3.6.32 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^2}{g+h x} \, dx\) [532]

Optimal. Leaf size=123 \[ \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2443, 2481, 2421, 6724, 2495} \begin {gather*} \frac {2 b p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {2 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2443

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

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 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]]

Rule 6724

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{g+h x} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}-\text {Subst}\left (\frac {(2 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}-\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac {f \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\text {Subst}\left (\frac {\left (2 b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(123)=246\).
time = 0.16, size = 313, normalized size = 2.54 \begin {gather*} \frac {a^2 \log (g+h x)}{h}+\frac {b \left (-2 a p q \log (e+f x) \log (g+h x)+b p^2 q^2 \log ^2(e+f x) \log (g+h x)+2 a \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-2 b p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+2 a p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-b p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 b p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-2 b p^2 q^2 \text {Li}_3\left (\frac {h (e+f x)}{-f g+e h}\right )\right )}{h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}}{h x +g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{g+h\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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