3.168 \(\int \frac{(c+d x^4)^3}{(a+b x^4)^2} \, dx\)

Optimal. Leaf size=317 \[ -\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{4 a b^3 \left (a+b x^4\right )}+\frac{d^3 x^5}{5 b^2} \]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^5)/(5*b^2) + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^4)) - (3*(b*c - a*d)^2*(
b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3
*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(13/4)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c +
3*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^(13/4))

________________________________________________________________________________________

Rubi [A]  time = 0.317178, antiderivative size = 317, 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} \[ -\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{4 a b^3 \left (a+b x^4\right )}+\frac{d^3 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^4)^3/(a + b*x^4)^2,x]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^5)/(5*b^2) + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^4)) - (3*(b*c - a*d)^2*(
b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3
*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(13/4)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c +
3*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^(13/4))

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

Mathematica [A]  time = 0.222286, size = 301, normalized size = 0.95 \[ \frac{-\frac{15 \sqrt{2} (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{15 \sqrt{2} (b c-a d)^2 (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}-\frac{30 \sqrt{2} (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{30 \sqrt{2} (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+160 \sqrt [4]{b} d^2 x (3 b c-2 a d)+\frac{40 \sqrt [4]{b} x (b c-a d)^3}{a \left (a+b x^4\right )}+32 b^{5/4} d^3 x^5}{160 b^{13/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^4)^3/(a + b*x^4)^2,x]

[Out]

(160*b^(1/4)*d^2*(3*b*c - 2*a*d)*x + 32*b^(5/4)*d^3*x^5 + (40*b^(1/4)*(b*c - a*d)^3*x)/(a*(a + b*x^4)) - (30*S
qrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (30*Sqrt[2]*(b*c - a*d)^
2*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) - (15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(160*b^(13/4))

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Maple [B]  time = 0.009, size = 669, normalized size = 2.1 \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)^3/(b*x^4+a)^2,x)

[Out]

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

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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((d*x^4+c)^3/(b*x^4+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.08882, size = 4251, normalized size = 13.41 \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)^3/(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/80*(16*a*b^2*d^3*x^9 + 48*(5*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^5 + 60*(a*b^4*x^4 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^
11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^
6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a
^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*arctan(-(a^5*b^10*x*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^1
0*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^
5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12
*d^12)/(a^7*b^13))^(3/4) - a^5*b^10*sqrt((a^4*b^6*sqrt(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 -
44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1
039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b
^13)) + (b^6*c^6 + 2*a*b^5*c^5*d - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 - 30*a^5*b*c*d^5
+ 9*a^6*d^6)*x^2)/(b^6*c^6 + 2*a*b^5*c^5*d - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 - 30*
a^5*b*c*d^5 + 9*a^6*d^6))*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4
*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a
^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(3/4))/(b^9*c^9 + 3*a*
b^8*c^8*d - 12*a^2*b^7*c^7*d^2 - 20*a^3*b^6*c^6*d^3 + 78*a^4*b^5*c^5*d^4 - 6*a^5*b^4*c^4*d^5 - 188*a^6*b^3*c^3
*d^6 + 252*a^7*b^2*c^2*d^7 - 135*a^8*b*c*d^8 + 27*a^9*d^9)) + 15*(a*b^4*x^4 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^11
*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*
c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^1
1*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*log(3*a^2*b^3*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d
^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^
7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(
a^7*b^13))^(1/4) + 3*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*x) - 15*(a*b^4*x^4 + a^2*b^3)*(-(b^12
*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^
5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^
2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*log(-3*a^2*b^3*(-(b^12*c^12 + 4*a*b^11*c^11*d - 1
4*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 3
28*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11
+ 81*a^12*d^12)/(a^7*b^13))^(1/4) + 3*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*x) + 20*(b^3*c^3 - 3
*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 9*a^3*d^3)*x)/(a*b^4*x^4 + a^2*b^3)

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Sympy [A]  time = 6.68376, size = 335, normalized size = 1.06 \begin{align*} - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{4 a^{2} b^{3} + 4 a b^{4} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{13} + 6561 a^{12} d^{12} - 43740 a^{11} b c d^{11} + 118098 a^{10} b^{2} c^{2} d^{10} - 156492 a^{9} b^{3} c^{3} d^{9} + 84159 a^{8} b^{4} c^{4} d^{8} + 26568 a^{7} b^{5} c^{5} d^{7} - 52164 a^{6} b^{6} c^{6} d^{6} + 11016 a^{5} b^{7} c^{7} d^{5} + 10287 a^{4} b^{8} c^{8} d^{4} - 3564 a^{3} b^{9} c^{9} d^{3} - 1134 a^{2} b^{10} c^{10} d^{2} + 324 a b^{11} c^{11} d + 81 b^{12} c^{12}, \left ( t \mapsto t \log{\left (\frac{16 t a^{2} b^{3}}{9 a^{3} d^{3} - 15 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 3 b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{5}}{5 b^{2}} - \frac{x \left (2 a d^{3} - 3 b c d^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**4+c)**3/(b*x**4+a)**2,x)

[Out]

-x*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/(4*a**2*b**3 + 4*a*b**4*x**4) + RootSum(65536*_
t**4*a**7*b**13 + 6561*a**12*d**12 - 43740*a**11*b*c*d**11 + 118098*a**10*b**2*c**2*d**10 - 156492*a**9*b**3*c
**3*d**9 + 84159*a**8*b**4*c**4*d**8 + 26568*a**7*b**5*c**5*d**7 - 52164*a**6*b**6*c**6*d**6 + 11016*a**5*b**7
*c**7*d**5 + 10287*a**4*b**8*c**8*d**4 - 3564*a**3*b**9*c**9*d**3 - 1134*a**2*b**10*c**10*d**2 + 324*a*b**11*c
**11*d + 81*b**12*c**12, Lambda(_t, _t*log(16*_t*a**2*b**3/(9*a**3*d**3 - 15*a**2*b*c*d**2 + 3*a*b**2*c**2*d +
3*b**3*c**3) + x))) + d**3*x**5/(5*b**2) - x*(2*a*d**3 - 3*b*c*d**2)/b**3

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Giac [A]  time = 1.1066, size = 670, normalized size = 2.11 \begin{align*} \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\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 )}{16 \, a^{2} b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\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 )}{16 \, a^{2} b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{4}} + \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{4 \,{\left (b x^{4} + a\right )} a b^{3}} + \frac{b^{8} d^{3} x^{5} + 15 \, b^{8} c d^{2} x - 10 \, a b^{7} d^{3} x}{5 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^3/(b*x^4+a)^2,x, algorithm="giac")

[Out]

3/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4
)*a^3*d^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 3/16*sqrt(2)*((a*b^3)^(1/4)
*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 3/32*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*
b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b)
)/(a^2*b^4) - 3/32*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 +
3*(a*b^3)^(1/4)*a^3*d^3)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4) + 1/4*(b^3*c^3*x - 3*a*b^2*c^2
*d*x + 3*a^2*b*c*d^2*x - a^3*d^3*x)/((b*x^4 + a)*a*b^3) + 1/5*(b^8*d^3*x^5 + 15*b^8*c*d^2*x - 10*a*b^7*d^3*x)/
b^10