\(\int \frac {1}{(c e+d e x)^2 (a+b (c+d x)^3)} \, dx\) [2891]

Optimal result

Integrand size = 24, antiderivative size = 166 \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=-\frac {1}{a d e^2 (c+d x)}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d e^2}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2} \]

[Out]

-1/a/d/e^2/(d*x+c)+1/3*b^(1/3)*ln(a^(1/3)+b^(1/3)*(d*x+c))/a^(4/3)/d/e^2-1/6*b^(1/3)*ln(a^(2/3)-a^(1/3)*b^(1/3
)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(4/3)/d/e^2+1/3*b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))
/a^(4/3)/d/e^2*3^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {379, 331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d e^2}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}-\frac {1}{a d e^2 (c+d x)} \]

[In]

Int[1/((c*e + d*e*x)^2*(a + b*(c + d*x)^3)),x]

[Out]

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

Rule 31

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 298

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 331

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 379

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 631

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 642

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 648

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {1}{a d e^2 (c+d x)}-\frac {b \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{a d e^2} \\ & = -\frac {1}{a d e^2 (c+d x)}+\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{4/3} d e^2}-\frac {b^{2/3} \text {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 )}{3 a^{4/3} d e^2} \\ & = -\frac {1}{a d e^2 (c+d x)}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}-\frac {\sqrt [3]{b} \text {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^{4/3} d e^2}-\frac {b^{2/3} \text {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 a d e^2} \\ & = -\frac {1}{a d e^2 (c+d x)}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{a^{4/3} d e^2} \\ & = -\frac {1}{a d e^2 (c+d x)}+\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} d e^2}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {-\frac {6 \sqrt [3]{a}}{c+d x}-2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2} \]

[In]

Integrate[1/((c*e + d*e*x)^2*(a + b*(c + d*x)^3)),x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.58

method	result	size
default	\(\frac {-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 d a}-\frac {1}{a d \left (d x +c \right )}}{e^{2}}\)	\(96\)
risch	\(-\frac {1}{a d \,e^{2} \left (d x +c \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} d^{3} e^{6} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{4} d^{4} e^{6} \textit {\_R}^{3}+3 b d \right ) x -4 a^{4} c \,d^{3} e^{6} \textit {\_R}^{3}-a^{3} d^{2} e^{4} \textit {\_R}^{2}+3 b c \right )\right )}{3}\)	\(101\)

[In]

int(1/(d*e*x+c*e)^2/(a+b*(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/e^2*(-1/3/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))/a-1/a/d/(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=-\frac {2 \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - {\left (a d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d x + b c + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, {\left (a d^{2} e^{2} x + a c d e^{2}\right )}} \]

[In]

integrate(1/(d*e*x+c*e)^2/(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/6*(2*sqrt(3)*(d*x + c)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*(d*x + c)*(b/a)^(1/3) - 1/3*sqrt(3)) + (d*x + c)*(b/a
)^(1/3)*log(b*d^2*x^2 + 2*b*c*d*x + b*c^2 - (a*d*x + a*c)*(b/a)^(2/3) + a*(b/a)^(1/3)) - 2*(d*x + c)*(b/a)^(1/
3)*log(b*d*x + b*c + a*(b/a)^(2/3)) + 6)/(a*d^2*e^2*x + a*c*d*e^2)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.33 \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=- \frac {1}{a c d e^{2} + a d^{2} e^{2} x} + \frac {\operatorname {RootSum} {\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} a^{3} + b c}{b d} \right )} \right )\right )}}{d e^{2}} \]

[In]

integrate(1/(d*e*x+c*e)**2/(a+b*(d*x+c)**3),x)

[Out]

-1/(a*c*d*e**2 + a*d**2*e**2*x) + RootSum(27*_t**3*a**4 - b, Lambda(_t, _t*log(x + (9*_t**2*a**3 + b*c)/(b*d))
))/(d*e**2)

Maxima [F]

\[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )} {\left (d e x + c e\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*e*x+c*e)^2/(a+b*(d*x+c)^3),x, algorithm="maxima")

[Out]

-1/(a*d^2*e^2*x + a*c*d*e^2) - b*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*e^2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {\left (\frac {b}{a d^{3} e^{6}}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {b}{a d^{3} e^{6}}\right )^{\frac {1}{3}} - \frac {1}{{\left (d e x + c e\right )} d e} \right |}\right )}{3 \, a} - \frac {\sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (\frac {b}{a d^{3} e^{6}}\right )^{\frac {1}{3}} - \frac {2}{{\left (d e x + c e\right )} d e}\right )}}{3 \, \left (\frac {b}{a d^{3} e^{6}}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} d e^{2}} - \frac {\left (a^{2} b\right )^{\frac {1}{3}} \log \left (\left (\frac {b}{a d^{3} e^{6}}\right )^{\frac {2}{3}} - \frac {\left (\frac {b}{a d^{3} e^{6}}\right )^{\frac {1}{3}}}{{\left (d e x + c e\right )} d e} + \frac {1}{{\left (d e x + c e\right )}^{2} d^{2} e^{2}}\right )}{6 \, a^{2} d e^{2}} - \frac {1}{{\left (d e x + c e\right )} a d e} \]

[In]

integrate(1/(d*e*x+c*e)^2/(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

1/3*(b/(a*d^3*e^6))^(1/3)*log(abs(-(b/(a*d^3*e^6))^(1/3) - 1/((d*e*x + c*e)*d*e)))/a - 1/3*sqrt(3)*(a^2*b)^(1/
3)*arctan(1/3*sqrt(3)*((b/(a*d^3*e^6))^(1/3) - 2/((d*e*x + c*e)*d*e))/(b/(a*d^3*e^6))^(1/3))/(a^2*d*e^2) - 1/6
*(a^2*b)^(1/3)*log((b/(a*d^3*e^6))^(2/3) - (b/(a*d^3*e^6))^(1/3)/((d*e*x + c*e)*d*e) + 1/((d*e*x + c*e)^2*d^2*
e^2))/(a^2*d*e^2) - 1/((d*e*x + c*e)*a*d*e)

Mupad [B] (verification not implemented)

Time = 5.88 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {b^{1/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{3\,a^{4/3}\,d\,e^2}-\frac {1}{a\,d\,\left (c\,e^2+d\,e^2\,x\right )}-\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,d\,e^2}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}\,d\,e^2} \]

[In]

int(1/((c*e + d*e*x)^2*(a + b*(c + d*x)^3)),x)

[Out]

(b^(1/3)*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x))/(3*a^(4/3)*d*e^2) - 1/(a*d*(c*e^2 + d*e^2*x)) - (b^(1/3)*log(
3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*c - a^(1/3) + 2*b^(1/3)*d*x)*((3^(1/2)*1i)/2 + 1/2))/(3*a^(4/3)*d*e^2) + (b^(1/
3)*log(2*b^(1/3)*c - 3^(1/2)*a^(1/3)*1i - a^(1/3) + 2*b^(1/3)*d*x)*((3^(1/2)*1i)/6 - 1/6))/(a^(4/3)*d*e^2)