3.5.50 \(\int \frac {(g+h x)^2}{(a+b \log (c (d (e+f x)^p)^q))^2} \, dx\) [450]

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

[Out]

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

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Rubi [A]
time = 0.92, antiderivative size = 326, normalized size of antiderivative = 1.00, 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 = {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^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 )} \end {gather*}

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 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 )^2} \, 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 )^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 )}+\text {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 )-\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 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 )}+\text {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 )-\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 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 )}+\text {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 )-\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 f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {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 )-\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 f^2 p q},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 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 )}+\text {Subst}\left (\frac {\left (3 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 )}{b f^3 p q},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 f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(6 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 f^3 p q},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 f^3 p q},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 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 )}+\text {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 ) \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 )}{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 )-\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 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 (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 ) \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 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 (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 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 (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 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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1310\) vs. \(2(326)=652\).
time = 0.55, size = 1310, normalized size = 4.02 \begin {gather*} \frac {e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (-b e e^{\frac {3 a}{b p q}} f^2 g^2 p q \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}-b e^{\frac {3 a}{b p q}} f^3 g^2 p q x \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}-2 b e e^{\frac {3 a}{b p q}} f^2 g h p q x \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}-2 b e^{\frac {3 a}{b p q}} f^3 g h p q x^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}-b e e^{\frac {3 a}{b p q}} f^2 h^2 p q x^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}-b e^{\frac {3 a}{b p q}} f^3 h^2 p q x^3 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+a e^{\frac {2 a}{b p q}} f^2 g^2 (e+f x) \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 )-2 a e e^{\frac {2 a}{b p q}} f g h (e+f x) \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 )+a e^2 e^{\frac {2 a}{b p q}} h^2 (e+f x) \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 )+4 a e^{\frac {a}{b p q}} f g h (e+f x)^2 \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 )-4 a e e^{\frac {a}{b p q}} h^2 (e+f x)^2 \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 )+3 a h^2 (e+f x)^3 \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 e^{\frac {2 a}{b p q}} f^2 g^2 (e+f x) \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 ) \log \left (c \left (d (e+f x)^p\right )^q\right )-2 b e e^{\frac {2 a}{b p q}} f g h (e+f x) \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 ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b e^2 e^{\frac {2 a}{b p q}} h^2 (e+f x) \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 ) \log \left (c \left (d (e+f x)^p\right )^q\right )+4 b e^{\frac {a}{b p q}} f g h (e+f x)^2 \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 ) \log \left (c \left (d (e+f x)^p\right )^q\right )-4 b e e^{\frac {a}{b p q}} h^2 (e+f x)^2 \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 ) \log \left (c \left (d (e+f x)^p\right )^q\right )+3 b h^2 (e+f x)^3 \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 ) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b^2 f^3 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \end {gather*}

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 = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (h x +g \right )^{2}}{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}}\, dx\]

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))^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 = 0.00, 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 595, normalized size = 1.83 \begin {gather*} \frac {{\left (4 \, {\left (a f g h - a h^{2} e + {\left (b f g h p q - b h^{2} p q e\right )} \log \left (f x + e\right ) + {\left (b f g h - b h^{2} e\right )} \log \left (c\right ) + {\left (b f g h q - b h^{2} q e\right )} \log \left (d\right )\right )} e^{\left (\frac {b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \operatorname {log\_integral}\left ({\left (f^{2} x^{2} + 2 \, f x e + 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 f g h e + a h^{2} e^{2} + {\left (b f^{2} g^{2} p q - 2 \, b f g h p q e + b h^{2} p q e^{2}\right )} \log \left (f x + e\right ) + {\left (b f^{2} g^{2} - 2 \, b f g h e + b h^{2} e^{2}\right )} \log \left (c\right ) + {\left (b f^{2} g^{2} q - 2 \, b f g h q e + b h^{2} q e^{2}\right )} \log \left (d\right )\right )} e^{\left (\frac {2 \, {\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )} \operatorname {log\_integral}\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} + 2 \, b f^{3} g h p q x^{2} + b f^{3} g^{2} p q x + {\left (b f^{2} h^{2} p q x^{2} + 2 \, b f^{2} g h p q x + b f^{2} g^{2} p q\right )} e\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 )} \operatorname {log\_integral}\left ({\left (f^{3} x^{3} + 3 \, f^{2} x^{2} e + 3 \, f x e^{2} + 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 {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))^2,x, algorithm="fricas")

[Out]

(4*(a*f*g*h - a*h^2*e + (b*f*g*h*p*q - b*h^2*p*q*e)*log(f*x + e) + (b*f*g*h - b*h^2*e)*log(c) + (b*f*g*h*q - b
*h^2*q*e)*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*f^2*g^2 - 2*a*f*g*h*e + a*h^2*e^2 + (b*f^2*g^2*p*q - 2*b*f*g*h*p*q*e + b*h^
2*p*q*e^2)*log(f*x + e) + (b*f^2*g^2 - 2*b*f*g*h*e + b*h^2*e^2)*log(c) + (b*f^2*g^2*q - 2*b*f*g*h*q*e + b*h^2*
q*e^2)*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*f^3*h^2*p*q*x^3 + 2*b*f^3*g*h*p*q*x^2 + b*f^3*g^2*p*q*x + (b*f^2*h^2*p*q*x^2 + 2*b*f^2*g*h*p*q
*x + b*f^2*g^2*p*q)*e)*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)*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^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 = 0.00, 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 )^{2}}\, 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))**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] Leaf count of result is larger than twice the leaf count of optimal. 4046 vs. \(2 (341) = 682\).
time = 4.92, size = 4046, normalized size = 12.41 \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))^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) + ...

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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 )}^2} \,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))^2,x)

[Out]

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

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