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

Optimal. Leaf size=357 \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}-\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{2 d^3 x^5 (2 b c-a d)}{5 b^3}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^4\right )}+\frac{d^4 x^9}{9 b^2} \]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^5)/(5*b^3) + (d^4*x^9)/(9*b^2) + ((b*
c - a*d)^4*x)/(4*a*b^4*(a + b*x^4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(8*Sqrt[2]*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sq
rt[2]*a^(7/4)*b^(17/4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^
2])/(16*Sqrt[2]*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(17/4))

________________________________________________________________________________________

Rubi [A]  time = 0.36656, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {390, 385, 211, 1165, 628, 1162, 617, 204} \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}-\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{2 d^3 x^5 (2 b c-a d)}{5 b^3}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^4\right )}+\frac{d^4 x^9}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^4)^4/(a + b*x^4)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^5)/(5*b^3) + (d^4*x^9)/(9*b^2) + ((b*
c - a*d)^4*x)/(4*a*b^4*(a + b*x^4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(8*Sqrt[2]*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sq
rt[2]*a^(7/4)*b^(17/4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^
2])/(16*Sqrt[2]*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(17/4))

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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 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])

Rubi steps

\begin{align*} \int \frac{\left (c+d x^4\right )^4}{\left (a+b x^4\right )^2} \, dx &=\int \left (\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac{2 d^3 (2 b c-a d) x^4}{b^3}+\frac{d^4 x^8}{b^2}+\frac{(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^4}{b^4 \left (a+b x^4\right )^2}\right ) \, dx\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac{d^4 x^9}{9 b^2}+\frac{\int \frac{(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^4}{\left (a+b x^4\right )^2} \, dx}{b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac{d^4 x^9}{9 b^2}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}+\frac{\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac{1}{a+b x^4} \, dx}{4 a b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac{d^4 x^9}{9 b^2}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}+\frac{\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^4}+\frac{\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac{d^4 x^9}{9 b^2}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}+\frac{\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{9/2}}+\frac{\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{9/2}}-\frac{\left ((b c-a d)^3 (3 b c+13 a d)\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^{7/4} b^{17/4}}-\frac{\left ((b c-a d)^3 (3 b c+13 a d)\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^{7/4} b^{17/4}}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac{d^4 x^9}{9 b^2}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}-\frac{(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{\left ((b c-a d)^3 (3 b c+13 a d)\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^{7/4} b^{17/4}}-\frac{\left ((b c-a d)^3 (3 b c+13 a d)\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^{7/4} b^{17/4}}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac{d^4 x^9}{9 b^2}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}-\frac{(b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}-\frac{(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}\\ \end{align*}

Mathematica [A]  time = 0.28772, size = 341, normalized size = 0.96 \[ \frac{1440 \sqrt [4]{b} d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )+\frac{45 \sqrt{2} (a d-b c)^3 (13 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{45 \sqrt{2} (b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{90 \sqrt{2} (a d-b c)^3 (13 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{90 \sqrt{2} (b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+576 b^{5/4} d^3 x^5 (2 b c-a d)+\frac{360 \sqrt [4]{b} x (b c-a d)^4}{a \left (a+b x^4\right )}+160 b^{9/4} d^4 x^9}{1440 b^{17/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1440*b^(1/4)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 576*b^(5/4)*d^3*(2*b*c - a*d)*x^5 + 160*b^(9/4)*d^4*
x^9 + (360*b^(1/4)*(b*c - a*d)^4*x)/(a*(a + b*x^4)) + (90*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (90*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4
)*x)/a^(1/4)])/a^(7/4) + (45*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/a^(7/4) + (45*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/a^(7/4))/(1440*b^(17/4))

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 885, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*d^4*x^9/b^2-8*d^3/b^3*c*a*x+1/4/b^4*a^3*x/(b*x^4+a)*d^4+1/8/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)))*c^3*d+9/4/b^3*a*(1/b*a)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c*d^3+1/4/b/a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c^3*d-
13/16/b^4*a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*d^4-15/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arcta
n(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^2*d^2-13/32/b^4*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)))*d^4-15/16/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)))*c^2*d^2-13/16/b^4*a^2*(1/b*a)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*d^4-15/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*
c^2*d^2-1/b^3*a^2*x/(b*x^4+a)*c*d^3+3/2/b^2*a*x/(b*x^4+a)*c^2*d^2-1/b*x/(b*x^4+a)*c^3*d+3/16/a^2*(1/b*a)^(1/4)
*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^4+3/32/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)))*c^4+3/16/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(1/b*a)^(1/4)*x+1)*c^4+9/4/b^3*a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c*d^3+1/4/b/a*(1/b*a)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^3*d+9/8/b^3*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)))*c*d^3+1/4/a*x/(b*x^4+a)*c^4-2/5*d^4/b^3*x
^5*a+4/5*d^3/b^2*x^5*c+3*d^4/b^4*a^2*x+6*d^2/b^2*c^2*x

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.21363, size = 6400, normalized size = 17.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/720*(80*a*b^3*d^4*x^13 + 16*(36*a*b^3*c*d^3 - 13*a^2*b^2*d^4)*x^9 + 144*(30*a*b^3*c^2*d^2 - 36*a^2*b^2*c*d^3
+ 13*a^3*b*d^4)*x^5 - 180*(a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2
- 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 7721
12*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a
^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*
a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4)*arctan((a^5*b^13*x*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 23
76*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6
*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*
c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b
^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(3/4) - a^5*b^13*sqrt((a^4*b^8*sqrt(-(81*b^1
6*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400
*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*
b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^1
3*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17)) + (9*b^8*c^8
+ 24*a*b^7*c^7*d - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 20
76*a^6*b^2*c^2*d^6 - 936*a^7*b*c*d^7 + 169*a^8*d^8)*x^2)/(9*b^8*c^8 + 24*a*b^7*c^7*d - 164*a^2*b^6*c^6*d^2 - 2
4*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 - 936*a^7*b*c*d^7 + 169
*a^8*d^8))*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b
^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^
8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^
4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*
b^17))^(3/4))/(27*b^12*c^12 + 108*a*b^11*c^11*d - 666*a^2*b^10*c^10*d^2 - 1124*a^3*b^9*c^9*d^3 + 8901*a^4*b^8*
c^8*d^4 - 7848*a^5*b^7*c^7*d^5 - 34860*a^6*b^6*c^6*d^6 + 113688*a^7*b^5*c^5*d^7 - 161451*a^8*b^4*c^4*d^8 + 132
924*a^9*b^3*c^3*d^9 - 65754*a^10*b^2*c^2*d^10 + 18252*a^11*b*c*d^11 - 2197*a^12*d^12)) - 45*(a*b^5*x^4 + a^2*b
^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^1
2*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 -
4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12
- 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^
(1/4)*log(a^2*b^4*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 4572
4*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8
*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12
*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16
)/(a^7*b^17))^(1/4) - (3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*a^4*d^4)*x) + 45*(
a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 4
5724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*
a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a
^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d
^16)/(a^7*b^17))^(1/4)*log(-a^2*b^4*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^
13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^
9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d
^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15
+ 28561*a^16*d^16)/(a^7*b^17))^(1/4) - (3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*
a^4*d^4)*x) + 180*(b^4*c^4 - 4*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 - 36*a^3*b*c*d^3 + 13*a^4*d^4)*x)/(a*b^5*x^4 +
a^2*b^4)

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Sympy [A]  time = 37.2598, size = 466, normalized size = 1.31 \begin{align*} \frac{x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{4 a^{2} b^{4} + 4 a b^{5} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{17} + 28561 a^{16} d^{16} - 316368 a^{15} b c d^{15} + 1577784 a^{14} b^{2} c^{2} d^{14} - 4651504 a^{13} b^{3} c^{3} d^{13} + 8923164 a^{12} b^{4} c^{4} d^{12} - 11486160 a^{11} b^{5} c^{5} d^{11} + 9723912 a^{10} b^{6} c^{6} d^{10} - 4810608 a^{9} b^{7} c^{7} d^{9} + 617958 a^{8} b^{8} c^{8} d^{8} + 772112 a^{7} b^{9} c^{9} d^{7} - 434808 a^{6} b^{10} c^{10} d^{6} + 20400 a^{5} b^{11} c^{11} d^{5} + 45724 a^{4} b^{12} c^{12} d^{4} - 8304 a^{3} b^{13} c^{13} d^{3} - 2376 a^{2} b^{14} c^{14} d^{2} + 432 a b^{15} c^{15} d + 81 b^{16} c^{16}, \left ( t \mapsto t \log{\left (- \frac{16 t a^{2} b^{4}}{13 a^{4} d^{4} - 36 a^{3} b c d^{3} + 30 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{9}}{9 b^{2}} - \frac{x^{5} \left (2 a d^{4} - 4 b c d^{3}\right )}{5 b^{3}} + \frac{x \left (3 a^{2} d^{4} - 8 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4)/(4*a**2*b**4 + 4*a*b**5*
x**4) + RootSum(65536*_t**4*a**7*b**17 + 28561*a**16*d**16 - 316368*a**15*b*c*d**15 + 1577784*a**14*b**2*c**2*
d**14 - 4651504*a**13*b**3*c**3*d**13 + 8923164*a**12*b**4*c**4*d**12 - 11486160*a**11*b**5*c**5*d**11 + 97239
12*a**10*b**6*c**6*d**10 - 4810608*a**9*b**7*c**7*d**9 + 617958*a**8*b**8*c**8*d**8 + 772112*a**7*b**9*c**9*d*
*7 - 434808*a**6*b**10*c**10*d**6 + 20400*a**5*b**11*c**11*d**5 + 45724*a**4*b**12*c**12*d**4 - 8304*a**3*b**1
3*c**13*d**3 - 2376*a**2*b**14*c**14*d**2 + 432*a*b**15*c**15*d + 81*b**16*c**16, Lambda(_t, _t*log(-16*_t*a**
2*b**4/(13*a**4*d**4 - 36*a**3*b*c*d**3 + 30*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - 3*b**4*c**4) + x))) + d**
4*x**9/(9*b**2) - x**5*(2*a*d**4 - 4*b*c*d**3)/(5*b**3) + x*(3*a**2*d**4 - 8*a*b*c*d**3 + 6*b**2*c**2*d**2)/b*
*4

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Giac [B]  time = 1.1142, size = 867, normalized size = 2.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 + 36*(a
*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4)
)/(a^2*b^5) + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)*a^2*b^2*c
^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4
))/(a/b)^(1/4))/(a^2*b^5) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(
1/4)*a^2*b^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4
) + sqrt(a/b))/(a^2*b^5) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1
/4)*a^2*b^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*log(x^2 - sqrt(2)*x*(a/b)^(1/4)
+ sqrt(a/b))/(a^2*b^5) + 1/4*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4*d^4*x
)/((b*x^4 + a)*a*b^4) + 1/45*(5*b^16*d^4*x^9 + 36*b^16*c*d^3*x^5 - 18*a*b^15*d^4*x^5 + 270*b^16*c^2*d^2*x - 36
0*a*b^15*c*d^3*x + 135*a^2*b^14*d^4*x)/b^18