3.5.55 \(\int \frac {(g+h x)^2}{(a+b \log (c (d (e+f x)^p)^q))^3} \, dx\) [455]

Optimal. Leaf size=432 \[ \frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^3 p^3 q^3}+\frac {9 e^{-\frac {3 a}{b p q}} h^2 (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]

[Out]

1/2*(-e*h+f*g)^2*(f*x+e)*Ei((a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b^3/exp(a/b/p/q)/f^3/p^3/q^3/((c*(d*(f*x+e)^p)^
q)^(1/p/q))+4*h*(-e*h+f*g)*(f*x+e)^2*Ei(2*(a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b^3/exp(2*a/b/p/q)/f^3/p^3/q^3/((
c*(d*(f*x+e)^p)^q)^(2/p/q))+9/2*h^2*(f*x+e)^3*Ei(3*(a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b^3/exp(3*a/b/p/q)/f^3/p
^3/q^3/((c*(d*(f*x+e)^p)^q)^(3/p/q))-1/2*(f*x+e)*(h*x+g)^2/b/f/p/q/(a+b*ln(c*(d*(f*x+e)^p)^q))^2+(-e*h+f*g)*(f
*x+e)*(h*x+g)/b^2/f^2/p^2/q^2/(a+b*ln(c*(d*(f*x+e)^p)^q))-3/2*(f*x+e)*(h*x+g)^2/b^2/f/p^2/q^2/(a+b*ln(c*(d*(f*
x+e)^p)^q))

________________________________________________________________________________________

Rubi [A]
time = 1.49, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347, 2495} \begin {gather*} \frac {4 h (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^3 p^3 q^3}+\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}+\frac {9 h^2 (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}+\frac {(e+f x) (g+h x) (f g-e h)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^2/(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

((f*g - e*h)^2*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(2*b^3*E^(a/(b*p*q))*f^3*p^3
*q^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (4*h*(f*g - e*h)*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x
)^p)^q]))/(b*p*q)])/(b^3*E^((2*a)/(b*p*q))*f^3*p^3*q^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (9*h^2*(e + f*x)^3*E
xpIntegralEi[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(2*b^3*E^((3*a)/(b*p*q))*f^3*p^3*q^3*(c*(d*(e + f*
x)^p)^q)^(3/(p*q))) - ((e + f*x)*(g + h*x)^2)/(2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) + ((f*g - e*h)*(e
+ f*x)*(g + h*x))/(b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])) - (3*(e + f*x)*(g + h*x)^2)/(2*b^2*f*p^2
*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

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

Rule 2446

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

Rule 2447

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(d
+ e*x)*(f + g*x)^q*((a + b*Log[c*(d + e*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[q*((e*f - d*g)/(b*e*n*(p + 1))), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx &=\text {Subst}\left (\int \frac {(g+h x)^2}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\text {Subst}\left (\frac {3 \int \frac {(g+h x)^2}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2} \, dx}{2 b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(f g-e h) \int \frac {g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\text {Subst}\left (\frac {9 \int \frac {(g+h x)^2}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{2 b^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(2 (f g-e h)) \int \frac {g+h x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 (f g-e h)) \int \frac {g+h x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h)^2 \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\text {Subst}\left (\frac {9 \int \left (\frac {(f g-e h)^2}{f^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}+\frac {2 h (f g-e h) (e+f x)}{f^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}+\frac {h^2 (e+f x)^2}{f^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right ) \, dx}{2 b^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(2 (f g-e h)) \int \left (\frac {f g-e h}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}+\frac {h (e+f x)}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right ) \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 (f g-e h)) \int \left (\frac {f g-e h}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}+\frac {h (e+f x)}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right ) \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h)^2 \text {Subst}\left (\int \frac {1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\text {Subst}\left (\frac {\left (9 h^2\right ) \int \frac {(e+f x)^2}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{2 b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(2 h (f g-e h)) \int \frac {e+f x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 h (f g-e h)) \int \frac {e+f x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(9 h (f g-e h)) \int \frac {e+f x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (2 (f g-e h)^2\right ) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 (f g-e h)^2\right ) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (9 (f g-e h)^2\right ) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{2 b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left ((f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^3 f^3 p^3 q^3}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\text {Subst}\left (\frac {\left (9 h^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(2 h (f g-e h)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 h (f g-e h)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(9 h (f g-e h)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (2 (f g-e h)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 (f g-e h)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (9 (f g-e h)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^3 f^3 p^3 q^3}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\text {Subst}\left (\frac {\left (9 h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{2 b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (2 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (9 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (2 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (9 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{2 b^2 f^3 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^3 p^3 q^3}+\frac {9 e^{-\frac {3 a}{b p q}} h^2 (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 1.45, size = 438, normalized size = 1.01 \begin {gather*} \frac {e^{-\frac {3 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (e^{\frac {2 a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-8 e^{\frac {a}{b p q}} h (-f g+e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+9 h^2 (e+f x)^2 \text {Ei}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-b e^{\frac {3 a}{b p q}} f p q \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}} (g+h x) \left (b f p q (g+h x)+a (f g+2 e h+3 f h x)+b (2 e h+f (g+3 h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{2 b^3 f^3 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^2/(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

((e + f*x)*(E^((2*a)/(b*p*q))*(f*g - e*h)^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*ExpIntegralEi[(a + b*Log[c*(d*(e +
f*x)^p)^q])/(b*p*q)]*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 - 8*E^(a/(b*p*q))*h*(-(f*g) + e*h)*(e + f*x)*(c*(d*(e
+ f*x)^p)^q)^(1/(p*q))*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)]*(a + b*Log[c*(d*(e + f*x)^
p)^q])^2 + 9*h^2*(e + f*x)^2*ExpIntegralEi[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)]*(a + b*Log[c*(d*(e +
f*x)^p)^q])^2 - b*E^((3*a)/(b*p*q))*f*p*q*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(g + h*x)*(b*f*p*q*(g + h*x) + a*(f*
g + 2*e*h + 3*f*h*x) + b*(2*e*h + f*(g + 3*h*x))*Log[c*(d*(e + f*x)^p)^q])))/(2*b^3*E^((3*a)/(b*p*q))*f^3*p^3*
q^3*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (h x +g \right )^{2}}{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2/(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

[Out]

int((h*x+g)^2/(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="maxima")

[Out]

-1/2*((3*a*f^2*h^2 + (f^2*h^2*p*q + 3*f^2*h^2*q*log(d) + 3*f^2*h^2*log(c))*b)*x^3 + (4*a*f^2*g*h + 2*(f^2*g*h*
p*q + 2*f^2*g*h*q*log(d) + 2*f^2*g*h*log(c))*b + (5*a*f*h^2 + (f*h^2*p*q + 5*f*h^2*q*log(d) + 5*f*h^2*log(c))*
b)*e)*x^2 + (a*f^2*g^2 + (f^2*g^2*p*q + f^2*g^2*q*log(d) + f^2*g^2*log(c))*b + 2*(a*h^2 + (h^2*q*log(d) + h^2*
log(c))*b)*e^2 + 2*(3*a*f*g*h + (f*g*h*p*q + 3*f*g*h*q*log(d) + 3*f*g*h*log(c))*b)*e)*x + 2*(a*g*h + (g*h*q*lo
g(d) + g*h*log(c))*b)*e^2 + (a*f*g^2 + (f*g^2*p*q + f*g^2*q*log(d) + f*g^2*log(c))*b)*e + (3*b*f^2*h^2*x^3 + b
*f*g^2*e + 2*b*g*h*e^2 + (4*b*f^2*g*h + 5*b*f*h^2*e)*x^2 + (b*f^2*g^2 + 6*b*f*g*h*e + 2*b*h^2*e^2)*x)*log(((f*
x + e)^p)^q))/(b^4*f^2*p^2*q^2*log(((f*x + e)^p)^q)^2 + a^2*b^2*f^2*p^2*q^2 + 2*(f^2*p^2*q^3*log(d) + f^2*p^2*
q^2*log(c))*a*b^3 + (f^2*p^2*q^4*log(d)^2 + 2*f^2*p^2*q^3*log(c)*log(d) + f^2*p^2*q^2*log(c)^2)*b^4 + 2*(a*b^3
*f^2*p^2*q^2 + (f^2*p^2*q^3*log(d) + f^2*p^2*q^2*log(c))*b^4)*log(((f*x + e)^p)^q)) + integrate(1/2*(9*f^2*h^2
*x^2 + f^2*g^2 + 6*f*g*h*e + 2*h^2*e^2 + 2*(4*f^2*g*h + 5*f*h^2*e)*x)/(b^3*f^2*p^2*q^2*log(((f*x + e)^p)^q) +
a*b^2*f^2*p^2*q^2 + (f^2*p^2*q^3*log(d) + f^2*p^2*q^2*log(c))*b^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1803 vs. \(2 (444) = 888\).
time = 0.39, size = 1803, normalized size = 4.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="fricas")

[Out]

1/2*(8*(a^2*f*g*h - a^2*h^2*e + (b^2*f*g*h*p^2*q^2 - b^2*h^2*p^2*q^2*e)*log(f*x + e)^2 + (b^2*f*g*h - b^2*h^2*
e)*log(c)^2 + (b^2*f*g*h*q^2 - b^2*h^2*q^2*e)*log(d)^2 + 2*(a*b*f*g*h*p*q - a*b*h^2*p*q*e + (b^2*f*g*h*p*q - b
^2*h^2*p*q*e)*log(c) + (b^2*f*g*h*p*q^2 - b^2*h^2*p*q^2*e)*log(d))*log(f*x + e) + 2*(a*b*f*g*h - a*b*h^2*e)*lo
g(c) + 2*(a*b*f*g*h*q - a*b*h^2*q*e + (b^2*f*g*h*q - b^2*h^2*q*e)*log(c))*log(d))*e^((b*q*log(d) + b*log(c) +
a)/(b*p*q))*log_integral((f^2*x^2 + 2*f*x*e + e^2)*e^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q))) + (a^2*f^2*g^2 -
2*a^2*f*g*h*e + a^2*h^2*e^2 + (b^2*f^2*g^2*p^2*q^2 - 2*b^2*f*g*h*p^2*q^2*e + b^2*h^2*p^2*q^2*e^2)*log(f*x + e
)^2 + (b^2*f^2*g^2 - 2*b^2*f*g*h*e + b^2*h^2*e^2)*log(c)^2 + (b^2*f^2*g^2*q^2 - 2*b^2*f*g*h*q^2*e + b^2*h^2*q^
2*e^2)*log(d)^2 + 2*(a*b*f^2*g^2*p*q - 2*a*b*f*g*h*p*q*e + a*b*h^2*p*q*e^2 + (b^2*f^2*g^2*p*q - 2*b^2*f*g*h*p*
q*e + b^2*h^2*p*q*e^2)*log(c) + (b^2*f^2*g^2*p*q^2 - 2*b^2*f*g*h*p*q^2*e + b^2*h^2*p*q^2*e^2)*log(d))*log(f*x
+ e) + 2*(a*b*f^2*g^2 - 2*a*b*f*g*h*e + a*b*h^2*e^2)*log(c) + 2*(a*b*f^2*g^2*q - 2*a*b*f*g*h*q*e + a*b*h^2*q*e
^2 + (b^2*f^2*g^2*q - 2*b^2*f*g*h*q*e + b^2*h^2*q*e^2)*log(c))*log(d))*e^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q
))*log_integral((f*x + e)*e^((b*q*log(d) + b*log(c) + a)/(b*p*q))) - ((b^2*f^3*h^2*p^2*q^2 + 3*a*b*f^3*h^2*p*q
)*x^3 + 2*(b^2*f^3*g*h*p^2*q^2 + 2*a*b*f^3*g*h*p*q)*x^2 + (b^2*f^3*g^2*p^2*q^2 + a*b*f^3*g^2*p*q)*x + 2*(a*b*f
*h^2*p*q*x + a*b*f*g*h*p*q)*e^2 + (b^2*f^2*g^2*p^2*q^2 + a*b*f^2*g^2*p*q + (b^2*f^2*h^2*p^2*q^2 + 5*a*b*f^2*h^
2*p*q)*x^2 + 2*(b^2*f^2*g*h*p^2*q^2 + 3*a*b*f^2*g*h*p*q)*x)*e + (3*b^2*f^3*h^2*p^2*q^2*x^3 + 4*b^2*f^3*g*h*p^2
*q^2*x^2 + b^2*f^3*g^2*p^2*q^2*x + 2*(b^2*f*h^2*p^2*q^2*x + b^2*f*g*h*p^2*q^2)*e^2 + (5*b^2*f^2*h^2*p^2*q^2*x^
2 + 6*b^2*f^2*g*h*p^2*q^2*x + b^2*f^2*g^2*p^2*q^2)*e)*log(f*x + e) + (3*b^2*f^3*h^2*p*q*x^3 + 4*b^2*f^3*g*h*p*
q*x^2 + b^2*f^3*g^2*p*q*x + 2*(b^2*f*h^2*p*q*x + b^2*f*g*h*p*q)*e^2 + (5*b^2*f^2*h^2*p*q*x^2 + 6*b^2*f^2*g*h*p
*q*x + b^2*f^2*g^2*p*q)*e)*log(c) + (3*b^2*f^3*h^2*p*q^2*x^3 + 4*b^2*f^3*g*h*p*q^2*x^2 + b^2*f^3*g^2*p*q^2*x +
2*(b^2*f*h^2*p*q^2*x + b^2*f*g*h*p*q^2)*e^2 + (5*b^2*f^2*h^2*p*q^2*x^2 + 6*b^2*f^2*g*h*p*q^2*x + b^2*f^2*g^2*
p*q^2)*e)*log(d))*e^(3*(b*q*log(d) + b*log(c) + a)/(b*p*q)) + 9*(b^2*h^2*p^2*q^2*log(f*x + e)^2 + b^2*h^2*q^2*
log(d)^2 + b^2*h^2*log(c)^2 + 2*a*b*h^2*log(c) + a^2*h^2 + 2*(b^2*h^2*p*q^2*log(d) + b^2*h^2*p*q*log(c) + a*b*
h^2*p*q)*log(f*x + e) + 2*(b^2*h^2*q*log(c) + a*b*h^2*q)*log(d))*log_integral((f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e
^2 + e^3)*e^(3*(b*q*log(d) + b*log(c) + a)/(b*p*q))))*e^(-3*(b*q*log(d) + b*log(c) + a)/(b*p*q))/(b^5*f^3*p^5*
q^5*log(f*x + e)^2 + b^5*f^3*p^3*q^5*log(d)^2 + b^5*f^3*p^3*q^3*log(c)^2 + 2*a*b^4*f^3*p^3*q^3*log(c) + a^2*b^
3*f^3*p^3*q^3 + 2*(b^5*f^3*p^4*q^5*log(d) + b^5*f^3*p^4*q^4*log(c) + a*b^4*f^3*p^4*q^4)*log(f*x + e) + 2*(b^5*
f^3*p^3*q^4*log(c) + a*b^4*f^3*p^3*q^4)*log(d))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (g + h x\right )^{2}}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**2/(a+b*ln(c*(d*(f*x+e)**p)**q))**3,x)

[Out]

Integral((g + h*x)**2/(a + b*log(c*(d*(e + f*x)**p)**q))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6028 vs. \(2 (444) = 888\).
time = 3.75, size = 6028, normalized size = 13.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="giac")

[Out]

-1/2*((f*x + e)*b^2*f^2*g^2*p^2*q^2*log(f*x + e) + 4*(f*x + e)^2*b^2*f*g*h*p^2*q^2*log(f*x + e) + 3*(f*x + e)^
3*b^2*h^2*p^2*q^2*log(f*x + e) - 2*(f*x + e)*b^2*f*g*h*p^2*q^2*e*log(f*x + e) - 4*(f*x + e)^2*b^2*h^2*p^2*q^2*
e*log(f*x + e) - b^2*f^2*g^2*p^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log
(f*x + e)^2/(c^(1/(p*q))*d^(1/p)) + (f*x + e)*b^2*f^2*g^2*p^2*q^2 + 2*(f*x + e)^2*b^2*f*g*h*p^2*q^2 + (f*x + e
)^3*b^2*h^2*p^2*q^2 - 2*(f*x + e)*b^2*f*g*h*p^2*q^2*e - 2*(f*x + e)^2*b^2*h^2*p^2*q^2*e + (f*x + e)*b^2*h^2*p^
2*q^2*e^2*log(f*x + e) + 2*b^2*f*g*h*p^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p
*q) + 1)*log(f*x + e)^2/(c^(1/(p*q))*d^(1/p)) - 8*b^2*f*g*h*p^2*q^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*
q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(f*x + e)^2/(c^(2/(p*q))*d^(2/p)) + (f*x + e)*b^2*f^2*g^2*p*q^2*log(d
) + 4*(f*x + e)^2*b^2*f*g*h*p*q^2*log(d) + 3*(f*x + e)^3*b^2*h^2*p*q^2*log(d) - 2*(f*x + e)*b^2*f*g*h*p*q^2*e*
log(d) - 4*(f*x + e)^2*b^2*h^2*p*q^2*e*log(d) - 2*b^2*f^2*g^2*p*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + l
og(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)*log(d)/(c^(1/(p*q))*d^(1/p)) + (f*x + e)*b^2*h^2*p^2*q^2*e^2 - b^2*h^
2*p^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 2)*log(f*x + e)^2/(c^(1/(p*q)
)*d^(1/p)) + 8*b^2*h^2*p^2*q^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q)
+ 1)*log(f*x + e)^2/(c^(2/(p*q))*d^(2/p)) - 9*b^2*h^2*p^2*q^2*Ei(3*log(d)/p + 3*log(c)/(p*q) + 3*a/(b*p*q) + 3
*log(f*x + e))*e^(-3*a/(b*p*q))*log(f*x + e)^2/(c^(3/(p*q))*d^(3/p)) + (f*x + e)*b^2*f^2*g^2*p*q*log(c) + 4*(f
*x + e)^2*b^2*f*g*h*p*q*log(c) + 3*(f*x + e)^3*b^2*h^2*p*q*log(c) - 2*(f*x + e)*b^2*f*g*h*p*q*e*log(c) - 4*(f*
x + e)^2*b^2*h^2*p*q*e*log(c) - 2*b^2*f^2*g^2*p*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a
/(b*p*q))*log(f*x + e)*log(c)/(c^(1/(p*q))*d^(1/p)) + (f*x + e)*b^2*h^2*p*q^2*e^2*log(d) + 4*b^2*f*g*h*p*q^2*E
i(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(f*x + e)*log(d)/(c^(1/(p*q))*d^(1
/p)) - 16*b^2*f*g*h*p*q^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(
f*x + e)*log(d)/(c^(2/(p*q))*d^(2/p)) - b^2*f^2*g^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))
*e^(-a/(b*p*q))*log(d)^2/(c^(1/(p*q))*d^(1/p)) + (f*x + e)*a*b*f^2*g^2*p*q + 4*(f*x + e)^2*a*b*f*g*h*p*q + 3*(
f*x + e)^3*a*b*h^2*p*q - 2*(f*x + e)*a*b*f*g*h*p*q*e - 4*(f*x + e)^2*a*b*h^2*p*q*e - 2*a*b*f^2*g^2*p*q*Ei(log(
d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)/(c^(1/(p*q))*d^(1/p)) + (f*x + e)*
b^2*h^2*p*q*e^2*log(c) + 4*b^2*f*g*h*p*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q)
+ 1)*log(f*x + e)*log(c)/(c^(1/(p*q))*d^(1/p)) - 16*b^2*f*g*h*p*q*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q)
+ 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(f*x + e)*log(c)/(c^(2/(p*q))*d^(2/p)) - 2*b^2*h^2*p*q^2*Ei(log(d)/p +
log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 2)*log(f*x + e)*log(d)/(c^(1/(p*q))*d^(1/p)) + 16*b^2
*h^2*p*q^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q) + 1)*log(f*x + e)*lo
g(d)/(c^(2/(p*q))*d^(2/p)) - 18*b^2*h^2*p*q^2*Ei(3*log(d)/p + 3*log(c)/(p*q) + 3*a/(b*p*q) + 3*log(f*x + e))*e
^(-3*a/(b*p*q))*log(f*x + e)*log(d)/(c^(3/(p*q))*d^(3/p)) - 2*b^2*f^2*g^2*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*
p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(c)*log(d)/(c^(1/(p*q))*d^(1/p)) + 2*b^2*f*g*h*q^2*Ei(log(d)/p + log(c)
/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(d)^2/(c^(1/(p*q))*d^(1/p)) - 8*b^2*f*g*h*q^2*Ei(2*lo
g(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(d)^2/(c^(2/(p*q))*d^(2/p)) + (f*x
+ e)*a*b*h^2*p*q*e^2 + 4*a*b*f*g*h*p*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) +
1)*log(f*x + e)/(c^(1/(p*q))*d^(1/p)) - 16*a*b*f*g*h*p*q*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log
(f*x + e))*e^(-2*a/(b*p*q))*log(f*x + e)/(c^(2/(p*q))*d^(2/p)) - 2*b^2*h^2*p*q*Ei(log(d)/p + log(c)/(p*q) + a/
(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 2)*log(f*x + e)*log(c)/(c^(1/(p*q))*d^(1/p)) + 16*b^2*h^2*p*q*Ei(2*log
(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q) + 1)*log(f*x + e)*log(c)/(c^(2/(p*q))*d
^(2/p)) - 18*b^2*h^2*p*q*Ei(3*log(d)/p + 3*log(c)/(p*q) + 3*a/(b*p*q) + 3*log(f*x + e))*e^(-3*a/(b*p*q))*log(f
*x + e)*log(c)/(c^(3/(p*q))*d^(3/p)) - b^2*f^2*g^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-
a/(b*p*q))*log(c)^2/(c^(1/(p*q))*d^(1/p)) - 2*a*b*f^2*g^2*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x +
e))*e^(-a/(b*p*q))*log(d)/(c^(1/(p*q))*d^(1/p)) + 4*b^2*f*g*h*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(
f*x + e))*e^(-a/(b*p*q) + 1)*log(c)*log(d)/(c^(1/(p*q))*d^(1/p)) - 16*b^2*f*g*h*q*Ei(2*log(d)/p + 2*log(c)/(p*
q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(c)*log(d)/(c^(2/(p*q))*d^(2/p)) - b^2*h^2*q^2*Ei(log(d
)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (g+h\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)^2/(a + b*log(c*(d*(e + f*x)^p)^q))^3,x)

[Out]

int((g + h*x)^2/(a + b*log(c*(d*(e + f*x)^p)^q))^3, x)

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