frac{a f}{b}\right ) x^4}{x^4 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{\left (\frac{b^2 c}{a^2}-\frac{b d}{a}+e-\frac{a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}-\frac{\int \frac{-8 a^2 b^2 c+8 a b^2 (2 b c-a d) x^2-b \left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x^4}{x^4 \left (a+b x^2\right )} \, dx}{8 a^4 b^2}\\ &=\frac{\left (\frac{b^2 c}{a^2}-\frac{b d}{a}+e-\frac{a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}-\frac{\int \left (-\frac{8 a b^2 c}{x^4}+\frac{8 b^2 (3 b c-a d)}{x^2}-\frac{b \left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right )}{a+b x^2}\right ) \, dx}{8 a^4 b^2}\\ &=-\frac{c}{3 a^3 x^3}+\frac{3 b c-a d}{a^4 x}+\frac{\left (\frac{b^2 c}{a^2}-\frac{b d}{a}+e-\frac{a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}+\frac{\left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^4 b}\\ &=-\frac{c}{3 a^3 x^3}+\frac{3 b c-a d}{a^4 x}+\frac{\left (\frac{b^2 c}{a^2}-\frac{b d}{a}+e-\frac{a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}+\frac{\left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.137575, size = 169, normalized size = 1.01 \[ \frac{a^2 b^2 x^2 \left (56 c-75 d x^2+9 e x^4\right )+a^3 b \left (3 x^2 \left (-8 d+5 e x^2+f x^4\right )-8 c\right )-3 a^4 f x^4+5 a b^3 x^4 \left (35 c-9 d x^2\right )+105 b^4 c x^6}{24 a^4 b x^3 \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 f-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^4*f*x^4 + 105*b^4*c*x^6 + 5*a*b^3*x^4*(35*c - 9*d*x^2) + a^2*b^2*x^2*(56*c - 75*d*x^2 + 9*e*x^4) + a^3*b
*(-8*c + 3*x^2*(-8*d + 5*e*x^2 + f*x^4)))/(24*a^4*b*x^3*(a + b*x^2)^2) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e +
a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))

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Maple [A]  time = 0.016, size = 264, normalized size = 1.6 \begin{align*} -{\frac{c}{3\,{a}^{3}{x}^{3}}}-{\frac{d}{{a}^{3}x}}+3\,{\frac{bc}{{a}^{4}x}}+{\frac{{x}^{3}f}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,{x}^{3}be}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{x}^{3}{b}^{2}d}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{x}^{3}{b}^{3}c}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{fx}{8\, \left ( b{x}^{2}+a \right ) ^{2}b}}+{\frac{5\,ex}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bdx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}xc}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{f}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,e}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,bd}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{b}^{2}c}{8\,{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^4/(b*x^2+a)^3,x)

[Out]

-1/3*c/a^3/x^3-1/a^3/x*d+3/a^4/x*b*c+1/8/a/(b*x^2+a)^2*x^3*f+3/8/a^2/(b*x^2+a)^2*x^3*b*e-7/8/a^3/(b*x^2+a)^2*x
^3*b^2*d+11/8/a^4/(b*x^2+a)^2*x^3*b^3*c-1/8/(b*x^2+a)^2/b*x*f+5/8/a/(b*x^2+a)^2*x*e-9/8/a^2/(b*x^2+a)^2*b*x*d+
13/8/a^3/(b*x^2+a)^2*b^2*x*c+1/8/a/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f+3/8/a^2/(a*b)^(1/2)*arctan(b*x/(a*b
)^(1/2))*e-15/8/a^3*b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+35/8/a^4*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51377, size = 1219, normalized size = 7.26 \begin{align*} \left [-\frac{16 \, a^{4} b^{2} c - 6 \,{\left (35 \, a b^{5} c - 15 \, a^{2} b^{4} d + 3 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{6} - 2 \,{\left (175 \, a^{2} b^{4} c - 75 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{4} - 16 \,{\left (7 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{2} + 3 \,{\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} +{\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{48 \,{\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}, -\frac{8 \, a^{4} b^{2} c - 3 \,{\left (35 \, a b^{5} c - 15 \, a^{2} b^{4} d + 3 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{6} -{\left (175 \, a^{2} b^{4} c - 75 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{4} - 8 \,{\left (7 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{2} - 3 \,{\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} +{\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{24 \,{\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\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^4/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/48*(16*a^4*b^2*c - 6*(35*a*b^5*c - 15*a^2*b^4*d + 3*a^3*b^3*e + a^4*b^2*f)*x^6 - 2*(175*a^2*b^4*c - 75*a^3
*b^3*d + 15*a^4*b^2*e - 3*a^5*b*f)*x^4 - 16*(7*a^3*b^3*c - 3*a^4*b^2*d)*x^2 + 3*((35*b^5*c - 15*a*b^4*d + 3*a^
2*b^3*e + a^3*b^2*f)*x^7 + 2*(35*a*b^4*c - 15*a^2*b^3*d + 3*a^3*b^2*e + a^4*b*f)*x^5 + (35*a^2*b^3*c - 15*a^3*
b^2*d + 3*a^4*b*e + a^5*f)*x^3)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^5*b^4*x^7 + 2*a^6
*b^3*x^5 + a^7*b^2*x^3), -1/24*(8*a^4*b^2*c - 3*(35*a*b^5*c - 15*a^2*b^4*d + 3*a^3*b^3*e + a^4*b^2*f)*x^6 - (1
75*a^2*b^4*c - 75*a^3*b^3*d + 15*a^4*b^2*e - 3*a^5*b*f)*x^4 - 8*(7*a^3*b^3*c - 3*a^4*b^2*d)*x^2 - 3*((35*b^5*c
- 15*a*b^4*d + 3*a^2*b^3*e + a^3*b^2*f)*x^7 + 2*(35*a*b^4*c - 15*a^2*b^3*d + 3*a^3*b^2*e + a^4*b*f)*x^5 + (35
*a^2*b^3*c - 15*a^3*b^2*d + 3*a^4*b*e + a^5*f)*x^3)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5*b^4*x^7 + 2*a^6*b^3*
x^5 + a^7*b^2*x^3)]

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Sympy [A]  time = 58.4529, size = 270, normalized size = 1.61 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log{\left (- a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log{\left (a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{16} + \frac{- 8 a^{3} b c + x^{6} \left (3 a^{3} b f + 9 a^{2} b^{2} e - 45 a b^{3} d + 105 b^{4} c\right ) + x^{4} \left (- 3 a^{4} f + 15 a^{3} b e - 75 a^{2} b^{2} d + 175 a b^{3} c\right ) + x^{2} \left (- 24 a^{3} b d + 56 a^{2} b^{2} c\right )}{24 a^{6} b x^{3} + 48 a^{5} b^{2} x^{5} + 24 a^{4} b^{3} 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**4/(b*x**2+a)**3,x)

[Out]

-sqrt(-1/(a**9*b**3))*(a**3*f + 3*a**2*b*e - 15*a*b**2*d + 35*b**3*c)*log(-a**5*b*sqrt(-1/(a**9*b**3)) + x)/16
+ sqrt(-1/(a**9*b**3))*(a**3*f + 3*a**2*b*e - 15*a*b**2*d + 35*b**3*c)*log(a**5*b*sqrt(-1/(a**9*b**3)) + x)/1
6 + (-8*a**3*b*c + x**6*(3*a**3*b*f + 9*a**2*b**2*e - 45*a*b**3*d + 105*b**4*c) + x**4*(-3*a**4*f + 15*a**3*b*
e - 75*a**2*b**2*d + 175*a*b**3*c) + x**2*(-24*a**3*b*d + 56*a**2*b**2*c))/(24*a**6*b*x**3 + 48*a**5*b**2*x**5
+ 24*a**4*b**3*x**7)

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Giac [A]  time = 1.16959, size = 230, normalized size = 1.37 \begin{align*} \frac{{\left (35 \, b^{3} c - 15 \, a b^{2} d + a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{4} b} + \frac{11 \, b^{4} c x^{3} - 7 \, a b^{3} d x^{3} + a^{3} b f x^{3} + 3 \, a^{2} b^{2} x^{3} e + 13 \, a b^{3} c x - 9 \, a^{2} b^{2} d x - a^{4} f x + 5 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{4} b} + \frac{9 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{4} 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)^3,x, algorithm="giac")

[Out]

1/8*(35*b^3*c - 15*a*b^2*d + a^3*f + 3*a^2*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4*b) + 1/8*(11*b^4*c*x^3 -
7*a*b^3*d*x^3 + a^3*b*f*x^3 + 3*a^2*b^2*x^3*e + 13*a*b^3*c*x - 9*a^2*b^2*d*x - a^4*f*x + 5*a^3*b*x*e)/((b*x^2
+ a)^2*a^4*b) + 1/3*(9*b*c*x^2 - 3*a*d*x^2 - a*c)/(a^4*x^3)