3.107 \(\int \frac{1}{x (a+b (c+d x)^3)} \, dx\)
Optimal. Leaf size=224 \[ -\frac{\left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} c+b^{2/3} c^2\right )}+\frac{\sqrt [3]{b} c \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} c+b^{2/3} c^2\right )}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )}+\frac{\log (x)}{a+b c^3} \]
[Out]
(b^(1/3)*c*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^(2/3) - a^(1/3)*b^(1
/3)*c + b^(2/3)*c^2)) + Log[x]/(a + b*c^3) - Log[a^(1/3) + b^(1/3)*(c + d*x)]/(3*a^(2/3)*(a^(1/3) + b^(1/3)*c)
) - ((2*a^(1/3) - b^(1/3)*c)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(6*a^(2/3)*(a^(2/
3) - a^(1/3)*b^(1/3)*c + b^(2/3)*c^2))
________________________________________________________________________________________
Rubi [A] time = 0.481797, antiderivative size = 238, normalized size of antiderivative =
1.06, number of steps used = 11, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588, Rules
used = {371, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}+\frac{\log (x)}{a+b c^3} \]
Antiderivative was successfully verified.
[In]
Int[1/(x*(a + b*(c + d*x)^3)),x]
[Out]
(b^(1/3)*c*(a^(1/3) + b^(1/3)*c)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(
a + b*c^3)) + Log[x]/(a + b*c^3) + (b^(1/3)*c*(a^(1/3) - b^(1/3)*c)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3*a^(2/
3)*(a + b*c^3)) - (b^(1/3)*c*(a^(1/3) - b^(1/3)*c)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)
^2])/(6*a^(2/3)*(a + b*c^3)) - Log[a + b*(c + d*x)^3]/(3*(a + b*c^3))
Rule 371
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Rule 6725
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rule 1871
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Rule 1860
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]
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]
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b (c+d x)^3\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(-c+x) \left (a+b x^3\right )} \, dx,x,c+d x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{\left (a+b c^3\right ) (c-x)}-\frac{b \left (c^2+c x+x^2\right )}{\left (a+b c^3\right ) \left (a+b x^3\right )}\right ) \, dx,x,c+d x\right )\\ &=\frac{\log (x)}{a+b c^3}-\frac{b \operatorname{Subst}\left (\int \frac{c^2+c x+x^2}{a+b x^3} \, dx,x,c+d x\right )}{a+b c^3}\\ &=\frac{\log (x)}{a+b c^3}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,c+d x\right )}{a+b c^3}-\frac{b \operatorname{Subst}\left (\int \frac{c^2+c x}{a+b x^3} \, dx,x,c+d x\right )}{a+b c^3}\\ &=\frac{\log (x)}{a+b c^3}-\frac{\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (\sqrt [3]{a} c+2 \sqrt [3]{b} c^2\right )+\sqrt [3]{b} \left (\sqrt [3]{a} c-\sqrt [3]{b} c^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 a^{2/3} \left (a+b c^3\right )}+\frac{\left (b^{2/3} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{2/3} \left (a+b c^3\right )}\\ &=\frac{\log (x)}{a+b c^3}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}-\frac{\left (\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )\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 )}{6 a^{2/3} \left (a+b c^3\right )}-\frac{\left (b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )\right ) \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 )}{2 \sqrt [3]{a} \left (a+b c^3\right )}\\ &=\frac{\log (x)}{a+b c^3}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}-\frac{\left (\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )\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 )}{a^{2/3} \left (a+b c^3\right )}\\ &=\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )}+\frac{\log (x)}{a+b c^3}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}\\ \end{align*}
Mathematica [C] time = 0.0484136, size = 119, normalized size = 0.53 \[ -\frac{\text{RootSum}\left [3 \text{$\#$1}^2 b c d^2+\text{$\#$1}^3 b d^3+3 \text{$\#$1} b c^2 d+a+b c^3\& ,\frac{\text{$\#$1}^2 d^2 \log (x-\text{$\#$1})+3 c^2 \log (x-\text{$\#$1})+3 \text{$\#$1} c d \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\& \right ]-3 \log (x)}{3 \left (a+b c^3\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x*(a + b*(c + d*x)^3)),x]
[Out]
-(-3*Log[x] + RootSum[a + b*c^3 + 3*b*c^2*d*#1 + 3*b*c*d^2*#1^2 + b*d^3*#1^3 & , (3*c^2*Log[x - #1] + 3*c*d*Lo
g[x - #1]*#1 + d^2*Log[x - #1]*#1^2)/(c^2 + 2*c*d*#1 + d^2*#1^2) & ])/(3*(a + b*c^3))
________________________________________________________________________________________
Maple [C] time = 0.007, size = 105, normalized size = 0.5 \begin{align*} -{\frac{1}{3\,b{c}^{3}+3\,a}\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 ({d}^{2}{{\it \_R}}^{2}+3\,cd{\it \_R}+3\,{c}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}+{\frac{\ln \left ( x \right ) }{b{c}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(a+b*(d*x+c)^3),x)
[Out]
-1/3*sum((_R^2*d^2+3*_R*c*d+3*c^2)/(_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))/(b*c^3+a)+ln(x)/(b*c^3+a)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(a+b*(d*x+c)^3),x, algorithm="maxima")
[Out]
Exception raised: AttributeError
________________________________________________________________________________________
Fricas [C] time = 8.20721, size = 9806, normalized size = 43.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(a+b*(d*x+c)^3),x, algorithm="fricas")
[Out]
1/12*(2*(b*c^3 + a)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^
2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(
3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^
(1/3) - 2/(b*c^3 + a))*log(b*c^2*d*x + b*c^3 + 1/4*(a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c
^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)
) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/
((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a))^2 - 1/2*(a*b*c^3 - 2*a^2)*(2*(1/2)^(2/
3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3
/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^
2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a)) + a) - ((
b*c^3 + a)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) -
1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(
b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2
/(b*c^3 + a)) + 3*sqrt(1/3)*(b*c^3 + a)*sqrt(-(16*b*c^3 + (a*b^2*c^6 + 2*a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*s
qrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c
^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(
a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a))^2 + 4*(a*b*c^3 +
a^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2
*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/
((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^
3 + a)) + 4*a)/(a*b^2*c^6 + 2*a^2*b*c^3 + a^3)) + 6)*log(2*b*c^2*d*x + 2*b*c^3 - 1/4*(a^2*b*c^3 + a^3)*(2*(1/2
)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3
) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)
^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a))^2 +
1/2*(a*b*c^3 - 2*a^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a
)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqr
t(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3
)^(1/3) - 2/(b*c^3 + a)) + 3/4*sqrt(1/3)*(2*a*b*c^3 + (a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*
b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 +
a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) +
3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a)) + 2*a^2)*sqrt(-(16*b*c^3 + (a*b^2*c
^6 + 2*a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 +
a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*s
qrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)
^3)^(1/3) - 2/(b*c^3 + a))^2 + 4*(a*b*c^3 + a^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3
+ a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)
^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*
c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a)) + 4*a)/(a*b^2*c^6 + 2*a^2*b*c^3 + a^3)) - a) - ((b*c^3 + a
)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*
c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b
*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 +
a)) - 3*sqrt(1/3)*(b*c^3 + a)*sqrt(-(16*b*c^3 + (a*b^2*c^6 + 2*a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) +
1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)
*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3
+ a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a))^2 + 4*(a*b*c^3 + a^2)*(2*(
1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 +
a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 +
a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a)) +
4*a)/(a*b^2*c^6 + 2*a^2*b*c^3 + a^3)) + 6)*log(2*b*c^2*d*x + 2*b*c^3 - 1/4*(a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(
-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a
*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) -
1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a))^2 + 1/2*(a*b*
c^3 - 2*a^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2)
- 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)
*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) -
2/(b*c^3 + a)) - 3/4*sqrt(1/3)*(2*a*b*c^3 + (a^2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a
^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/
(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*
c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a)) + 2*a^2)*sqrt(-(16*b*c^3 + (a*b^2*c^6 + 2*a^
2*b*c^3 + a^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/(b*c^3/((b*c^3 + a)^2*a^2
) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) - (1/2)^(1/3)*(I*sqrt(3) +
1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3)
- 2/(b*c^3 + a))^2 + 4*(a*b*c^3 + a^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a*b*c^3 + a^2) - 1/(b*c^3 + a)^2)/
(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a)) - 2/(b*c^3 + a)^3)^(1/3) -
(1/2)^(1/3)*(I*sqrt(3) + 1)*(b*c^3/((b*c^3 + a)^2*a^2) - 1/(a^2*b*c^3 + a^3) + 3/((a*b*c^3 + a^2)*(b*c^3 + a))
- 2/(b*c^3 + a)^3)^(1/3) - 2/(b*c^3 + a)) + 4*a)/(a*b^2*c^6 + 2*a^2*b*c^3 + a^3)) - a) + 12*log(x))/(b*c^3 +
a)
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Sympy [B] time = 17.9407, size = 559, normalized size = 2.5 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} + 27 a^{2} b c^{3}\right ) + 27 t^{2} a^{2} + 9 t a + 1, \left ( t \mapsto t \log{\left (x + \frac{- 432 t^{3} a^{6} - 837 t^{3} a^{5} b c^{3} - 405 t^{3} a^{4} b^{2} c^{6} - 27 t^{3} a^{3} b^{3} c^{9} - 27 t^{3} a^{2} b^{4} c^{12} + 144 t^{2} a^{5} + 270 t^{2} a^{4} b c^{3} + 108 t^{2} a^{3} b^{2} c^{6} - 18 t^{2} a^{2} b^{3} c^{9} + 240 t a^{4} - 261 t a^{3} b c^{3} - 27 t a^{2} b^{2} c^{6} - 12 t a b^{3} c^{9} + 48 a^{3} + 60 a^{2} b c^{3} + 12 a b^{2} c^{6}}{64 a^{2} b c^{2} d + 11 a b^{2} c^{5} d + b^{3} c^{8} d} \right )} \right )\right )} + \frac{\log{\left (x + \frac{- \frac{432 a^{6}}{\left (a + b c^{3}\right )^{3}} - \frac{837 a^{5} b c^{3}}{\left (a + b c^{3}\right )^{3}} + \frac{144 a^{5}}{\left (a + b c^{3}\right )^{2}} - \frac{405 a^{4} b^{2} c^{6}}{\left (a + b c^{3}\right )^{3}} + \frac{270 a^{4} b c^{3}}{\left (a + b c^{3}\right )^{2}} + \frac{240 a^{4}}{a + b c^{3}} - \frac{27 a^{3} b^{3} c^{9}}{\left (a + b c^{3}\right )^{3}} + \frac{108 a^{3} b^{2} c^{6}}{\left (a + b c^{3}\right )^{2}} - \frac{261 a^{3} b c^{3}}{a + b c^{3}} + 48 a^{3} - \frac{27 a^{2} b^{4} c^{12}}{\left (a + b c^{3}\right )^{3}} - \frac{18 a^{2} b^{3} c^{9}}{\left (a + b c^{3}\right )^{2}} - \frac{27 a^{2} b^{2} c^{6}}{a + b c^{3}} + 60 a^{2} b c^{3} - \frac{12 a b^{3} c^{9}}{a + b c^{3}} + 12 a b^{2} c^{6}}{64 a^{2} b c^{2} d + 11 a b^{2} c^{5} d + b^{3} c^{8} d} \right )}}{a + b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(a+b*(d*x+c)**3),x)
[Out]
RootSum(_t**3*(27*a**3 + 27*a**2*b*c**3) + 27*_t**2*a**2 + 9*_t*a + 1, Lambda(_t, _t*log(x + (-432*_t**3*a**6
- 837*_t**3*a**5*b*c**3 - 405*_t**3*a**4*b**2*c**6 - 27*_t**3*a**3*b**3*c**9 - 27*_t**3*a**2*b**4*c**12 + 144*
_t**2*a**5 + 270*_t**2*a**4*b*c**3 + 108*_t**2*a**3*b**2*c**6 - 18*_t**2*a**2*b**3*c**9 + 240*_t*a**4 - 261*_t
*a**3*b*c**3 - 27*_t*a**2*b**2*c**6 - 12*_t*a*b**3*c**9 + 48*a**3 + 60*a**2*b*c**3 + 12*a*b**2*c**6)/(64*a**2*
b*c**2*d + 11*a*b**2*c**5*d + b**3*c**8*d)))) + log(x + (-432*a**6/(a + b*c**3)**3 - 837*a**5*b*c**3/(a + b*c*
*3)**3 + 144*a**5/(a + b*c**3)**2 - 405*a**4*b**2*c**6/(a + b*c**3)**3 + 270*a**4*b*c**3/(a + b*c**3)**2 + 240
*a**4/(a + b*c**3) - 27*a**3*b**3*c**9/(a + b*c**3)**3 + 108*a**3*b**2*c**6/(a + b*c**3)**2 - 261*a**3*b*c**3/
(a + b*c**3) + 48*a**3 - 27*a**2*b**4*c**12/(a + b*c**3)**3 - 18*a**2*b**3*c**9/(a + b*c**3)**2 - 27*a**2*b**2
*c**6/(a + b*c**3) + 60*a**2*b*c**3 - 12*a*b**3*c**9/(a + b*c**3) + 12*a*b**2*c**6)/(64*a**2*b*c**2*d + 11*a*b
**2*c**5*d + b**3*c**8*d))/(a + b*c**3)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left ({\left (d x + c\right )}^{3} b + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(a+b*(d*x+c)^3),x, algorithm="giac")
[Out]
integrate(1/(((d*x + c)^3*b + a)*x), x)