3.302 \(\int \frac{1}{x^5 (d+e x^2) (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=268 \[ -\frac{\left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (a e^2-b d e+c d^2\right )}+\frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{\log (x) \left (a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 d^3}+\frac{a e+b d}{2 a^2 d^2 x^2}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2-b d e+c d^2\right )}-\frac{1}{4 a d x^4} \]

[Out]

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

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Rubi [A]  time = 0.596883, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1251, 893, 634, 618, 206, 628} \[ -\frac{\left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (a e^2-b d e+c d^2\right )}+\frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{\log (x) \left (a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 d^3}+\frac{a e+b d}{2 a^2 d^2 x^2}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2-b d e+c d^2\right )}-\frac{1}{4 a d x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

Rule 1251

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

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a d x^3}+\frac{-b d-a e}{a^2 d^2 x^2}+\frac{b^2 d^2+a b d e-a \left (c d^2-a e^2\right )}{a^3 d^3 x}-\frac{e^5}{d^3 \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{-b^3 c d+2 a b c^2 d+b^4 e-3 a b^2 c e+a^2 c^2 e-c \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{a^3 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 a d x^4}+\frac{b d+a e}{2 a^2 d^2 x^2}+\frac{\left (b^2 d^2+a b d e-a \left (c d^2-a e^2\right )\right ) \log (x)}{a^3 d^3}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2-b d e+a e^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 c d+2 a b c^2 d+b^4 e-3 a b^2 c e+a^2 c^2 e-c \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{4 a d x^4}+\frac{b d+a e}{2 a^2 d^2 x^2}+\frac{\left (b^2 d^2+a b d e-a \left (c d^2-a e^2\right )\right ) \log (x)}{a^3 d^3}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{4 a d x^4}+\frac{b d+a e}{2 a^2 d^2 x^2}+\frac{\left (b^2 d^2+a b d e-a \left (c d^2-a e^2\right )\right ) \log (x)}{a^3 d^3}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{4 a d x^4}+\frac{b d+a e}{2 a^2 d^2 x^2}+\frac{\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^2 d^2+a b d e-a \left (c d^2-a e^2\right )\right ) \log (x)}{a^3 d^3}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.426105, size = 426, normalized size = 1.59 \[ \frac{1}{4} \left (-\frac{\left (a c^2 \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 \left (e \sqrt{b^2-4 a c}-c d\right )-b^2 c \left (d \sqrt{b^2-4 a c}+4 a e\right )+a b c \left (3 c d-2 e \sqrt{b^2-4 a c}\right )+b^4 e\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^3 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}-\frac{\left (a c^2 \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 \left (e \sqrt{b^2-4 a c}+c d\right )+b^2 c \left (4 a e-d \sqrt{b^2-4 a c}\right )-a b c \left (2 e \sqrt{b^2-4 a c}+3 c d\right )+b^4 (-e)\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^3 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}+\frac{4 \log (x) \left (a b d e+a \left (a e^2-c d^2\right )+b^2 d^2\right )}{a^3 d^3}+\frac{2 (a e+b d)}{a^2 d^2 x^2}-\frac{2 e^4 \log \left (d+e x^2\right )}{d^3 e (a e-b d)+c d^5}-\frac{1}{a d x^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

(-(1/(a*d*x^4)) + (2*(b*d + a*e))/(a^2*d^2*x^2) + (4*(b^2*d^2 + a*b*d*e + a*(-(c*d^2) + a*e^2))*Log[x])/(a^3*d
^3) - ((b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*S
qrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^3*Sqrt[b^2
- 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) - ((-(b^4*e) + a*c^2*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 -
4*a*c]*d) + 4*a*e) + b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*Log[b + Sqrt[b^2
- 4*a*c] + 2*c*x^2])/(a^3*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) - (2*e^4*Log[d + e*x^2])/(c*d^5 + d^3
*e*(-(b*d) + a*e)))/4

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Maple [B]  time = 0.019, size = 584, normalized size = 2.2 \begin{align*} -{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) eb}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){a}^{2}}}+{\frac{{c}^{2}\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){a}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e{b}^{3}}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){a}^{3}}}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d{b}^{2}}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){a}^{3}}}+{\frac{e{c}^{2}}{ \left ( a{e}^{2}-deb+c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{ec{b}^{2}}{ \left ( a{e}^{2}-deb+c{d}^{2} \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,d{c}^{2}b}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}e}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}cd}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{4\,ad{x}^{4}}}+{\frac{e}{2\,a{d}^{2}{x}^{2}}}+{\frac{b}{2\,d{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ){e}^{2}}{a{d}^{3}}}+{\frac{\ln \left ( x \right ) be}{{d}^{2}{a}^{2}}}-{\frac{\ln \left ( x \right ) c}{d{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}d}}-{\frac{{e}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{3} \left ( a{e}^{2}-deb+c{d}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(e*x^2+d)/(c*x^4+b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [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^5/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [A]  time = 1.18515, size = 448, normalized size = 1.67 \begin{align*} -\frac{{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )}} - \frac{e^{5} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{5} e - b d^{4} e^{2} + a d^{3} e^{3}\right )}} - \frac{{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (b^{2} d^{2} - a c d^{2} + a b d e + a^{2} e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} d^{3}} - \frac{3 \, b^{2} d^{2} x^{4} - 3 \, a c d^{2} x^{4} + 3 \, a b d x^{4} e + 3 \, a^{2} x^{4} e^{2} - 2 \, a b d^{2} x^{2} - 2 \, a^{2} d x^{2} e + a^{2} d^{2}}{4 \, a^{3} d^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/4*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*log(c*x^4 + b*x^2 + a)/(a^3*c*d^2 - a^3*b*d*e + a^4*e^2) - 1/2*e^
5*log(abs(x^2*e + d))/(c*d^5*e - b*d^4*e^2 + a*d^3*e^3) - 1/2*(b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2
*a^2*c^2*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*sqrt(-b^2 + 4*a*c)) +
1/2*(b^2*d^2 - a*c*d^2 + a*b*d*e + a^2*e^2)*log(x^2)/(a^3*d^3) - 1/4*(3*b^2*d^2*x^4 - 3*a*c*d^2*x^4 + 3*a*b*d*
x^4*e + 3*a^2*x^4*e^2 - 2*a*b*d^2*x^2 - 2*a^2*d*x^2*e + a^2*d^2)/(a^3*d^3*x^4)