3.2880 \(\int \frac{c+d x}{(a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=202 \[ -\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[Out]

(c + d*x)^2/(6*a*d*(a + b*(c + d*x)^3)^2) + (2*(c + d*x)^2)/(9*a^2*d*(a + b*(c + d*x)^3)) - (2*ArcTan[(a^(1/3)
- 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)*d) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)
])/(27*a^(7/3)*b^(2/3)*d) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(27*a^(7/3)*b^(2/3)
*d)

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Rubi [A]  time = 0.166015, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {372, 290, 292, 31, 634, 617, 204, 628} \[ -\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^2/(6*a*d*(a + b*(c + d*x)^3)^2) + (2*(c + d*x)^2)/(9*a^2*d*(a + b*(c + d*x)^3)) - (2*ArcTan[(a^(1/3)
- 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)*d) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)
])/(27*a^(7/3)*b^(2/3)*d) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(27*a^(7/3)*b^(2/3)
*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 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+d x}{\left (a+b (c+d x)^3\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,c+d x\right )}{9 a^2 d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{7/3} \sqrt [3]{b} d}+\frac{2 \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^{7/3} \sqrt [3]{b} d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\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^{7/3} b^{2/3} d}+\frac{\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^2 \sqrt [3]{b} d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}+\frac{2 \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^{7/3} b^{2/3} d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}\\ \end{align*}

Mathematica [A]  time = 0.119765, size = 180, normalized size = 0.89 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}+\frac{9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac{12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}}{54 a^{7/3} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((9*a^(4/3)*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 + (12*a^(1/3)*(c + d*x)^2)/(a + b*(c + d*x)^3) + (4*Sqrt[3]*Arc
Tan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*(c + d*x)])/b^(2/3
) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/b^(2/3))/(54*a^(7/3)*d)

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Maple [C]  time = 0.015, size = 214, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}} \left ({\frac{2\,b{d}^{4}{x}^{5}}{9\,{a}^{2}}}+{\frac{10\,bc{d}^{3}{x}^{4}}{9\,{a}^{2}}}+{\frac{20\,{c}^{2}{d}^{2}b{x}^{3}}{9\,{a}^{2}}}+{\frac{d \left ( 40\,b{c}^{3}+7\,a \right ){x}^{2}}{18\,{a}^{2}}}+{\frac{c \left ( 10\,b{c}^{3}+7\,a \right ) x}{9\,{a}^{2}}}+{\frac{{c}^{2} \left ( 4\,b{c}^{3}+7\,a \right ) }{18\,{a}^{2}d}} \right ) }+{\frac{2}{27\,b{a}^{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*x+c)/(a+b*(d*x+c)^3)^3,x)

[Out]

(2/9*b*d^4/a^2*x^5+10/9*b*c*d^3/a^2*x^4+20/9*c^2*d^2*b/a^2*x^3+1/18*d*(40*b*c^3+7*a)/a^2*x^2+1/9*c*(10*b*c^3+7
*a)/a^2*x+1/18*c^2/d*(4*b*c^3+7*a)/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+2/27/a^2/b/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{4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 4 \, b c^{5} +{\left (40 \, b c^{3} + 7 \, a\right )} d^{2} x^{2} + 7 \, a c^{2} + 2 \,{\left (10 \, b c^{4} + 7 \, a c\right )} d x}{18 \,{\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} + \frac{2 \,{\left (-\frac{1}{3} \, \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 ) - \frac{1}{6} \, \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 ) + \frac{1}{3} \, \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 )}}{9 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(4*b*d^5*x^5 + 20*b*c*d^4*x^4 + 40*b*c^2*d^3*x^3 + 4*b*c^5 + (40*b*c^3 + 7*a)*d^2*x^2 + 7*a*c^2 + 2*(10*b
*c^4 + 7*a*c)*d*x)/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b
)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6 + 2*a^3*b
*c^3 + a^4)*d) + 2/9*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^2

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Fricas [B]  time = 1.9354, size = 3613, normalized size = 17.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(12*a*b^3*d^5*x^5 + 60*a*b^3*c*d^4*x^4 + 120*a*b^3*c^2*d^3*x^3 + 12*a*b^3*c^5 + 21*a^2*b^2*c^2 + 3*(40*a
*b^3*c^3 + 7*a^2*b^2)*d^2*x^2 + 6*(10*a*b^3*c^4 + 7*a^2*b^2*c)*d*x + 6*sqrt(1/3)*(a*b^3*d^6*x^6 + 6*a*b^3*c*d^
5*x^5 + 15*a*b^3*c^2*d^4*x^4 + a*b^3*c^6 + 2*a^2*b^2*c^3 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^3*x^3 + 3*(5*a*b^3*c^4
+ 2*a^2*b^2*c)*d^2*x^2 + a^3*b + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d*x)*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)) + 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b
^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*(-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)) - 4*(b^2*d^6*
x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4
+ 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a^3*
b^4*d^7*x^6 + 6*a^3*b^4*c*d^6*x^5 + 15*a^3*b^4*c^2*d^5*x^4 + 2*(10*a^3*b^4*c^3 + a^4*b^3)*d^4*x^3 + 3*(5*a^3*b
^4*c^4 + 2*a^4*b^3*c)*d^3*x^2 + 6*(a^3*b^4*c^5 + a^4*b^3*c^2)*d^2*x + (a^3*b^4*c^6 + 2*a^4*b^3*c^3 + a^5*b^2)*
d), 1/54*(12*a*b^3*d^5*x^5 + 60*a*b^3*c*d^4*x^4 + 120*a*b^3*c^2*d^3*x^3 + 12*a*b^3*c^5 + 21*a^2*b^2*c^2 + 3*(4
0*a*b^3*c^3 + 7*a^2*b^2)*d^2*x^2 + 6*(10*a*b^3*c^4 + 7*a^2*b^2*c)*d*x + 12*sqrt(1/3)*(a*b^3*d^6*x^6 + 6*a*b^3*
c*d^5*x^5 + 15*a*b^3*c^2*d^4*x^4 + a*b^3*c^6 + 2*a^2*b^2*c^3 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^3*x^3 + 3*(5*a*b^3
*c^4 + 2*a^2*b^2*c)*d^2*x^2 + a^3*b + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d*x)*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) + 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^
2*d^4*x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5
+ a*b*c^2)*d*x + a^2)*(-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)) - 4*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x
^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*(-a*b^2)^(2/3)*log(b*d*x +
b*c - (-a*b^2)^(1/3)))/(a^3*b^4*d^7*x^6 + 6*a^3*b^4*c*d^6*x^5 + 15*a^3*b^4*c^2*d^5*x^4 + 2*(10*a^3*b^4*c^3 +
a^4*b^3)*d^4*x^3 + 3*(5*a^3*b^4*c^4 + 2*a^4*b^3*c)*d^3*x^2 + 6*(a^3*b^4*c^5 + a^4*b^3*c^2)*d^2*x + (a^3*b^4*c^
6 + 2*a^4*b^3*c^3 + a^5*b^2)*d)]

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Sympy [A]  time = 17.3841, size = 296, normalized size = 1.47 \begin{align*} \frac{7 a c^{2} + 4 b c^{5} + 40 b c^{2} d^{3} x^{3} + 20 b c d^{4} x^{4} + 4 b d^{5} x^{5} + x^{2} \left (7 a d^{2} + 40 b c^{3} d^{2}\right ) + x \left (14 a c d + 20 b c^{4} d\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{5} b + 4 c}{4 d} \right )} \right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*a*c**2 + 4*b*c**5 + 40*b*c**2*d**3*x**3 + 20*b*c*d**4*x**4 + 4*b*d**5*x**5 + x**2*(7*a*d**2 + 40*b*c**3*d**
2) + x*(14*a*c*d + 20*b*c**4*d))/(18*a**4*d + 36*a**3*b*c**3*d + 18*a**2*b**2*c**6*d + 270*a**2*b**2*c**2*d**5
*x**4 + 108*a**2*b**2*c*d**6*x**5 + 18*a**2*b**2*d**7*x**6 + x**3*(36*a**3*b*d**4 + 360*a**2*b**2*c**3*d**4) +
x**2*(108*a**3*b*c*d**3 + 270*a**2*b**2*c**4*d**3) + x*(108*a**3*b*c**2*d**2 + 108*a**2*b**2*c**5*d**2)) + Ro
otSum(19683*_t**3*a**7*b**2 + 8, Lambda(_t, _t*log(x + (729*_t**2*a**5*b + 4*c)/(4*d))))/d

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Giac [A]  time = 1.19586, size = 370, normalized size = 1.83 \begin{align*} -\frac{2}{27} \, \sqrt{3} \left (-\frac{1}{a^{7} 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 ) - \frac{1}{27} \, \left (-\frac{1}{a^{7} 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 ) + \frac{2}{27} \, \left (-\frac{1}{a^{7} b^{2} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | 9 \, a^{3} b d x + 9 \, a^{3} b c + 9 \, \left (-a^{2} b\right )^{\frac{2}{3}} a^{2} \right |}\right ) + \frac{4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 40 \, b c^{3} d^{2} x^{2} + 20 \, b c^{4} d x + 4 \, b c^{5} + 7 \, a d^{2} x^{2} + 14 \, a c d x + 7 \, a c^{2}}{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^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*sqrt(3)*(-1/(a^7*b^2*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
)) - 1/27*(-1/(a^7*b^2*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)) + 2/27*(-1
/(a^7*b^2*d^3))^(1/3)*log(abs(9*a^3*b*d*x + 9*a^3*b*c + 9*(-a^2*b)^(2/3)*a^2)) + 1/18*(4*b*d^5*x^5 + 20*b*c*d^
4*x^4 + 40*b*c^2*d^3*x^3 + 40*b*c^3*d^2*x^2 + 20*b*c^4*d*x + 4*b*c^5 + 7*a*d^2*x^2 + 14*a*c*d*x + 7*a*c^2)/((b
*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)^2*a^2*d)