3.2884 \(\int \frac{1}{(c+d x)^3 (a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=219 \[ -\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} d}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]

[Out]

-10/(9*a^3*d*(c + d*x)^2) + 1/(6*a*d*(c + d*x)^2*(a + b*(c + d*x)^3)^2) + 4/(9*a^2*d*(c + d*x)^2*(a + b*(c + d
*x)^3)) + (20*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*d) - (20*
b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(11/3)*d) + (10*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x
) + b^(2/3)*(c + d*x)^2])/(27*a^(11/3)*d)

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Rubi [A]  time = 0.17914, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {372, 290, 325, 200, 31, 634, 617, 204, 628} \[ -\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} d}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^3*(a + b*(c + d*x)^3)^3),x]

[Out]

-10/(9*a^3*d*(c + d*x)^2) + 1/(6*a*d*(c + d*x)^2*(a + b*(c + d*x)^3)^2) + 4/(9*a^2*d*(c + d*x)^2*(a + b*(c + d
*x)^3)) + (20*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*d) - (20*
b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(11/3)*d) + (10*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x
) + b^(2/3)*(c + d*x)^2])/(27*a^(11/3)*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 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 325

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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{1}{(c+d x)^3 \left (a+b (c+d x)^3\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac{20 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{9 a^2 d}\\ &=-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{(20 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,c+d x\right )}{9 a^3 d}\\ &=-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{(20 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{11/3} d}-\frac{(20 b) \operatorname{Subst}\left (\int \frac{2 \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^{11/3} d}\\ &=-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac{\left (10 b^{2/3}\right ) \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 )}{27 a^{11/3} d}-\frac{(10 b) \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 )}{9 a^{10/3} d}\\ &=-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}-\frac{\left (20 b^{2/3}\right ) \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^{11/3} d}\\ &=-\frac{10}{9 a^3 d (c+d x)^2}+\frac{1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{11/3} d}-\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}\\ \end{align*}

Mathematica [A]  time = 0.144645, size = 192, normalized size = 0.88 \[ \frac{20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{9 a^{5/3} b (c+d x)}{\left (a+b (c+d x)^3\right )^2}-\frac{33 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-\frac{27 a^{2/3}}{(c+d x)^2}-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-40 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{54 a^{11/3} d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^3*(a + b*(c + d*x)^3)^3),x]

[Out]

((-27*a^(2/3))/(c + d*x)^2 - (9*a^(5/3)*b*(c + d*x))/(a + b*(c + d*x)^3)^2 - (33*a^(2/3)*b*(c + d*x))/(a + b*(
c + d*x)^3) - 40*Sqrt[3]*b^(2/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] - 40*b^(2/3)*Log[a
^(1/3) + b^(1/3)*(c + d*x)] + 20*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(54*a
^(11/3)*d)

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Maple [C]  time = 0.02, size = 419, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,{a}^{3}d \left ( dx+c \right ) ^{2}}}-{\frac{11\,{b}^{2}{d}^{3}{x}^{4}}{18\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{22\,c{d}^{2}{b}^{2}{x}^{3}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{2}d{x}^{2}}{3\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{22\,{b}^{2}x{c}^{3}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{7\,bx}{9\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{4}}{18\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{7\,bc}{9\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{20}{27\,{a}^{3}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{\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(1/(d*x+c)^3/(a+b*(d*x+c)^3)^3,x)

[Out]

-1/2/a^3/d/(d*x+c)^2-11/18*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d^3*x^4-22/9*b^2/a^3/(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^2*x^3-11/3*b^2/a^3/(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*x^2-22/9*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*x*c^3-7/9*b/a^2/(b*d^3*x^3+3*b*c*d^2
*x^2+3*b*c^2*d*x+b*c^3+a)^2*x-11/18*b^2/a^3/(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/d-7/9*b/a^2/(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-20/27/a^3/d*sum(1/(_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{20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{6} + 16 \,{\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 32 \, a b c^{3} + 12 \,{\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 24 \,{\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x + 9 \, a^{2}}{18 \,{\left (a^{3} b^{2} d^{9} x^{8} + 8 \, a^{3} b^{2} c d^{8} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} x^{6} + 2 \,{\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} x^{5} + 10 \,{\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} x^{4} + 4 \,{\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} x^{3} +{\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} x^{2} + 2 \,{\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} x +{\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d\right )}} - \frac{\frac{10}{3} \,{\left (2 \, \sqrt{3} \left (\frac{1}{a^{2} b d^{3}}\right )^{\frac{1}{3}} \arctan \left (-\frac{b d x + b c + \left (a b^{2}\right )^{\frac{1}{3}}}{\sqrt{3} b d x + \sqrt{3} b c - \sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}}}\right ) - \left (\frac{1}{a^{2} b d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (\sqrt{3} b d x + \sqrt{3} b c - \sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}}\right )}^{2} +{\left (b d x + b c + \left (a b^{2}\right )^{\frac{1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac{1}{a^{2} b d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac{1}{3}} \right |}\right )\right )} b}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+b*(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-1/18*(20*b^2*d^6*x^6 + 120*b^2*c*d^5*x^5 + 300*b^2*c^2*d^4*x^4 + 20*b^2*c^6 + 16*(25*b^2*c^3 + 2*a*b)*d^3*x^3
+ 32*a*b*c^3 + 12*(25*b^2*c^4 + 8*a*b*c)*d^2*x^2 + 24*(5*b^2*c^5 + 4*a*b*c^2)*d*x + 9*a^2)/(a^3*b^2*d^9*x^8 +
8*a^3*b^2*c*d^8*x^7 + 28*a^3*b^2*c^2*d^7*x^6 + 2*(28*a^3*b^2*c^3 + a^4*b)*d^6*x^5 + 10*(7*a^3*b^2*c^4 + a^4*b
*c)*d^5*x^4 + 4*(14*a^3*b^2*c^5 + 5*a^4*b*c^2)*d^4*x^3 + (28*a^3*b^2*c^6 + 20*a^4*b*c^3 + a^5)*d^3*x^2 + 2*(4*
a^3*b^2*c^7 + 5*a^4*b*c^4 + a^5*c)*d^2*x + (a^3*b^2*c^8 + 2*a^4*b*c^5 + a^5*c^2)*d) - 20/9*b*integrate(1/(b*d^
3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/a^3

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Fricas [B]  time = 2.3773, size = 2176, normalized size = 9.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+b*(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

-1/54*(60*b^2*d^6*x^6 + 360*b^2*c*d^5*x^5 + 900*b^2*c^2*d^4*x^4 + 60*b^2*c^6 + 48*(25*b^2*c^3 + 2*a*b)*d^3*x^3
+ 96*a*b*c^3 + 36*(25*b^2*c^4 + 8*a*b*c)*d^2*x^2 + 72*(5*b^2*c^5 + 4*a*b*c^2)*d*x - 40*sqrt(3)*(b^2*d^8*x^8 +
8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 + 10*(7*b^2*c^4 + a*b*c)*d^4*x^
4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*
b^2*c^7 + 5*a*b*c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*(a*d*x + a*c)*(-b^2/a^2)^(2/3) - sqrt
(3)*b)/b) + 20*(b^2*d^8*x^8 + 8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 +
10*(7*b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a
^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2*c^7 + 5*a*b*c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*log(b^2*d^2*x^2 + 2*b^2*c*d*x
+ b^2*c^2 + a^2*(-b^2/a^2)^(2/3) + (a*b*d*x + a*b*c)*(-b^2/a^2)^(1/3)) - 40*(b^2*d^8*x^8 + 8*b^2*c*d^7*x^7 + 2
8*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 + 10*(7*b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(1
4*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2*c^7 + 5*a*b*c^4
+ a^2*c)*d*x)*(-b^2/a^2)^(1/3)*log(b*d*x + b*c - a*(-b^2/a^2)^(1/3)) + 27*a^2)/(a^3*b^2*d^9*x^8 + 8*a^3*b^2*c*
d^8*x^7 + 28*a^3*b^2*c^2*d^7*x^6 + 2*(28*a^3*b^2*c^3 + a^4*b)*d^6*x^5 + 10*(7*a^3*b^2*c^4 + a^4*b*c)*d^5*x^4 +
4*(14*a^3*b^2*c^5 + 5*a^4*b*c^2)*d^4*x^3 + (28*a^3*b^2*c^6 + 20*a^4*b*c^3 + a^5)*d^3*x^2 + 2*(4*a^3*b^2*c^7 +
5*a^4*b*c^4 + a^5*c)*d^2*x + (a^3*b^2*c^8 + 2*a^4*b*c^5 + a^5*c^2)*d)

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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(1/(d*x+c)**3/(a+b*(d*x+c)**3)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.30227, size = 494, normalized size = 2.26 \begin{align*} \frac{20}{27} \, \sqrt{3} \left (-\frac{b^{2}}{a^{11} d^{3}}\right )^{\frac{1}{3}} \arctan \left (-\frac{b d x + b c - \left (-a b^{2}\right )^{\frac{1}{3}}}{\sqrt{3} b d x + \sqrt{3} b c + \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}}\right ) - \frac{10}{27} \, \left (-\frac{b^{2}}{a^{11} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (\sqrt{3} b d x + \sqrt{3} b c + \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}^{2} +{\left (b d x + b c - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}^{2}\right ) + \frac{20}{27} \, \left (-\frac{b^{2}}{a^{11} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | -9 \, a^{3} b d x - 9 \, a^{3} b c + 9 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} \right |}\right ) - \frac{20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 400 \, b^{2} c^{3} d^{3} x^{3} + 300 \, b^{2} c^{4} d^{2} x^{2} + 120 \, b^{2} c^{5} d x + 20 \, b^{2} c^{6} + 32 \, a b d^{3} x^{3} + 96 \, a b c d^{2} x^{2} + 96 \, a b c^{2} d x + 32 \, a b c^{3} + 9 \, a^{2}}{18 \,{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a d x + a c\right )}^{2} a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+b*(d*x+c)^3)^3,x, algorithm="giac")

[Out]

20/27*sqrt(3)*(-b^2/(a^11*d^3))^(1/3)*arctan(-(b*d*x + b*c - (-a*b^2)^(1/3))/(sqrt(3)*b*d*x + sqrt(3)*b*c + sq
rt(3)*(-a*b^2)^(1/3))) - 10/27*(-b^2/(a^11*d^3))^(1/3)*log((sqrt(3)*b*d*x + sqrt(3)*b*c + sqrt(3)*(-a*b^2)^(1/
3))^2 + (b*d*x + b*c - (-a*b^2)^(1/3))^2) + 20/27*(-b^2/(a^11*d^3))^(1/3)*log(abs(-9*a^3*b*d*x - 9*a^3*b*c + 9
*(-a*b^2)^(1/3)*a^3)) - 1/18*(20*b^2*d^6*x^6 + 120*b^2*c*d^5*x^5 + 300*b^2*c^2*d^4*x^4 + 400*b^2*c^3*d^3*x^3 +
300*b^2*c^4*d^2*x^2 + 120*b^2*c^5*d*x + 20*b^2*c^6 + 32*a*b*d^3*x^3 + 96*a*b*c*d^2*x^2 + 96*a*b*c^2*d*x + 32*
a*b*c^3 + 9*a^2)/((b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a*d*x + a*c)^2*a^3*d)