3.634 \(\int \log (c (d+\frac{e}{(f+g x)^3})^q) \, dx\)

Optimal. Leaf size=165 \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}-\frac{\sqrt [3]{e} q \log \left (d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}\right )}{2 \sqrt [3]{d} g}+\frac{\sqrt [3]{e} q \log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{\sqrt [3]{d} g}-\frac{\sqrt{3} \sqrt [3]{e} q \tan ^{-1}\left (\frac{\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{\sqrt{3} \sqrt [3]{e}}\right )}{\sqrt [3]{d} g} \]

[Out]

-((Sqrt[3]*e^(1/3)*q*ArcTan[(e^(1/3) - 2*d^(1/3)*(f + g*x))/(Sqrt[3]*e^(1/3))])/(d^(1/3)*g)) + ((f + g*x)*Log[
c*(d + e/(f + g*x)^3)^q])/g + (e^(1/3)*q*Log[e^(1/3) + d^(1/3)*(f + g*x)])/(d^(1/3)*g) - (e^(1/3)*q*Log[e^(2/3
) - d^(1/3)*e^(1/3)*(f + g*x) + d^(2/3)*(f + g*x)^2])/(2*d^(1/3)*g)

________________________________________________________________________________________

Rubi [A]  time = 0.172958, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {2483, 2448, 263, 200, 31, 634, 617, 204, 628} \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}-\frac{\sqrt [3]{e} q \log \left (d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}\right )}{2 \sqrt [3]{d} g}+\frac{\sqrt [3]{e} q \log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{\sqrt [3]{d} g}-\frac{\sqrt{3} \sqrt [3]{e} q \tan ^{-1}\left (\frac{\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{\sqrt{3} \sqrt [3]{e}}\right )}{\sqrt [3]{d} g} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e/(f + g*x)^3)^q],x]

[Out]

-((Sqrt[3]*e^(1/3)*q*ArcTan[(e^(1/3) - 2*d^(1/3)*(f + g*x))/(Sqrt[3]*e^(1/3))])/(d^(1/3)*g)) + ((f + g*x)*Log[
c*(d + e/(f + g*x)^3)^q])/g + (e^(1/3)*q*Log[e^(1/3) + d^(1/3)*(f + g*x)])/(d^(1/3)*g) - (e^(1/3)*q*Log[e^(2/3
) - d^(1/3)*e^(1/3)*(f + g*x) + d^(2/3)*(f + g*x)^2])/(2*d^(1/3)*g)

Rule 2483

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_.))^(q_.), x_Symbol] :> Dist[1/g, Su
bst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q
, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^3}\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}+\frac{(3 e q) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^3}\right ) x^3} \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}+\frac{(3 e q) \operatorname{Subst}\left (\int \frac{1}{e+d x^3} \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}+\frac{\left (\sqrt [3]{e} q\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{e}+\sqrt [3]{d} x} \, dx,x,f+g x\right )}{g}+\frac{\left (\sqrt [3]{e} q\right ) \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{e}-\sqrt [3]{d} x}{e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3} x^2} \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}+\frac{\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac{\left (\sqrt [3]{e} q\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{d} \sqrt [3]{e}+2 d^{2/3} x}{e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3} x^2} \, dx,x,f+g x\right )}{2 \sqrt [3]{d} g}+\frac{\left (3 e^{2/3} q\right ) \operatorname{Subst}\left (\int \frac{1}{e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3} x^2} \, dx,x,f+g x\right )}{2 g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}+\frac{\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac{\sqrt [3]{e} q \log \left (e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{d} g}+\frac{\left (3 \sqrt [3]{e} q\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}\right )}{\sqrt [3]{d} g}\\ &=-\frac{\sqrt{3} \sqrt [3]{e} q \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}}{\sqrt{3}}\right )}{\sqrt [3]{d} g}+\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}+\frac{\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac{\sqrt [3]{e} q \log \left (e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{d} g}\\ \end{align*}

Mathematica [C]  time = 0.333291, size = 66, normalized size = 0.4 \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^3}\right )^q\right )}{g}-\frac{3 e q \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{e}{d (f+g x)^3}\right )}{2 d g (f+g x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e/(f + g*x)^3)^q],x]

[Out]

(-3*e*q*Hypergeometric2F1[2/3, 1, 5/3, -(e/(d*(f + g*x)^3))])/(2*d*g*(f + g*x)^2) + ((f + g*x)*Log[c*(d + e/(f
+ g*x)^3)^q])/g

________________________________________________________________________________________

Maple [C]  time = 0.554, size = 157, normalized size = 1. \begin{align*} \ln \left ( c \left ({\frac{d{g}^{3}{x}^{3}+3\,df{g}^{2}{x}^{2}+3\,d{f}^{2}gx+d{f}^{3}+e}{ \left ( gx+f \right ) ^{3}}} \right ) ^{q} \right ) x-3\,{\frac{qf\ln \left ( gx+f \right ) }{g}}+{\frac{q}{dg}\sum _{{\it \_R}={\it RootOf} \left ( d{g}^{3}{{\it \_Z}}^{3}+3\,df{g}^{2}{{\it \_Z}}^{2}+3\,d{f}^{2}g{\it \_Z}+d{f}^{3}+e \right ) }{\frac{ \left ({{\it \_R}}^{2}df{g}^{2}+2\,{\it \_R}\,d{f}^{2}g+d{f}^{3}+e \right ) \ln \left ( x-{\it \_R} \right ) }{{g}^{2}{{\it \_R}}^{2}+2\,fg{\it \_R}+{f}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e/(g*x+f)^3)^q),x)

[Out]

ln(c*((d*g^3*x^3+3*d*f*g^2*x^2+3*d*f^2*g*x+d*f^3+e)/(g*x+f)^3)^q)*x-3/g*q*f*ln(g*x+f)+1/g*q/d*sum((_R^2*d*f*g^
2+2*_R*d*f^2*g+d*f^3+e)/(_R^2*g^2+2*_R*f*g+f^2)*ln(x-_R),_R=RootOf(_Z^3*d*g^3+3*_Z^2*d*f*g^2+3*_Z*d*f^2*g+d*f^
3+e))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 3 \, q \int \frac{d f g^{2} x^{2} + 2 \, d f^{2} g x + d f^{3} + e}{d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e}\,{d x} - \frac{3 \, f q \log \left (g x + f\right ) - g x \log \left ({\left (d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e\right )}^{q}\right ) + 3 \, g x \log \left ({\left (g x + f\right )}^{q}\right ) - g x \log \left (c\right )}{g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="maxima")

[Out]

3*q*integrate((d*f*g^2*x^2 + 2*d*f^2*g*x + d*f^3 + e)/(d*g^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 + e), x
) - (3*f*q*log(g*x + f) - g*x*log((d*g^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 + e)^q) + 3*g*x*log((g*x +
f)^q) - g*x*log(c))/g

________________________________________________________________________________________

Fricas [C]  time = 9.93869, size = 3146, normalized size = 19.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="fricas")

[Out]

1/4*(4*g*q*x*log((d*g^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 + e)/(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^
3)) - 4*sqrt(3)*g*sqrt((((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqr
t(3) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)
*(I*sqrt(3) + 1) - 2*f*q/g)*f*g*q + 4*f^2*q^2)/g^2)*arctan(-1/24*(2*sqrt(3)*sqrt(4*g^2*q^2*x^2 + 12*f*g*q^2*x
+ ((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2
*g^2 + 12*f^2*q^2 + 2*(g^2*q*x + 3*f*g*q)*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*
g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g))*(((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g
^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*d*g^2 + 2*d*f*g*q)*sqrt((((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d
*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) +
1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*f*g*q + 4*f^2*q^2)/g^2) - sqrt(3)*(8*d*f*g^
2*q^2*x + ((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*
f*q/g)^2*d*g^3 + 12*d*f^2*g*q^2 + 4*(d*g^3*q*x + 2*d*f*g^2*q)*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*
f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g))*sqrt((((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2
*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3
) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*f*g*q + 4*f^2*q^2)/g^2))/(e*q^3)) - 12*f
*q*log(g*x + f) - 2*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3)
+ 1) - 2*f*q/g)*g*log(q*x - 1/2*(-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3
)*(I*sqrt(3) + 1) + f*q/g) + 4*g*x*log(c) + (((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/
(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*g + 6*f*q)*log(4*g^2*q^2*x^2 + 12*f*g*q^2*x + ((-1/2*f^3*q^3/g^3 + 1
/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2*g^2 + 12*f^2*q^2 + 2*(g
^2*q*x + 3*f*g*q)*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) +
1) - 2*f*q/g)))/g

________________________________________________________________________________________

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(ln(c*(d+e/(g*x+f)**3)**q),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="giac")

[Out]

Timed out