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

Optimal. Leaf size=127 \[ -\frac{x \left (a e^2-b d e+c d^2\right )}{4 d^2 e \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 e (b d-5 a e)+c d^2\right )}{8 d^{7/2} e^{3/2}}+\frac{x \left (e (3 b d-7 a e)+c d^2\right )}{8 d^3 e \left (d+e x^2\right )}-\frac{a}{d^3 x} \]

[Out]

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

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Rubi [A]  time = 0.204063, antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1259, 456, 453, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 e (b d-5 a e)+c d^2\right )}{8 d^{7/2} e^{3/2}}+\frac{x \left (e (3 b d-7 a e)+c d^2\right )}{8 d^3 e \left (d+e x^2\right )}-\frac{x \left (\frac{c}{e}-\frac{b d-a e}{d^2}\right )}{4 \left (d+e x^2\right )^2}-\frac{a}{d^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 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{a+b x^2+c x^4}{x^2 \left (d+e x^2\right )^3} \, dx &=-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}-\frac{\int \frac{-4 a d e^2-e \left (c d^2+3 e (b d-a e)\right ) x^2}{x^2 \left (d+e x^2\right )^2} \, dx}{4 d^2 e^2}\\ &=-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}+\frac{\left (c d^2+e (3 b d-7 a e)\right ) x}{8 d^3 e \left (d+e x^2\right )}+\frac{\int \frac{8 a e^2+e \left (c d+e \left (3 b-\frac{7 a e}{d}\right )\right ) x^2}{x^2 \left (d+e x^2\right )} \, dx}{8 d^2 e^2}\\ &=-\frac{a}{d^3 x}-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}+\frac{\left (c d^2+e (3 b d-7 a e)\right ) x}{8 d^3 e \left (d+e x^2\right )}+\frac{\left (c d^2+3 e (b d-5 a e)\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^3 e}\\ &=-\frac{a}{d^3 x}-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}+\frac{\left (c d^2+e (3 b d-7 a e)\right ) x}{8 d^3 e \left (d+e x^2\right )}+\frac{\left (c d^2+3 e (b d-5 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{7/2} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.140273, size = 124, normalized size = 0.98 \[ \frac{\frac{\sqrt{d} \left (d x^2 \left (b e \left (5 d+3 e x^2\right )+c d \left (e x^2-d\right )\right )-a e \left (8 d^2+25 d e x^2+15 e^2 x^4\right )\right )}{e x \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 e (b d-5 a e)+c d^2\right )}{e^{3/2}}}{8 d^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[d]*(-(a*e*(8*d^2 + 25*d*e*x^2 + 15*e^2*x^4)) + d*x^2*(c*d*(-d + e*x^2) + b*e*(5*d + 3*e*x^2))))/(e*x*(d
+ e*x^2)^2) + ((c*d^2 + 3*e*(b*d - 5*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2))/(8*d^(7/2))

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Maple [A]  time = 0.012, size = 182, normalized size = 1.4 \begin{align*} -{\frac{a}{{d}^{3}x}}-{\frac{7\,{x}^{3}a{e}^{2}}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{x}^{3}be}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{{x}^{3}c}{8\,d \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{9\,aex}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,bx}{8\,d \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{xc}{8\, \left ( e{x}^{2}+d \right ) ^{2}e}}-{\frac{15\,ae}{8\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,b}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c}{8\,de}\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^2/(e*x^2+d)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77073, size = 887, normalized size = 6.98 \begin{align*} \left [-\frac{16 \, a d^{3} e^{2} - 2 \,{\left (c d^{3} e^{2} + 3 \, b d^{2} e^{3} - 15 \, a d e^{4}\right )} x^{4} + 2 \,{\left (c d^{4} e - 5 \, b d^{3} e^{2} + 25 \, a d^{2} e^{3}\right )} x^{2} -{\left ({\left (c d^{2} e^{2} + 3 \, b d e^{3} - 15 \, a e^{4}\right )} x^{5} + 2 \,{\left (c d^{3} e + 3 \, b d^{2} e^{2} - 15 \, a d e^{3}\right )} x^{3} +{\left (c d^{4} + 3 \, b d^{3} e - 15 \, a d^{2} e^{2}\right )} x\right )} \sqrt{-d e} \log \left (\frac{e x^{2} + 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right )}{16 \,{\left (d^{4} e^{4} x^{5} + 2 \, d^{5} e^{3} x^{3} + d^{6} e^{2} x\right )}}, -\frac{8 \, a d^{3} e^{2} -{\left (c d^{3} e^{2} + 3 \, b d^{2} e^{3} - 15 \, a d e^{4}\right )} x^{4} +{\left (c d^{4} e - 5 \, b d^{3} e^{2} + 25 \, a d^{2} e^{3}\right )} x^{2} -{\left ({\left (c d^{2} e^{2} + 3 \, b d e^{3} - 15 \, a e^{4}\right )} x^{5} + 2 \,{\left (c d^{3} e + 3 \, b d^{2} e^{2} - 15 \, a d e^{3}\right )} x^{3} +{\left (c d^{4} + 3 \, b d^{3} e - 15 \, a d^{2} e^{2}\right )} x\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right )}{8 \,{\left (d^{4} e^{4} x^{5} + 2 \, d^{5} e^{3} x^{3} + d^{6} e^{2} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.42076, size = 202, normalized size = 1.59 \begin{align*} \frac{\sqrt{- \frac{1}{d^{7} e^{3}}} \left (15 a e^{2} - 3 b d e - c d^{2}\right ) \log{\left (- d^{4} e \sqrt{- \frac{1}{d^{7} e^{3}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{d^{7} e^{3}}} \left (15 a e^{2} - 3 b d e - c d^{2}\right ) \log{\left (d^{4} e \sqrt{- \frac{1}{d^{7} e^{3}}} + x \right )}}{16} - \frac{8 a d^{2} e + x^{4} \left (15 a e^{3} - 3 b d e^{2} - c d^{2} e\right ) + x^{2} \left (25 a d e^{2} - 5 b d^{2} e + c d^{3}\right )}{8 d^{5} e x + 16 d^{4} e^{2} x^{3} + 8 d^{3} e^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09502, size = 149, normalized size = 1.17 \begin{align*} \frac{{\left (c d^{2} + 3 \, b d e - 15 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{3}{2}\right )}}{8 \, d^{\frac{7}{2}}} - \frac{a}{d^{3} x} + \frac{{\left (c d^{2} x^{3} e + 3 \, b d x^{3} e^{2} - c d^{3} x - 7 \, a x^{3} e^{3} + 5 \, b d^{2} x e - 9 \, a d x e^{2}\right )} e^{\left (-1\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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