3.235 \(\int \frac{1}{x^3 (d+e x^2) (a+c x^4)} \, dx\)

Optimal. Leaf size=129 \[ -\frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2+c d^2\right )}+\frac{c e \log \left (a+c x^4\right )}{4 a \left (a e^2+c d^2\right )}-\frac{e \log (x)}{a d^2}-\frac{1}{2 a d x^2} \]

[Out]

-1/(2*a*d*x^2) - (c^(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2)*(c*d^2 + a*e^2)) - (e*Log[x])/(a*d^2) +
(e^3*Log[d + e*x^2])/(2*d^2*(c*d^2 + a*e^2)) + (c*e*Log[a + c*x^4])/(4*a*(c*d^2 + a*e^2))

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Rubi [A]  time = 0.149769, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1252, 894, 635, 205, 260} \[ -\frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2+c d^2\right )}+\frac{c e \log \left (a+c x^4\right )}{4 a \left (a e^2+c d^2\right )}-\frac{e \log (x)}{a d^2}-\frac{1}{2 a d x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x^2)*(a + c*x^4)),x]

[Out]

-1/(2*a*d*x^2) - (c^(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2)*(c*d^2 + a*e^2)) - (e*Log[x])/(a*d^2) +
(e^3*Log[d + e*x^2])/(2*d^2*(c*d^2 + a*e^2)) + (c*e*Log[a + c*x^4])/(4*a*(c*d^2 + a*e^2))

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a d x^2}-\frac{e}{a d^2 x}+\frac{e^4}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac{c^2 (d-e x)}{a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a d x^2}-\frac{e \log (x)}{a d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}-\frac{c^2 \operatorname{Subst}\left (\int \frac{d-e x}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac{1}{2 a d x^2}-\frac{e \log (x)}{a d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}-\frac{\left (c^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}+\frac{\left (c^2 e\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac{1}{2 a d x^2}-\frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )}-\frac{e \log (x)}{a d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}+\frac{c e \log \left (a+c x^4\right )}{4 a \left (c d^2+a e^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.103158, size = 169, normalized size = 1.31 \[ \frac{2 c^{3/2} d^3 x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 c^{3/2} d^3 x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{a} \left (-4 e x^2 \log (x) \left (a e^2+c d^2\right )+c d^2 e x^2 \log \left (a+c x^4\right )+2 a e^3 x^2 \log \left (d+e x^2\right )-2 a d e^2-2 c d^3\right )}{4 a^{3/2} d^2 x^2 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x^2)*(a + c*x^4)),x]

[Out]

(2*c^(3/2)*d^3*x^2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*c^(3/2)*d^3*x^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/
a^(1/4)] + Sqrt[a]*(-2*c*d^3 - 2*a*d*e^2 - 4*e*(c*d^2 + a*e^2)*x^2*Log[x] + 2*a*e^3*x^2*Log[d + e*x^2] + c*d^2
*e*x^2*Log[a + c*x^4]))/(4*a^(3/2)*d^2*(c*d^2 + a*e^2)*x^2)

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Maple [A]  time = 0.01, size = 119, normalized size = 0.9 \begin{align*}{\frac{ec\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) a}}-{\frac{{c}^{2}d}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{1}{2\,ad{x}^{2}}}-{\frac{\ln \left ( x \right ) e}{{d}^{2}a}}+{\frac{{e}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(e*x^2+d)/(c*x^4+a),x)

[Out]

1/4*c*e*ln(c*x^4+a)/a/(a*e^2+c*d^2)-1/2*c^2/(a*e^2+c*d^2)/a*d/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))-1/2/a/d/x^
2-e*ln(x)/a/d^2+1/2*e^3*ln(e*x^2+d)/d^2/(a*e^2+c*d^2)

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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(1/x^3/(e*x^2+d)/(c*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 178.174, size = 574, normalized size = 4.45 \begin{align*} \left [\frac{c d^{3} x^{2} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) + c d^{2} e x^{2} \log \left (c x^{4} + a\right ) + 2 \, a e^{3} x^{2} \log \left (e x^{2} + d\right ) - 2 \, c d^{3} - 2 \, a d e^{2} - 4 \,{\left (c d^{2} e + a e^{3}\right )} x^{2} \log \left (x\right )}{4 \,{\left (a c d^{4} + a^{2} d^{2} e^{2}\right )} x^{2}}, \frac{2 \, c d^{3} x^{2} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) + c d^{2} e x^{2} \log \left (c x^{4} + a\right ) + 2 \, a e^{3} x^{2} \log \left (e x^{2} + d\right ) - 2 \, c d^{3} - 2 \, a d e^{2} - 4 \,{\left (c d^{2} e + a e^{3}\right )} x^{2} \log \left (x\right )}{4 \,{\left (a c d^{4} + a^{2} d^{2} e^{2}\right )} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x^2+d)/(c*x^4+a),x, algorithm="fricas")

[Out]

[1/4*(c*d^3*x^2*sqrt(-c/a)*log((c*x^4 - 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) + c*d^2*e*x^2*log(c*x^4 + a) + 2*
a*e^3*x^2*log(e*x^2 + d) - 2*c*d^3 - 2*a*d*e^2 - 4*(c*d^2*e + a*e^3)*x^2*log(x))/((a*c*d^4 + a^2*d^2*e^2)*x^2)
, 1/4*(2*c*d^3*x^2*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) + c*d^2*e*x^2*log(c*x^4 + a) + 2*a*e^3*x^2*log(e*x^2
+ d) - 2*c*d^3 - 2*a*d*e^2 - 4*(c*d^2*e + a*e^3)*x^2*log(x))/((a*c*d^4 + a^2*d^2*e^2)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(e*x**2+d)/(c*x**4+a),x)

[Out]

Timed out

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Giac [A]  time = 1.10962, size = 178, normalized size = 1.38 \begin{align*} -\frac{c^{2} d \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \,{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt{a c}} + \frac{c e \log \left (c x^{4} + a\right )}{4 \,{\left (a c d^{2} + a^{2} e^{2}\right )}} + \frac{e^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{4} e + a d^{2} e^{3}\right )}} - \frac{e \log \left (x^{2}\right )}{2 \, a d^{2}} + \frac{x^{2} e - d}{2 \, a d^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x^2+d)/(c*x^4+a),x, algorithm="giac")

[Out]

-1/2*c^2*d*arctan(c*x^2/sqrt(a*c))/((a*c*d^2 + a^2*e^2)*sqrt(a*c)) + 1/4*c*e*log(c*x^4 + a)/(a*c*d^2 + a^2*e^2
) + 1/2*e^4*log(abs(x^2*e + d))/(c*d^4*e + a*d^2*e^3) - 1/2*e*log(x^2)/(a*d^2) + 1/2*(x^2*e - d)/(a*d^2*x^2)