3.2902 \(\int \frac{(c e+d e x)^4}{(a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=220 \[ -\frac{e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}-\frac{e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3} d}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[Out]

-(e^4*(c + d*x)^2)/(6*b*d*(a + b*(c + d*x)^3)^2) + (e^4*(c + d*x)^2)/(9*a*b*d*(a + b*(c + d*x)^3)) - (e^4*ArcT
an[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(5/3)*d) - (e^4*Log[a^(1/3) + b^(1
/3)*(c + d*x)])/(27*a^(4/3)*b^(5/3)*d) + (e^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/
(54*a^(4/3)*b^(5/3)*d)

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Rubi [A]  time = 0.159712, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {372, 288, 290, 292, 31, 634, 617, 204, 628} \[ -\frac{e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}-\frac{e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3} d}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^4*(c + d*x)^2)/(6*b*d*(a + b*(c + d*x)^3)^2) + (e^4*(c + d*x)^2)/(9*a*b*d*(a + b*(c + d*x)^3)) - (e^4*ArcT
an[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(5/3)*d) - (e^4*Log[a^(1/3) + b^(1
/3)*(c + d*x)])/(27*a^(4/3)*b^(5/3)*d) + (e^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/
(54*a^(4/3)*b^(5/3)*d)

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 290

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

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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 e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{e^4 \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}+\frac{e^4 \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,c+d x\right )}{9 a b d}\\ &=-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{e^4 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{4/3} b^{4/3} d}+\frac{e^4 \operatorname{Subst}\left (\int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{4/3} b^{4/3} d}\\ &=-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{e^4 \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{54 a^{4/3} b^{5/3} d}+\frac{e^4 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{18 a b^{4/3} d}\\ &=-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}+\frac{e^4 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{9 a^{4/3} b^{5/3} d}\\ &=-\frac{e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{e^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3} d}-\frac{e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}\\ \end{align*}

Mathematica [A]  time = 0.0144404, size = 185, normalized size = 0.84 \[ \frac{e^4 \left (\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{4/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac{6 b^{2/3} (c+d x)^2}{a \left (a+b (c+d x)^3\right )}-\frac{9 b^{2/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}\right )}{54 b^{5/3} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*((-9*b^(2/3)*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 + (6*b^(2/3)*(c + d*x)^2)/(a*(a + b*(c + d*x)^3)) + (2*Sq
rt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/a^(4/3) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)
])/a^(4/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/a^(4/3)))/(54*b^(5/3)*d)

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Maple [C]  time = 0.018, size = 521, normalized size = 2.4 \begin{align*}{\frac{{e}^{4}{d}^{4}{x}^{5}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{5\,{e}^{4}c{d}^{3}{x}^{4}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{10\,{e}^{4}{c}^{2}{d}^{2}{x}^{3}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{10\,{e}^{4}d{x}^{2}{c}^{3}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}-{\frac{{e}^{4}d{x}^{2}}{18\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}b}}+{\frac{5\,{e}^{4}{c}^{4}x}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}-{\frac{{e}^{4}xc}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}b}}+{\frac{{e}^{4}{c}^{5}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}da}}-{\frac{{e}^{4}{c}^{2}}{18\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}bd}}+{\frac{{e}^{4}}{27\,a{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4/(a+b*(d*x+c)^3)^3,x)

[Out]

1/9*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d^4/a*x^5+5/9*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x
+b*c^3+a)^2*c*d^3/a*x^4+10/9*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^2*d^2/a*x^3+10/9*e^4/(b*d^3
*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d/a*x^2*c^3-1/18*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2
/b*x^2*d+5/9*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^4/a*x-1/9*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*
c^2*d*x+b*c^3+a)^2/b*x*c+1/9*e^4/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^5/d/a-1/18*e^4/(b*d^3*x^3+3
*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^2/b/d+1/27*e^4/b^2/a/d*sum((_R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=
RootOf(_Z^3*b*d^3+3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{1}{6} \,{\left (2 \, \sqrt{3} \left (-\frac{1}{a b^{2} d^{3}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac{2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac{2}{3}}}\right ) + \left (-\frac{1}{a b^{2} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac{2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac{4}{3}}\right ) - 2 \, \left (-\frac{1}{a b^{2} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac{2}{3}} \right |}\right )\right )} e^{4}}{9 \, a b} + \frac{2 \, b d^{5} e^{4} x^{5} + 10 \, b c d^{4} e^{4} x^{4} + 20 \, b c^{2} d^{3} e^{4} x^{3} +{\left (20 \, b c^{3} - a\right )} d^{2} e^{4} x^{2} + 2 \,{\left (5 \, b c^{4} - a c\right )} d e^{4} x +{\left (2 \, b c^{5} - a c^{2}\right )} e^{4}}{18 \,{\left (a b^{3} d^{7} x^{6} + 6 \, a b^{3} c d^{6} x^{5} + 15 \, a b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a b^{3} c^{3} + a^{2} b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, a b^{3} c^{4} + 2 \, a^{2} b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (a b^{3} c^{5} + a^{2} b^{2} c^{2}\right )} d^{2} x +{\left (a b^{3} c^{6} + 2 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*e^4*integrate((d*x + c)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/(a*b) + 1/18*(2*b*d^5*e^
4*x^5 + 10*b*c*d^4*e^4*x^4 + 20*b*c^2*d^3*e^4*x^3 + (20*b*c^3 - a)*d^2*e^4*x^2 + 2*(5*b*c^4 - a*c)*d*e^4*x + (
2*b*c^5 - a*c^2)*e^4)/(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 15*a*b^3*c^2*d^5*x^4 + 2*(10*a*b^3*c^3 + a^2*b^2)*d
^4*x^3 + 3*(5*a*b^3*c^4 + 2*a^2*b^2*c)*d^3*x^2 + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d^2*x + (a*b^3*c^6 + 2*a^2*b^2*c^
3 + a^3*b)*d)

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Fricas [B]  time = 1.97284, size = 3895, normalized size = 17.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(6*a*b^3*d^5*e^4*x^5 + 30*a*b^3*c*d^4*e^4*x^4 + 60*a*b^3*c^2*d^3*e^4*x^3 + 3*(20*a*b^3*c^3 - a^2*b^2)*d^
2*e^4*x^2 + 6*(5*a*b^3*c^4 - a^2*b^2*c)*d*e^4*x + 3*(2*a*b^3*c^5 - a^2*b^2*c^2)*e^4 + 3*sqrt(1/3)*(a*b^3*d^6*e
^4*x^6 + 6*a*b^3*c*d^5*e^4*x^5 + 15*a*b^3*c^2*d^4*e^4*x^4 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^3*e^4*x^3 + 3*(5*a*b^
3*c^4 + 2*a^2*b^2*c)*d^2*e^4*x^2 + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d*e^4*x + (a*b^3*c^6 + 2*a^2*b^2*c^3 + a^3*b)*e
^4)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*d^3*x^3 + 6*b^2*c*d^2*x^2 + 6*b^2*c^2*d*x + 2*b^2*c^3 - a*b + 3*sqrt(1/3
)*(a*b*d*x + a*b*c + 2*(d^2*x^2 + 2*c*d*x + c^2)*(-a*b^2)^(2/3) + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3
*(-a*b^2)^(2/3)*(d*x + c))/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)) + (b^2*d^6*e^4*x^6 + 6*b^2*c
*d^5*e^4*x^5 + 15*b^2*c^2*d^4*e^4*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e^4*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e^4*x^2
+ 6*(b^2*c^5 + a*b*c^2)*d*e^4*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e^4)*(-a*b^2)^(2/3)*log(b^2*d^2*x^2 + 2*b^2*c*d
*x + b^2*c^2 + (-a*b^2)^(1/3)*(b*d*x + b*c) + (-a*b^2)^(2/3)) - 2*(b^2*d^6*e^4*x^6 + 6*b^2*c*d^5*e^4*x^5 + 15*
b^2*c^2*d^4*e^4*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e^4*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e^4*x^2 + 6*(b^2*c^5 + a*
b*c^2)*d*e^4*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e^4)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a^2*b^5*d
^7*x^6 + 6*a^2*b^5*c*d^6*x^5 + 15*a^2*b^5*c^2*d^5*x^4 + 2*(10*a^2*b^5*c^3 + a^3*b^4)*d^4*x^3 + 3*(5*a^2*b^5*c^
4 + 2*a^3*b^4*c)*d^3*x^2 + 6*(a^2*b^5*c^5 + a^3*b^4*c^2)*d^2*x + (a^2*b^5*c^6 + 2*a^3*b^4*c^3 + a^4*b^3)*d), 1
/54*(6*a*b^3*d^5*e^4*x^5 + 30*a*b^3*c*d^4*e^4*x^4 + 60*a*b^3*c^2*d^3*e^4*x^3 + 3*(20*a*b^3*c^3 - a^2*b^2)*d^2*
e^4*x^2 + 6*(5*a*b^3*c^4 - a^2*b^2*c)*d*e^4*x + 3*(2*a*b^3*c^5 - a^2*b^2*c^2)*e^4 + 6*sqrt(1/3)*(a*b^3*d^6*e^4
*x^6 + 6*a*b^3*c*d^5*e^4*x^5 + 15*a*b^3*c^2*d^4*e^4*x^4 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^3*e^4*x^3 + 3*(5*a*b^3*
c^4 + 2*a^2*b^2*c)*d^2*e^4*x^2 + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d*e^4*x + (a*b^3*c^6 + 2*a^2*b^2*c^3 + a^3*b)*e^4
)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*d*x + 2*b*c + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + (b^
2*d^6*e^4*x^6 + 6*b^2*c*d^5*e^4*x^5 + 15*b^2*c^2*d^4*e^4*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e^4*x^3 + 3*(5*b^2*c^4
+ 2*a*b*c)*d^2*e^4*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*e^4*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e^4)*(-a*b^2)^(2/3)*log(
b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (-a*b^2)^(1/3)*(b*d*x + b*c) + (-a*b^2)^(2/3)) - 2*(b^2*d^6*e^4*x^6 + 6*
b^2*c*d^5*e^4*x^5 + 15*b^2*c^2*d^4*e^4*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e^4*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e^
4*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*e^4*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e^4)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*
b^2)^(1/3)))/(a^2*b^5*d^7*x^6 + 6*a^2*b^5*c*d^6*x^5 + 15*a^2*b^5*c^2*d^5*x^4 + 2*(10*a^2*b^5*c^3 + a^3*b^4)*d^
4*x^3 + 3*(5*a^2*b^5*c^4 + 2*a^3*b^4*c)*d^3*x^2 + 6*(a^2*b^5*c^5 + a^3*b^4*c^2)*d^2*x + (a^2*b^5*c^6 + 2*a^3*b
^4*c^3 + a^4*b^3)*d)]

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Sympy [A]  time = 28.3115, size = 332, normalized size = 1.51 \begin{align*} \frac{- a c^{2} e^{4} + 2 b c^{5} e^{4} + 20 b c^{2} d^{3} e^{4} x^{3} + 10 b c d^{4} e^{4} x^{4} + 2 b d^{5} e^{4} x^{5} + x^{2} \left (- a d^{2} e^{4} + 20 b c^{3} d^{2} e^{4}\right ) + x \left (- 2 a c d e^{4} + 10 b c^{4} d e^{4}\right )}{18 a^{3} b d + 36 a^{2} b^{2} c^{3} d + 18 a b^{3} c^{6} d + 270 a b^{3} c^{2} d^{5} x^{4} + 108 a b^{3} c d^{6} x^{5} + 18 a b^{3} d^{7} x^{6} + x^{3} \left (36 a^{2} b^{2} d^{4} + 360 a b^{3} c^{3} d^{4}\right ) + x^{2} \left (108 a^{2} b^{2} c d^{3} + 270 a b^{3} c^{4} d^{3}\right ) + x \left (108 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{5} d^{2}\right )} + \frac{e^{4} \operatorname{RootSum}{\left (19683 t^{3} a^{4} b^{5} + 1, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{3} b^{3} e^{8} + c e^{8}}{d e^{8}} \right )} \right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a*c**2*e**4 + 2*b*c**5*e**4 + 20*b*c**2*d**3*e**4*x**3 + 10*b*c*d**4*e**4*x**4 + 2*b*d**5*e**4*x**5 + x**2*(
-a*d**2*e**4 + 20*b*c**3*d**2*e**4) + x*(-2*a*c*d*e**4 + 10*b*c**4*d*e**4))/(18*a**3*b*d + 36*a**2*b**2*c**3*d
+ 18*a*b**3*c**6*d + 270*a*b**3*c**2*d**5*x**4 + 108*a*b**3*c*d**6*x**5 + 18*a*b**3*d**7*x**6 + x**3*(36*a**2
*b**2*d**4 + 360*a*b**3*c**3*d**4) + x**2*(108*a**2*b**2*c*d**3 + 270*a*b**3*c**4*d**3) + x*(108*a**2*b**2*c**
2*d**2 + 108*a*b**3*c**5*d**2)) + e**4*RootSum(19683*_t**3*a**4*b**5 + 1, Lambda(_t, _t*log(x + (729*_t**2*a**
3*b**3*e**8 + c*e**8)/(d*e**8))))/d

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Giac [A]  time = 1.27184, size = 410, normalized size = 1.86 \begin{align*} -\frac{1}{27} \, \sqrt{3} \left (-\frac{e^{12}}{a^{4} b^{5} d^{3}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac{2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac{2}{3}}}\right ) - \frac{1}{54} \, \left (-\frac{e^{12}}{a^{4} b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac{2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac{4}{3}}\right ) + \frac{1}{27} \, \left (-\frac{e^{12}}{a^{4} b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | 9 \, a^{2} b^{2} d x + 9 \, a^{2} b^{2} c + 9 \, \left (-a^{2} b\right )^{\frac{2}{3}} a b \right |}\right ) + \frac{2 \, b d^{5} x^{5} e^{4} + 10 \, b c d^{4} x^{4} e^{4} + 20 \, b c^{2} d^{3} x^{3} e^{4} + 20 \, b c^{3} d^{2} x^{2} e^{4} + 10 \, b c^{4} d x e^{4} + 2 \, b c^{5} e^{4} - a d^{2} x^{2} e^{4} - 2 \, a c d x e^{4} - a c^{2} e^{4}}{18 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*sqrt(3)*(-e^12/(a^4*b^5*d^3))^(1/3)*arctan(1/3*sqrt(3)*(2*a*b*d*x + 2*a*b*c - (-a^2*b)^(2/3))/(-a^2*b)^(
2/3)) - 1/54*(-e^12/(a^4*b^5*d^3))^(1/3)*log((2*a*b*d*x + 2*a*b*c - (-a^2*b)^(2/3))^2 + 3*(-a^2*b)^(4/3)) + 1/
27*(-e^12/(a^4*b^5*d^3))^(1/3)*log(abs(9*a^2*b^2*d*x + 9*a^2*b^2*c + 9*(-a^2*b)^(2/3)*a*b)) + 1/18*(2*b*d^5*x^
5*e^4 + 10*b*c*d^4*x^4*e^4 + 20*b*c^2*d^3*x^3*e^4 + 20*b*c^3*d^2*x^2*e^4 + 10*b*c^4*d*x*e^4 + 2*b*c^5*e^4 - a*
d^2*x^2*e^4 - 2*a*c*d*x*e^4 - a*c^2*e^4)/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)^2*a*b*d)