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

Optimal. Leaf size=171 \[ -\frac{e x \left (a e^2-b d e+c d^2\right )}{4 d^4 \left (d+e x^2\right )^2}-\frac{6 a e^2-3 b d e+c d^2}{d^5 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (63 a e^2-35 b d e+15 c d^2\right )}{8 d^{11/2}}-\frac{e x \left (7 c d^2-e (11 b d-15 a e)\right )}{8 d^5 \left (d+e x^2\right )}-\frac{b d-3 a e}{3 d^4 x^3}-\frac{a}{5 d^3 x^5} \]

[Out]

-a/(5*d^3*x^5) - (b*d - 3*a*e)/(3*d^4*x^3) - (c*d^2 - 3*b*d*e + 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e + a*e^2)*
x)/(4*d^4*(d + e*x^2)^2) - (e*(7*c*d^2 - e*(11*b*d - 15*a*e))*x)/(8*d^5*(d + e*x^2)) - (Sqrt[e]*(15*c*d^2 - 35
*b*d*e + 63*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(11/2))

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Rubi [A]  time = 0.373194, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1259, 1805, 1802, 205} \[ -\frac{e x \left (a e^2-b d e+c d^2\right )}{4 d^4 \left (d+e x^2\right )^2}-\frac{6 a e^2-3 b d e+c d^2}{d^5 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (63 a e^2-35 b d e+15 c d^2\right )}{8 d^{11/2}}-\frac{e x \left (7 c d^2-e (11 b d-15 a e)\right )}{8 d^5 \left (d+e x^2\right )}-\frac{b d-3 a e}{3 d^4 x^3}-\frac{a}{5 d^3 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(5*d^3*x^5) - (b*d - 3*a*e)/(3*d^4*x^3) - (c*d^2 - 3*b*d*e + 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e + a*e^2)*
x)/(4*d^4*(d + e*x^2)^2) - (e*(7*c*d^2 - e*(11*b*d - 15*a*e))*x)/(8*d^5*(d + e*x^2)) - (Sqrt[e]*(15*c*d^2 - 35
*b*d*e + 63*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(11/2))

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 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 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{a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^3} \, dx &=-\frac{e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac{\int \frac{-4 a d^3 e^2-4 d^2 e^2 (b d-a e) x^2-4 d e^2 \left (c d^2-b d e+a e^2\right ) x^4+3 e^3 \left (c d^2-b d e+a e^2\right ) x^6}{x^6 \left (d+e x^2\right )^2} \, dx}{4 d^4 e^2}\\ &=-\frac{e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac{e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}+\frac{\int \frac{8 a d^3 e^2+8 d^2 e^2 (b d-2 a e) x^2+8 d e^2 \left (c d^2-e (2 b d-3 a e)\right ) x^4-e^3 \left (7 c d^2-e (11 b d-15 a e)\right ) x^6}{x^6 \left (d+e x^2\right )} \, dx}{8 d^5 e^2}\\ &=-\frac{e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac{e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}+\frac{\int \left (\frac{8 a d^2 e^2}{x^6}+\frac{8 d e^2 (b d-3 a e)}{x^4}+\frac{8 e^2 \left (c d^2-3 b d e+6 a e^2\right )}{x^2}-\frac{e^3 \left (15 c d^2-35 b d e+63 a e^2\right )}{d+e x^2}\right ) \, dx}{8 d^5 e^2}\\ &=-\frac{a}{5 d^3 x^5}-\frac{b d-3 a e}{3 d^4 x^3}-\frac{c d^2-3 b d e+6 a e^2}{d^5 x}-\frac{e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac{e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}-\frac{\left (e \left (15 c d^2-35 b d e+63 a e^2\right )\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^5}\\ &=-\frac{a}{5 d^3 x^5}-\frac{b d-3 a e}{3 d^4 x^3}-\frac{c d^2-3 b d e+6 a e^2}{d^5 x}-\frac{e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac{e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}-\frac{\sqrt{e} \left (15 c d^2-35 b d e+63 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.11355, size = 173, normalized size = 1.01 \[ -\frac{x \left (15 a e^3-11 b d e^2+7 c d^2 e\right )}{8 d^5 \left (d+e x^2\right )}-\frac{e x \left (a e^2-b d e+c d^2\right )}{4 d^4 \left (d+e x^2\right )^2}+\frac{-6 a e^2+3 b d e-c d^2}{d^5 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (63 a e^2-35 b d e+15 c d^2\right )}{8 d^{11/2}}+\frac{3 a e-b d}{3 d^4 x^3}-\frac{a}{5 d^3 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(5*d^3*x^5) + (-(b*d) + 3*a*e)/(3*d^4*x^3) + (-(c*d^2) + 3*b*d*e - 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e + a
*e^2)*x)/(4*d^4*(d + e*x^2)^2) - ((7*c*d^2*e - 11*b*d*e^2 + 15*a*e^3)*x)/(8*d^5*(d + e*x^2)) - (Sqrt[e]*(15*c*
d^2 - 35*b*d*e + 63*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(11/2))

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Maple [A]  time = 0.016, size = 245, normalized size = 1.4 \begin{align*} -{\frac{a}{5\,{d}^{3}{x}^{5}}}+{\frac{ae}{{d}^{4}{x}^{3}}}-{\frac{b}{3\,{d}^{3}{x}^{3}}}-6\,{\frac{a{e}^{2}}{{d}^{5}x}}+3\,{\frac{be}{{d}^{4}x}}-{\frac{c}{{d}^{3}x}}-{\frac{15\,{e}^{4}{x}^{3}a}{8\,{d}^{5} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{11\,{e}^{3}{x}^{3}b}{8\,{d}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{7\,{e}^{2}{x}^{3}c}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{17\,{e}^{3}ax}{8\,{d}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{13\,{e}^{2}bx}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{9\,cex}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{63\,{e}^{3}a}{8\,{d}^{5}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{e}^{2}b}{8\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,ce}{8\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x)

[Out]

-1/5*a/d^3/x^5+1/d^4/x^3*a*e-1/3/d^3/x^3*b-6/d^5/x*a*e^2+3/d^4/x*e*b-1/d^3/x*c-15/8*e^4/d^5/(e*x^2+d)^2*x^3*a+
11/8*e^3/d^4/(e*x^2+d)^2*x^3*b-7/8*e^2/d^3/(e*x^2+d)^2*x^3*c-17/8*e^3/d^4/(e*x^2+d)^2*a*x+13/8*e^2/d^3/(e*x^2+
d)^2*b*x-9/8*e/d^2/(e*x^2+d)^2*c*x-63/8*e^3/d^5/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*a+35/8*e^2/d^4/(d*e)^(1/2)
*arctan(e*x/(d*e)^(1/2))*b-15/8*e/d^3/(d*e)^(1/2)*arctan(e*x/(d*e)^(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((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85279, size = 1143, normalized size = 6.68 \begin{align*} \left [-\frac{30 \,{\left (15 \, c d^{2} e^{2} - 35 \, b d e^{3} + 63 \, a e^{4}\right )} x^{8} + 50 \,{\left (15 \, c d^{3} e - 35 \, b d^{2} e^{2} + 63 \, a d e^{3}\right )} x^{6} + 48 \, a d^{4} + 16 \,{\left (15 \, c d^{4} - 35 \, b d^{3} e + 63 \, a d^{2} e^{2}\right )} x^{4} + 16 \,{\left (5 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2} - 15 \,{\left ({\left (15 \, c d^{2} e^{2} - 35 \, b d e^{3} + 63 \, a e^{4}\right )} x^{9} + 2 \,{\left (15 \, c d^{3} e - 35 \, b d^{2} e^{2} + 63 \, a d e^{3}\right )} x^{7} +{\left (15 \, c d^{4} - 35 \, b d^{3} e + 63 \, a d^{2} e^{2}\right )} x^{5}\right )} \sqrt{-\frac{e}{d}} \log \left (\frac{e x^{2} - 2 \, d x \sqrt{-\frac{e}{d}} - d}{e x^{2} + d}\right )}{240 \,{\left (d^{5} e^{2} x^{9} + 2 \, d^{6} e x^{7} + d^{7} x^{5}\right )}}, -\frac{15 \,{\left (15 \, c d^{2} e^{2} - 35 \, b d e^{3} + 63 \, a e^{4}\right )} x^{8} + 25 \,{\left (15 \, c d^{3} e - 35 \, b d^{2} e^{2} + 63 \, a d e^{3}\right )} x^{6} + 24 \, a d^{4} + 8 \,{\left (15 \, c d^{4} - 35 \, b d^{3} e + 63 \, a d^{2} e^{2}\right )} x^{4} + 8 \,{\left (5 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2} + 15 \,{\left ({\left (15 \, c d^{2} e^{2} - 35 \, b d e^{3} + 63 \, a e^{4}\right )} x^{9} + 2 \,{\left (15 \, c d^{3} e - 35 \, b d^{2} e^{2} + 63 \, a d e^{3}\right )} x^{7} +{\left (15 \, c d^{4} - 35 \, b d^{3} e + 63 \, a d^{2} e^{2}\right )} x^{5}\right )} \sqrt{\frac{e}{d}} \arctan \left (x \sqrt{\frac{e}{d}}\right )}{120 \,{\left (d^{5} e^{2} x^{9} + 2 \, d^{6} e x^{7} + d^{7} x^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

[-1/240*(30*(15*c*d^2*e^2 - 35*b*d*e^3 + 63*a*e^4)*x^8 + 50*(15*c*d^3*e - 35*b*d^2*e^2 + 63*a*d*e^3)*x^6 + 48*
a*d^4 + 16*(15*c*d^4 - 35*b*d^3*e + 63*a*d^2*e^2)*x^4 + 16*(5*b*d^4 - 9*a*d^3*e)*x^2 - 15*((15*c*d^2*e^2 - 35*
b*d*e^3 + 63*a*e^4)*x^9 + 2*(15*c*d^3*e - 35*b*d^2*e^2 + 63*a*d*e^3)*x^7 + (15*c*d^4 - 35*b*d^3*e + 63*a*d^2*e
^2)*x^5)*sqrt(-e/d)*log((e*x^2 - 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)))/(d^5*e^2*x^9 + 2*d^6*e*x^7 + d^7*x^5), -1
/120*(15*(15*c*d^2*e^2 - 35*b*d*e^3 + 63*a*e^4)*x^8 + 25*(15*c*d^3*e - 35*b*d^2*e^2 + 63*a*d*e^3)*x^6 + 24*a*d
^4 + 8*(15*c*d^4 - 35*b*d^3*e + 63*a*d^2*e^2)*x^4 + 8*(5*b*d^4 - 9*a*d^3*e)*x^2 + 15*((15*c*d^2*e^2 - 35*b*d*e
^3 + 63*a*e^4)*x^9 + 2*(15*c*d^3*e - 35*b*d^2*e^2 + 63*a*d*e^3)*x^7 + (15*c*d^4 - 35*b*d^3*e + 63*a*d^2*e^2)*x
^5)*sqrt(e/d)*arctan(x*sqrt(e/d)))/(d^5*e^2*x^9 + 2*d^6*e*x^7 + d^7*x^5)]

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Sympy [B]  time = 4.76643, size = 330, normalized size = 1.93 \begin{align*} \frac{\sqrt{- \frac{e}{d^{11}}} \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right ) \log{\left (- \frac{d^{6} \sqrt{- \frac{e}{d^{11}}} \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right )}{63 a e^{3} - 35 b d e^{2} + 15 c d^{2} e} + x \right )}}{16} - \frac{\sqrt{- \frac{e}{d^{11}}} \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right ) \log{\left (\frac{d^{6} \sqrt{- \frac{e}{d^{11}}} \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right )}{63 a e^{3} - 35 b d e^{2} + 15 c d^{2} e} + x \right )}}{16} - \frac{24 a d^{4} + x^{8} \left (945 a e^{4} - 525 b d e^{3} + 225 c d^{2} e^{2}\right ) + x^{6} \left (1575 a d e^{3} - 875 b d^{2} e^{2} + 375 c d^{3} e\right ) + x^{4} \left (504 a d^{2} e^{2} - 280 b d^{3} e + 120 c d^{4}\right ) + x^{2} \left (- 72 a d^{3} e + 40 b d^{4}\right )}{120 d^{7} x^{5} + 240 d^{6} e x^{7} + 120 d^{5} e^{2} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**6/(e*x**2+d)**3,x)

[Out]

sqrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)*log(-d**6*sqrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)/
(63*a*e**3 - 35*b*d*e**2 + 15*c*d**2*e) + x)/16 - sqrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)*log(d**6*s
qrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)/(63*a*e**3 - 35*b*d*e**2 + 15*c*d**2*e) + x)/16 - (24*a*d**4
+ x**8*(945*a*e**4 - 525*b*d*e**3 + 225*c*d**2*e**2) + x**6*(1575*a*d*e**3 - 875*b*d**2*e**2 + 375*c*d**3*e) +
x**4*(504*a*d**2*e**2 - 280*b*d**3*e + 120*c*d**4) + x**2*(-72*a*d**3*e + 40*b*d**4))/(120*d**7*x**5 + 240*d*
*6*e*x**7 + 120*d**5*e**2*x**9)

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Giac [A]  time = 1.12647, size = 221, normalized size = 1.29 \begin{align*} -\frac{{\left (15 \, c d^{2} e - 35 \, b d e^{2} + 63 \, a e^{3}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{8 \, d^{\frac{11}{2}}} - \frac{7 \, c d^{2} x^{3} e^{2} - 11 \, b d x^{3} e^{3} + 9 \, c d^{3} x e + 15 \, a x^{3} e^{4} - 13 \, b d^{2} x e^{2} + 17 \, a d x e^{3}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{5}} - \frac{15 \, c d^{2} x^{4} - 45 \, b d x^{4} e + 90 \, a x^{4} e^{2} + 5 \, b d^{2} x^{2} - 15 \, a d x^{2} e + 3 \, a d^{2}}{15 \, d^{5} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x, algorithm="giac")

[Out]

-1/8*(15*c*d^2*e - 35*b*d*e^2 + 63*a*e^3)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(11/2) - 1/8*(7*c*d^2*x^3*e^2 -
11*b*d*x^3*e^3 + 9*c*d^3*x*e + 15*a*x^3*e^4 - 13*b*d^2*x*e^2 + 17*a*d*x*e^3)/((x^2*e + d)^2*d^5) - 1/15*(15*c
*d^2*x^4 - 45*b*d*x^4*e + 90*a*x^4*e^2 + 5*b*d^2*x^2 - 15*a*d*x^2*e + 3*a*d^2)/(d^5*x^5)