3.196 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{(a+b x^4)^2} \, dx\)

Optimal. Leaf size=395 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )} \]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(4*a*b*(a + b*x^4)) + ((b*d + a*h)*ArcTan[(Sqrt[b]
*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(
1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 + (S
qrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) -
Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4))

________________________________________________________________________________________

Rubi [A]  time = 0.492387, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1858, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^2,x]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(4*a*b*(a + b*x^4)) + ((b*d + a*h)*ArcTan[(Sqrt[b]
*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(
1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 + (S
qrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) -
Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4))

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+196 x^6}{\left (a+b x^4\right )^2} \, dx &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac{\int \frac{-b (3 b c+a g)-2 b (b d+a h) x-b (588 a+b e) x^2}{a+b x^4} \, dx}{4 a b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac{\int \left (-\frac{2 b (b d+a h) x}{a+b x^4}+\frac{-b (3 b c+a g)-b (588 a+b e) x^2}{a+b x^4}\right ) \, dx}{4 a b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac{\int \frac{-b (3 b c+a g)-b (588 a+b e) x^2}{a+b x^4} \, dx}{4 a b^2}+\frac{(b d+a h) \int \frac{x}{a+b x^4} \, dx}{2 a b}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{8 a b^2}+\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{8 a b^2}+\frac{(b d+a h) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{4 a b}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac{(b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\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}{16 \sqrt{2} a^{5/4} b^{7/4}}+\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\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}{16 \sqrt{2} a^{5/4} b^{7/4}}+\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b^2}+\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac{(b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{5/4} b^{7/4}}-\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{5/4} b^{7/4}}+\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(196 a-b e) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac{(b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}-\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}+\frac{\left (588 a+b e+\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}+\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{5/4} b^{7/4}}-\frac{\left (588 a+b e-\frac{\sqrt{b} (3 b c+a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{5/4} b^{7/4}}\\ \end{align*}

Mathematica [A]  time = 0.344446, size = 415, normalized size = 1.05 \[ \frac{-\frac{8 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{a+b x^4}-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i+4 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i-4 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right )+\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 a^{3/2} i+\sqrt{a} b e-a \sqrt{b} g-3 b^{3/2} c\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-3 a^{3/2} i-\sqrt{a} b e+a \sqrt{b} g+3 b^{3/2} c\right )}{32 a^{7/4} b^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^2,x]

[Out]

((-8*a^(3/4)*b^(3/4)*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x)))))/(a + b*x^4) - 2*(3*Sqrt[2]*b^(3
/2)*c + 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3
/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b^(3/2)*c - 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a
]*b*e + Sqrt[2]*a*Sqrt[b]*g - 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4
)] + Sqrt[2]*(-3*b^(3/2)*c + Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2] + Sqrt[2]*(3*b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(7/4))

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 654, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x)

[Out]

(-1/4*(a*i-b*e)/a/b*x^3-1/4*(a*h-b*d)/a/b*x^2-1/4*(a*g-b*c)/a/b*x-1/4*f/b)/(b*x^4+a)+1/16/b/a*(1/b*a)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*g+3/16*c/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+1
/16/b/a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*g+3/16*c/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(1/b*a)^(1/4)*x-1)+1/32/b/a*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b
*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*g+3/32*c/a^2*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)
^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))+1/4/b/(a*b)^(1/2)*arctan(x^2*(b/a)^(1/2))*h+1/4*d/a/(a*b)
^(1/2)*arctan(x^2*(b/a)^(1/2))+3/32/b^2/(1/b*a)^(1/4)*2^(1/2)*ln((x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(
x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*i+1/32*e/a/b/(1/b*a)^(1/4)*2^(1/2)*ln((x^2-(1/b*a)^(1/4)*x*2^(1/2)
+(1/b*a)^(1/2))/(x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))+3/16/b^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/
b*a)^(1/4)*x+1)*i+1/16*e/a/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+3/16/b^2/(1/b*a)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*i+1/16*e/a/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)

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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((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,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((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,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((i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.09558, size = 795, normalized size = 2.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x, algorithm="giac")

[Out]

3/32*i*(2*sqrt(2)*(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) - sqrt(2)*
(a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^4)) + 3/32*i*(2*sqrt(2)*(a*b^3)^(3/4)*arctan(1
/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + sqrt(2)*(a*b^3)^(3/4)*log(x^2 - sqrt(2)*x*(a/b)^
(1/4) + sqrt(a/b))/(a*b^4)) - 1/4*(a*i*x^3 - b*x^3*e - b*d*x^2 + a*h*x^2 - b*c*x + a*g*x + a*f)/((b*x^4 + a)*a
*b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 2*sqrt(2)*sqrt(a*b)*a*b*h + 3*(a*b^3)^(1/4)*b^2*c + (a*b^3)^(1
/4)*a*b*g + (a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^3) + 1/16*sqrt
(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 2*sqrt(2)*sqrt(a*b)*a*b*h + 3*(a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g + (a*
b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^3) + 1/32*sqrt(2)*(3*(a*b^3)^
(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g - (a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^3) -
1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g - (a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) +
sqrt(a/b))/(a^2*b^3)