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

Optimal. Leaf size=417 \[ -\frac{c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{\sqrt{a} d^3 \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} \left (a d^4+b c^4\right )}+\frac{c^2 d \log (c+d x)}{a d^4+b c^4}-\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.546728, antiderivative size = 417, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6725, 1461, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 205, 260} \[ -\frac{c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{\sqrt{a} d^3 \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} \left (a d^4+b c^4\right )}+\frac{c^2 d \log (c+d x)}{a d^4+b c^4}-\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((c + d*x)*(a + b*x^4)),x]

[Out]

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 1461

Int[((A_) + (B_.)*(x_)^(m_.))*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Dis
t[A, Int[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] + Dist[B, Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /;
FreeQ[{a, c, d, e, A, B, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[m - n + 1, 0]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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]

Rule 1248

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

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

Mathematica [A]  time = 0.250189, size = 370, normalized size = 0.89 \[ \frac{-2 \left (2 a^{3/4} d^3-\sqrt{2} \sqrt{a} \sqrt [4]{b} c d^2+\sqrt{2} b^{3/4} c^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (-2 a^{3/4} d^3-\sqrt{2} \sqrt{a} \sqrt [4]{b} c d^2+\sqrt{2} b^{3/4} c^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )+\sqrt [4]{b} c \left (\sqrt{2} \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\sqrt{2} \sqrt{b} c^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-2 \sqrt [4]{a} \sqrt [4]{b} c d \log \left (a+b x^4\right )+8 \sqrt [4]{a} \sqrt [4]{b} c d \log (c+d x)-\sqrt{2} \sqrt{a} d^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )\right )}{8 \sqrt [4]{a} \sqrt{b} \left (a d^4+b c^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(Sqrt[2]*b^(3/4)*c^3 - Sqrt[2]*Sqrt[a]*b^(1/4)*c*d^2 + 2*a^(3/4)*d^3)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/
4)] + 2*(Sqrt[2]*b^(3/4)*c^3 - Sqrt[2]*Sqrt[a]*b^(1/4)*c*d^2 - 2*a^(3/4)*d^3)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a
^(1/4)] + b^(1/4)*c*(8*a^(1/4)*b^(1/4)*c*d*Log[c + d*x] + Sqrt[2]*(Sqrt[b]*c^2 + Sqrt[a]*d^2)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - Sqrt[2]*Sqrt[b]*c^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]
*x^2] - Sqrt[2]*Sqrt[a]*d^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - 2*a^(1/4)*b^(1/4)*c*d*Log
[a + b*x^4]))/(8*a^(1/4)*Sqrt[b]*(b*c^4 + a*d^4))

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Maple [A]  time = 0.013, size = 422, normalized size = 1. \begin{align*} -{\frac{c{d}^{2}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{c{d}^{2}\sqrt{2}}{8\,a{d}^{4}+8\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{c{d}^{2}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{a{d}^{3}}{2\,a{d}^{4}+2\,b{c}^{4}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}\sqrt{2}}{8\,a{d}^{4}+8\,b{c}^{4}}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{c}^{3}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{c}^{3}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}d\ln \left ( b{x}^{4}+a \right ) }{4\,a{d}^{4}+4\,b{c}^{4}}}+{\frac{{c}^{2}d\ln \left ( dx+c \right ) }{a{d}^{4}+b{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(a*d^4+b*c^4)*c*d^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)-1/8/(a*d^4+b*c^4)*c*d^2*(a/b)^(1/
4)*2^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))-1/4/(a*d^4+b*c^
4)*c*d^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/2/(a*d^4+b*c^4)*a*d^3/(a*b)^(1/2)*arctan(x^2*(b
/a)^(1/2))+1/8/(a*d^4+b*c^4)*c^3/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/
4)*x*2^(1/2)+(a/b)^(1/2)))+1/4/(a*d^4+b*c^4)*c^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/4/(a*d^
4+b*c^4)*c^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)-1/4*c^2*d*ln(b*x^4+a)/(a*d^4+b*c^4)+c^2*d*ln(
d*x+c)/(a*d^4+b*c^4)

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

[Out]

Timed out

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Giac [A]  time = 1.20294, size = 575, normalized size = 1.38 \begin{align*} \frac{c^{2} d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{4} d + a d^{5}} - \frac{c^{2} d \log \left ({\left | b x^{4} + a \right |}\right )}{4 \,{\left (b c^{4} + a d^{4}\right )}} - \frac{{\left (\sqrt{2} a^{2} b^{3} d - \left (a b^{3}\right )^{\frac{3}{4}} a b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} b^{4} c^{2} + \sqrt{2} \sqrt{a b} a^{2} b^{3} d^{2} - 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} c d\right )}} + \frac{{\left (\sqrt{2} a^{2} b^{3} d + \left (a b^{3}\right )^{\frac{3}{4}} a b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} b^{4} c^{2} + \sqrt{2} \sqrt{a b} a^{2} b^{3} d^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} c d\right )}} - \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} c^{3}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{4 \,{\left (\sqrt{2} a b^{3} c^{4} + \sqrt{2} a^{2} b^{2} d^{4}\right )}} + \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} c^{3}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{4 \,{\left (\sqrt{2} a b^{3} c^{4} + \sqrt{2} a^{2} b^{2} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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