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

Optimal. Leaf size=202 \[ \frac{x^5 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{x^3 \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{6 a b^4}+\frac{x \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^5}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^{11/2}}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^7}{7 b^2} \]

[Out]

((3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*x)/(2*b^5) - ((3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*x^3)/(6
*a*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^7)/(7*b^2) + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^5)/(2*a*(a
+ b*x^2)) - (Sqrt[a]*(3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Rubi [A]  time = 0.234874, antiderivative size = 202, 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 = {1804, 1585, 1261, 205} \[ \frac{x^5 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{x^3 \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{6 a b^4}+\frac{x \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^5}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^{11/2}}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*x)/(2*b^5) - ((3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*x^3)/(6
*a*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^7)/(7*b^2) + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^5)/(2*a*(a
+ b*x^2)) - (Sqrt[a]*(3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

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{x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac{\int \frac{x^3 \left (\left (3 b c-5 a d+\frac{5 a^2 e}{b}-\frac{5 a^3 f}{b^2}\right ) x-2 a \left (e-\frac{a f}{b}\right ) x^3-2 a f x^5\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac{\int \frac{x^4 \left (3 b c-5 a d+\frac{5 a^2 e}{b}-\frac{5 a^3 f}{b^2}-2 a \left (e-\frac{a f}{b}\right ) x^2-2 a f x^4\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac{\int \left (-\frac{a \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right )}{b^4}+\frac{\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^2}{b^3}-\frac{2 a (b e-2 a f) x^4}{b^2}-\frac{2 a f x^6}{b}+\frac{3 a^2 b^3 c-5 a^3 b^2 d+7 a^4 b e-9 a^5 f}{b^4 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=\frac{\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac{\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^7}{7 b^2}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac{\left (a \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^5}\\ &=\frac{\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac{\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^7}{7 b^2}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac{\sqrt{a} \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.101106, size = 187, normalized size = 0.93 \[ \frac{x \left (-a^2 b^2 d+a^3 b e+a^4 (-f)+a b^3 c\right )}{2 b^5 \left (a+b x^2\right )}+\frac{x \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{b^5}+\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-7 a^2 b e+9 a^3 f+5 a b^2 d-3 b^3 c\right )}{2 b^{11/2}}+\frac{x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^3)/(3*b^4) + ((b*e - 2*a*f)
*x^5)/(5*b^3) + (f*x^7)/(7*b^2) + ((a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/(2*b^5*(a + b*x^2)) + (Sqrt[a]*(
-3*b^3*c + 5*a*b^2*d - 7*a^2*b*e + 9*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Maple [A]  time = 0.01, size = 258, normalized size = 1.3 \begin{align*}{\frac{f{x}^{7}}{7\,{b}^{2}}}-{\frac{2\,{x}^{5}af}{5\,{b}^{3}}}+{\frac{{x}^{5}e}{5\,{b}^{2}}}+{\frac{{x}^{3}{a}^{2}f}{{b}^{4}}}-{\frac{2\,a{x}^{3}e}{3\,{b}^{3}}}+{\frac{{x}^{3}d}{3\,{b}^{2}}}-4\,{\frac{{a}^{3}fx}{{b}^{5}}}+3\,{\frac{{a}^{2}ex}{{b}^{4}}}-2\,{\frac{adx}{{b}^{3}}}+{\frac{cx}{{b}^{2}}}-{\frac{{a}^{4}xf}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{3}xe}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}xd}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{axc}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{a}^{4}f}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{a}^{3}e}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}d}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,ac}{2\,{b}^{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(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^2,x)

[Out]

1/7*f*x^7/b^2-2/5/b^3*x^5*a*f+1/5/b^2*x^5*e+1/b^4*x^3*a^2*f-2/3/b^3*x^3*a*e+1/3/b^2*x^3*d-4/b^5*a^3*f*x+3/b^4*
a^2*e*x-2/b^3*a*d*x+1/b^2*c*x-1/2*a^4/b^5*x/(b*x^2+a)*f+1/2*a^3/b^4*x/(b*x^2+a)*e-1/2*a^2/b^3*x/(b*x^2+a)*d+1/
2*a/b^2*x/(b*x^2+a)*c+9/2*a^4/b^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f-7/2*a^3/b^4/(a*b)^(1/2)*arctan(b*x/(a*
b)^(1/2))*e+5/2*a^2/b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d-3/2*a/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(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.49243, size = 1041, normalized size = 5.15 \begin{align*} \left [\frac{60 \, b^{4} f x^{9} + 12 \,{\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 28 \,{\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 140 \,{\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \,{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f +{\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 210 \,{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{420 \,{\left (b^{6} x^{2} + a b^{5}\right )}}, \frac{30 \, b^{4} f x^{9} + 6 \,{\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 14 \,{\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 70 \,{\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \,{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f +{\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 105 \,{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{210 \,{\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/420*(60*b^4*f*x^9 + 12*(7*b^4*e - 9*a*b^3*f)*x^7 + 28*(5*b^4*d - 7*a*b^3*e + 9*a^2*b^2*f)*x^5 + 140*(3*b^4*
c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^3 - 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f + (3*b^4*c -
5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(3
*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f)*x)/(b^6*x^2 + a*b^5), 1/210*(30*b^4*f*x^9 + 6*(7*b^4*e - 9*a*b^3
*f)*x^7 + 14*(5*b^4*d - 7*a*b^3*e + 9*a^2*b^2*f)*x^5 + 70*(3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^3
- 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f + (3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^2)*sq
rt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f)*x)/(b^6*x^2 + a*b^5)]

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Sympy [A]  time = 2.5218, size = 250, normalized size = 1.24 \begin{align*} - \frac{x \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{\sqrt{- \frac{a}{b^{11}}} \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log{\left (- b^{5} \sqrt{- \frac{a}{b^{11}}} + x \right )}}{4} + \frac{\sqrt{- \frac{a}{b^{11}}} \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log{\left (b^{5} \sqrt{- \frac{a}{b^{11}}} + x \right )}}{4} + \frac{f x^{7}}{7 b^{2}} - \frac{x^{5} \left (2 a f - b e\right )}{5 b^{3}} + \frac{x^{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{3 b^{4}} - \frac{x \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**2,x)

[Out]

-x*(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c)/(2*a*b**5 + 2*b**6*x**2) - sqrt(-a/b**11)*(9*a**3*f - 7*a**2*b
*e + 5*a*b**2*d - 3*b**3*c)*log(-b**5*sqrt(-a/b**11) + x)/4 + sqrt(-a/b**11)*(9*a**3*f - 7*a**2*b*e + 5*a*b**2
*d - 3*b**3*c)*log(b**5*sqrt(-a/b**11) + x)/4 + f*x**7/(7*b**2) - x**5*(2*a*f - b*e)/(5*b**3) + x**3*(3*a**2*f
- 2*a*b*e + b**2*d)/(3*b**4) - x*(4*a**3*f - 3*a**2*b*e + 2*a*b**2*d - b**3*c)/b**5

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Giac [A]  time = 1.1531, size = 271, normalized size = 1.34 \begin{align*} -\frac{{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d - 9 \, a^{4} f + 7 \, a^{3} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{5}} + \frac{a b^{3} c x - a^{2} b^{2} d x - a^{4} f x + a^{3} b x e}{2 \,{\left (b x^{2} + a\right )} b^{5}} + \frac{15 \, b^{12} f x^{7} - 42 \, a b^{11} f x^{5} + 21 \, b^{12} x^{5} e + 35 \, b^{12} d x^{3} + 105 \, a^{2} b^{10} f x^{3} - 70 \, a b^{11} x^{3} e + 105 \, b^{12} c x - 210 \, a b^{11} d x - 420 \, a^{3} b^{9} f x + 315 \, a^{2} b^{10} x e}{105 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(3*a*b^3*c - 5*a^2*b^2*d - 9*a^4*f + 7*a^3*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/2*(a*b^3*c*x -
a^2*b^2*d*x - a^4*f*x + a^3*b*x*e)/((b*x^2 + a)*b^5) + 1/105*(15*b^12*f*x^7 - 42*a*b^11*f*x^5 + 21*b^12*x^5*e
+ 35*b^12*d*x^3 + 105*a^2*b^10*f*x^3 - 70*a*b^11*x^3*e + 105*b^12*c*x - 210*a*b^11*d*x - 420*a^3*b^9*f*x + 315
*a^2*b^10*x*e)/b^14