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

Optimal. Leaf size=326 \[ \frac{4 h (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Ei}\left (\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^3 p^2 q^2}+\frac{(e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^3 p^2 q^2}+\frac{3 h^2 (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Ei}\left (\frac{3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^3 p^2 q^2}-\frac{(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.29182, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310, 2445} \[ \frac{4 h (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Ei}\left (\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^3 p^2 q^2}+\frac{(e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^3 p^2 q^2}+\frac{3 h^2 (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Ei}\left (\frac{3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^3 p^2 q^2}-\frac{(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2400

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 2399

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 2389

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 2300

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 2178

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

Rule 2390

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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2445

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 )^2} \, dx &=\operatorname{Subst}\left (\int \frac{(g+h x)^2}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2} \, 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}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{3 \int \frac{(g+h x)^2}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{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 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}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{3 \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}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{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 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}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{\left (3 h^2\right ) \int \frac{(e+f x)^2}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{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 f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(6 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 f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{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 f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{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 f^2 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}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{\left (3 h^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(2 h (f g-e h)) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(6 h (f g-e h)) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (2 (f g-e h)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (3 (f g-e h)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^3 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}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{\left (3 h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac{3}{p q}}\right ) \operatorname{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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{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 ) \operatorname{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 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac{2}{p q}}\right ) \operatorname{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 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{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 ) \operatorname{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 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{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 ) \operatorname{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 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^2 f^3 p^2 q^2}+\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^2 f^3 p^2 q^2}+\frac{3 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 )}{b^2 f^3 p^2 q^2}-\frac{(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\\ \end{align*}

Mathematica [B]  time = 0.939192, size = 1310, normalized size = 4.02 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.49, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( hx+g \right ) ^{2}}{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}}}\, dx \end{align*}

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))^2,x)

[Out]

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

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

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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.02772, size = 1366, normalized size = 4.19 \begin{align*} \frac{{\left (4 \,{\left (a f g h - a e h^{2} +{\left (b f g h - b e h^{2}\right )} p q \log \left (f x + e\right ) +{\left (b f g h - b e h^{2}\right )} q \log \left (d\right ) +{\left (b f g h - b e h^{2}\right )} \log \left (c\right )\right )} e^{\left (\frac{b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \logintegral \left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} e^{\left (\frac{2 \,{\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )}\right ) +{\left (a f^{2} g^{2} - 2 \, a e f g h + a e^{2} h^{2} +{\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} p q \log \left (f x + e\right ) +{\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} q \log \left (d\right ) +{\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} \log \left (c\right )\right )} e^{\left (\frac{2 \,{\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )} \logintegral \left ({\left (f x + e\right )} e^{\left (\frac{b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )}\right ) -{\left (b f^{3} h^{2} p q x^{3} + b e f^{2} g^{2} p q +{\left (2 \, b f^{3} g h + b e f^{2} h^{2}\right )} p q x^{2} +{\left (b f^{3} g^{2} + 2 \, b e f^{2} g h\right )} p q x\right )} e^{\left (\frac{3 \,{\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )} + 3 \,{\left (b h^{2} p q \log \left (f x + e\right ) + b h^{2} q \log \left (d\right ) + b h^{2} \log \left (c\right ) + a h^{2}\right )} \logintegral \left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )} e^{\left (\frac{3 \,{\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )}\right )\right )} e^{\left (-\frac{3 \,{\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )}}{b^{3} f^{3} p^{3} q^{3} \log \left (f x + e\right ) + b^{3} f^{3} p^{2} q^{3} \log \left (d\right ) + b^{3} f^{3} p^{2} q^{2} \log \left (c\right ) + a b^{2} f^{3} p^{2} q^{2}} \end{align*}

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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{2}}\, dx \end{align*}

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))**2,x)

[Out]

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

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Giac [B]  time = 1.82072, size = 5462, normalized size = 16.75 \begin{align*} \text{result too large to display} \end{align*}

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))^2,x, algorithm="giac")

[Out]

-(f*x + e)*b*f^2*g^2*p*q/(b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b
^2*f^3*p^2*q^2) - 2*(f*x + e)^2*b*f*g*h*p*q/(b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p
^2*q^2*log(c) + a*b^2*f^3*p^2*q^2) - (f*x + e)^3*b*h^2*p*q/(b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log
(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2) + 2*(f*x + e)*b*f*g*h*p*q*e/(b^3*f^3*p^3*q^3*log(f*x + e) +
b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2) + 2*(f*x + e)^2*b*h^2*p*q*e/(b^3*f^3*p^3*
q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2) + 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)/((b^3*f^3*p^3*q^3*log(f*x + e) +
b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*d^(1/p)) - (f*x + e)*b*h^2*p*
q*e^2/(b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2) - 2
*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)/((b^3*f^3*
p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*d^(1/p
)) + 4*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
)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(2/(
p*q))*d^(2/p)) + 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)/((b^
3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*
d^(1/p)) + b*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)/((
b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q)
)*d^(1/p)) - 4*b*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)*l
og(f*x + e)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q
^2)*c^(2/(p*q))*d^(2/p)) + 3*b*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)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2
*f^3*p^2*q^2)*c^(3/(p*q))*d^(3/p)) + b*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)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2
*q^2)*c^(1/(p*q))*d^(1/p)) - 2*b*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(d)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q
^2)*c^(1/(p*q))*d^(1/p)) + 4*b*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(d)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p
^2*q^2)*c^(2/(p*q))*d^(2/p)) + a*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))
/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p
*q))*d^(1/p)) - 2*b*f*g*h*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(c)/((b
^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))
*d^(1/p)) + 4*b*f*g*h*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)/(
(b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(2/(p*q
))*d^(2/p)) + b*h^2*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 2)*log(d)/((b^3*f
^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*d^(
1/p)) - 4*b*h^2*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(d)/(
(b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(2/(p*q
))*d^(2/p)) + 3*b*h^2*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(d)
/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(3/(p
*q))*d^(3/p)) - 2*a*f*g*h*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)/((b^3*f^3*
p^3*q^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*d^(1/p
)) + 4*a*f*g*h*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))/((b^3*f^3*p^3*q
^3*log(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(2/(p*q))*d^(2/p)) +
b*h^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 2)*log(c)/((b^3*f^3*p^3*q^3*log(f
*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*d^(1/p)) - 4*b*h^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(c)/((b^3*f^3*p^3*q^3*l
og(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(2/(p*q))*d^(2/p)) + 3*b*
h^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(c)/((b^3*f^3*p^3*q^3*l
og(f*x + e) + b^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(3/(p*q))*d^(3/p)) + a*h^
2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 2)/((b^3*f^3*p^3*q^3*log(f*x + e) + b
^3*f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(1/(p*q))*d^(1/p)) - 4*a*h^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)/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*
f^3*p^2*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(2/(p*q))*d^(2/p)) + 3*a*h^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))/((b^3*f^3*p^3*q^3*log(f*x + e) + b^3*f^3*p^2
*q^3*log(d) + b^3*f^3*p^2*q^2*log(c) + a*b^2*f^3*p^2*q^2)*c^(3/(p*q))*d^(3/p))