3.160 \(\int \frac{(c+d x^4)^4}{a+b x^4} \, dx\)
Optimal. Leaf size=332 \[ \frac{d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{5 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d^3 x^9 (4 b c-a d)}{9 b^2}+\frac{d^4 x^{13}}{13 b} \]
[Out]
(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^5)/(5*b^3
) + (d^3*(4*b*c - a*d)*x^9)/(9*b^2) + (d^4*x^13)/(13*b) - ((b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4
)])/(2*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*
b^(17/4)) - ((b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4)
) + ((b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4))
________________________________________________________________________________________
Rubi [A] time = 0.266637, antiderivative size = 332, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used
= {390, 211, 1165, 628, 1162, 617, 204} \[ \frac{d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{5 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d^3 x^9 (4 b c-a d)}{9 b^2}+\frac{d^4 x^{13}}{13 b} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x^4)^4/(a + b*x^4),x]
[Out]
(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^5)/(5*b^3
) + (d^3*(4*b*c - a*d)*x^9)/(9*b^2) + (d^4*x^13)/(13*b) - ((b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4
)])/(2*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*
b^(17/4)) - ((b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4)
) + ((b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*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 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}{a+b x^4} \, dx &=\int \left (\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{b^3}+\frac{d^3 (4 b c-a d) x^8}{b^2}+\frac{d^4 x^{12}}{b}+\frac{b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+a^4 d^4}{b^4 \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^3 (4 b c-a d) x^9}{9 b^2}+\frac{d^4 x^{13}}{13 b}+\frac{(b c-a d)^4 \int \frac{1}{a+b x^4} \, dx}{b^4}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^3 (4 b c-a d) x^9}{9 b^2}+\frac{d^4 x^{13}}{13 b}+\frac{(b c-a d)^4 \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{2 \sqrt{a} b^4}+\frac{(b c-a d)^4 \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{2 \sqrt{a} b^4}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^3 (4 b c-a d) x^9}{9 b^2}+\frac{d^4 x^{13}}{13 b}+\frac{(b c-a d)^4 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt{a} b^{9/2}}+\frac{(b c-a d)^4 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt{a} b^{9/2}}-\frac{(b c-a d)^4 \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} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \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} a^{3/4} b^{17/4}}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^3 (4 b c-a d) x^9}{9 b^2}+\frac{d^4 x^{13}}{13 b}-\frac{(b c-a d)^4 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \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} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \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} a^{3/4} b^{17/4}}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^3 (4 b c-a d) x^9}{9 b^2}+\frac{d^4 x^{13}}{13 b}-\frac{(b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}\\ \end{align*}
Mathematica [A] time = 0.192085, size = 322, normalized size = 0.97 \[ \frac{936 b^{5/4} d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )+4680 \sqrt [4]{b} d x \left (4 a^2 b c d^2-a^3 d^3-6 a b^2 c^2 d+4 b^3 c^3\right )-\frac{585 \sqrt{2} (b c-a d)^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}+\frac{585 \sqrt{2} (b c-a d)^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}-\frac{1170 \sqrt{2} (b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{1170 \sqrt{2} (b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}+520 b^{9/4} d^3 x^9 (4 b c-a d)+360 b^{13/4} d^4 x^{13}}{4680 b^{17/4}} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x^4)^4/(a + b*x^4),x]
[Out]
(4680*b^(1/4)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 936*b^(5/4)*d^2*(6*b^2*c^2 - 4*a*b*c
*d + a^2*d^2)*x^5 + 520*b^(9/4)*d^3*(4*b*c - a*d)*x^9 + 360*b^(13/4)*d^4*x^13 - (1170*Sqrt[2]*(b*c - a*d)^4*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (1170*Sqrt[2]*(b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^
(1/4)])/a^(3/4) - (585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) +
(585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4))/(4680*b^(17/4))
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Maple [B] time = 0.007, size = 837, 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),x)
[Out]
-d^4/b^4*a^3*x+4*d/b*c^3*x-1/2/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)))*c^3*d+1/4/b^4*(1/b*a)^(1/4)*a^3*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)
*d^4-1/b*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c^3*d+1/4*(1/b*a)^(1/4)/a*2^(1/2)*arctan(2^(1
/2)/(1/b*a)^(1/4)*x-1)*c^4+1/8*(1/b*a)^(1/4)/a*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^4-4/5*d^3/b^2*x^5*a*c+1/4*(1/b*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/b*a
)^(1/4)*x+1)*c^4+1/13*d^4*x^13/b+1/4/b^4*(1/b*a)^(1/4)*a^3*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*d^4-1/b*(
1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^3*d+1/8/b^4*(1/b*a)^(1/4)*a^3*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+4*d^3/b^3*a^2*c*x-6*d^2/b^2*a
*c^2*x-1/b^3*(1/b*a)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c*d^3+3/2/b^2*(1/b*a)^(1/4)*a*2^(1/2)
*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^2*d^2-1/2/b^3*(1/b*a)^(1/4)*a^2*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+3/4/b^2*(1/b*a)^(1/4)*a*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^2-1/b^3*(1/b*a)^(1/4)*a^2
*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c*d^3+3/2/b^2*(1/b*a)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*
x+1)*c^2*d^2+6/5*d^2/b*x^5*c^2-1/9*d^4/b^2*x^9*a+4/9*d^3/b*x^9*c+1/5*d^4/b^3*x^5*a^2
________________________________________________________________________________________
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),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.72795, size = 5701, normalized size = 17.17 \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),x, algorithm="fricas")
[Out]
1/2340*(180*b^3*d^4*x^13 + 260*(4*b^3*c*d^3 - a*b^2*d^4)*x^9 + 468*(6*b^3*c^2*d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)
*x^5 + 2340*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^1
2*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 -
11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b
^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4)*arctan(-(a^2*b^13*x*(-(b
^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^
5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d
^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^
14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(3/4) - a^2*b^13*sqrt((a^2*b^8*sqrt(-(b^16*c^16 -
16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11
*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a
^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*
d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17)) + (b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*
c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)*x^2)/(b^8*c^
8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6
*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8))*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^1
3*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7
+ 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*
b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(3/4)
)/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b
^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b
^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)) + 585*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2
- 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^
7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^1
1 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^
3*b^17))^(1/4)*log(a*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 182
0*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*
c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 5
60*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4) + (b^4*c^4 - 4*
a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x) - 585*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*
a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^
10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368
*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15
+ a^16*d^16)/(a^3*b^17))^(1/4)*log(-a*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b
^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^
7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^1
2*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/
4) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x) + 2340*(4*b^3*c^3*d - 6*a*b^2*
c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*x)/b^4
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Sympy [A] time = 2.90635, size = 430, normalized size = 1.3 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{17} + a^{16} d^{16} - 16 a^{15} b c d^{15} + 120 a^{14} b^{2} c^{2} d^{14} - 560 a^{13} b^{3} c^{3} d^{13} + 1820 a^{12} b^{4} c^{4} d^{12} - 4368 a^{11} b^{5} c^{5} d^{11} + 8008 a^{10} b^{6} c^{6} d^{10} - 11440 a^{9} b^{7} c^{7} d^{9} + 12870 a^{8} b^{8} c^{8} d^{8} - 11440 a^{7} b^{9} c^{9} d^{7} + 8008 a^{6} b^{10} c^{10} d^{6} - 4368 a^{5} b^{11} c^{11} d^{5} + 1820 a^{4} b^{12} c^{12} d^{4} - 560 a^{3} b^{13} c^{13} d^{3} + 120 a^{2} b^{14} c^{14} d^{2} - 16 a b^{15} c^{15} d + b^{16} c^{16}, \left ( t \mapsto t \log{\left (\frac{4 t a b^{4}}{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}} + x \right )} \right )\right )} + \frac{d^{4} x^{13}}{13 b} - \frac{x^{9} \left (a d^{4} - 4 b c d^{3}\right )}{9 b^{2}} + \frac{x^{5} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{5 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\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),x)
[Out]
RootSum(256*_t**4*a**3*b**17 + a**16*d**16 - 16*a**15*b*c*d**15 + 120*a**14*b**2*c**2*d**14 - 560*a**13*b**3*c
**3*d**13 + 1820*a**12*b**4*c**4*d**12 - 4368*a**11*b**5*c**5*d**11 + 8008*a**10*b**6*c**6*d**10 - 11440*a**9*
b**7*c**7*d**9 + 12870*a**8*b**8*c**8*d**8 - 11440*a**7*b**9*c**9*d**7 + 8008*a**6*b**10*c**10*d**6 - 4368*a**
5*b**11*c**11*d**5 + 1820*a**4*b**12*c**12*d**4 - 560*a**3*b**13*c**13*d**3 + 120*a**2*b**14*c**14*d**2 - 16*a
*b**15*c**15*d + b**16*c**16, Lambda(_t, _t*log(4*_t*a*b**4/(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) + x))) + d**4*x**13/(13*b) - x**9*(a*d**4 - 4*b*c*d**3)/(9*b**2) + x**5*(a**
2*d**4 - 4*a*b*c*d**3 + 6*b**2*c**2*d**2)/(5*b**3) - x*(a**3*d**4 - 4*a**2*b*c*d**3 + 6*a*b**2*c**2*d**2 - 4*b
**3*c**3*d)/b**4
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Giac [B] time = 1.11049, size = 833, normalized size = 2.51 \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),x, algorithm="giac")
[Out]
1/4*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)
^(1/4)*a^3*b*c*d^3 + (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*b^5
) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*
b^3)^(1/4)*a^3*b*c*d^3 + (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
*b^5) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4
*(a*b^3)^(1/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^5) - 1/8
*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)^(1
/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^5) + 1/585*(45*b^12
*d^4*x^13 + 260*b^12*c*d^3*x^9 - 65*a*b^11*d^4*x^9 + 702*b^12*c^2*d^2*x^5 - 468*a*b^11*c*d^3*x^5 + 117*a^2*b^1
0*d^4*x^5 + 2340*b^12*c^3*d*x - 3510*a*b^11*c^2*d^2*x + 2340*a^2*b^10*c*d^3*x - 585*a^3*b^9*d^4*x)/b^13