3.5.34 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^2}{(g+h x)^3} \, dx\) [434]

Optimal. Leaf size=222 \[ -\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}+\frac {b^2 f^2 p^2 q^2 \log (g+h x)}{h (f g-e h)^2}-\frac {b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (1+\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac {b^2 f^2 p^2 q^2 \text {Li}_2\left (-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2} \]

[Out]

-b*f*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/(-e*h+f*g)^2/(h*x+g)-1/2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/h/(h*x+g)^
2+b^2*f^2*p^2*q^2*ln(h*x+g)/h/(-e*h+f*g)^2-b*f^2*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(1+(-e*h+f*g)/h/(f*x+e))/h/
(-e*h+f*g)^2+b^2*f^2*p^2*q^2*polylog(2,(e*h-f*g)/h/(f*x+e))/h/(-e*h+f*g)^2

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Rubi [A]
time = 0.49, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2445, 2458, 2389, 2379, 2438, 2351, 31, 2495} \begin {gather*} \frac {b^2 f^2 p^2 q^2 \text {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}-\frac {b f^2 p q \log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}-\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(g+h x) (f g-e h)^2}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}+\frac {b^2 f^2 p^2 q^2 \log (g+h x)}{h (f g-e h)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*f*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/((f*g - e*h)^2*(g + h*x))) - (a + b*Log[c*(d*(e + f*x)^
p)^q])^2/(2*h*(g + h*x)^2) + (b^2*f^2*p^2*q^2*Log[g + h*x])/(h*(f*g - e*h)^2) - (b*f^2*p*q*(a + b*Log[c*(d*(e
+ f*x)^p)^q])*Log[1 + (f*g - e*h)/(h*(e + f*x))])/(h*(f*g - e*h)^2) + (b^2*f^2*p^2*q^2*PolyLog[2, -((f*g - e*h
)/(h*(e + f*x)))])/(h*(f*g - e*h)^2)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2351

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

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, 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)^3} \, 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}{2 h (g+h x)^2}+\text {Subst}\left (\frac {(b f p q) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (g+h x)^2} \, 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}{2 h (g+h x)^2}+\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{x \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^2} \, 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}{2 h (g+h x)^2}-\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^2} \, dx,x,e+f x\right )}{f g-e h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b f p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{x \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )} \, dx,x,e+f x\right )}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}-\text {Subst}\left (\frac {(b f p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{\frac {f g-e h}{f}+\frac {h x}{f}} \, dx,x,e+f x\right )}{(f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b f^2 p q\right ) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b^2 f p^2 q^2\right ) \text {Subst}\left (\int \frac {1}{\frac {f g-e h}{f}+\frac {h x}{f}} \, dx,x,e+f x\right )}{(f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f g-e h)^2 (g+h x)}+\frac {f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (f g-e h)^2}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}+\frac {b^2 f^2 p^2 q^2 \log (g+h x)}{h (f g-e h)^2}-\frac {b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}+\text {Subst}\left (\frac {\left (b^2 f^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f g-e h)^2 (g+h x)}+\frac {f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (f g-e h)^2}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}+\frac {b^2 f^2 p^2 q^2 \log (g+h x)}{h (f g-e h)^2}-\frac {b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {b^2 f^2 p^2 q^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 316, normalized size = 1.42 \begin {gather*} -\frac {\left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+\frac {2 b p q \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (h (e+f x) (e h-f (2 g+h x)) \log (e+f x)+f (g+h x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )\right )}{(f g-e h)^2}+\frac {b^2 p^2 q^2 \left (h (e+f x) (e h-f (2 g+h x)) \log ^2(e+f x)-2 f^2 (g+h x)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 f (g+h x) \log (e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+2 f^2 (g+h x)^2 \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )\right )}{(f g-e h)^2}}{2 h (g+h x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2 + (2*b*p*q*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*
(e + f*x)^p)^q])*(h*(e + f*x)*(e*h - f*(2*g + h*x))*Log[e + f*x] + f*(g + h*x)*(h*(e + f*x) + f*(g + h*x)*Log[
(f*(g + h*x))/(f*g - e*h)])))/(f*g - e*h)^2 + (b^2*p^2*q^2*(h*(e + f*x)*(e*h - f*(2*g + h*x))*Log[e + f*x]^2 -
2*f^2*(g + h*x)^2*Log[(f*(g + h*x))/(f*g - e*h)] + 2*f*(g + h*x)*Log[e + f*x]*(h*(e + f*x) + f*(g + h*x)*Log[
(f*(g + h*x))/(f*g - e*h)]) + 2*f^2*(g + h*x)^2*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)]))/(f*g - e*h)^2)/(h*(
g + h*x)^2)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}}{\left (h x +g \right )^{3}}\, dx\]

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)^3,x)

[Out]

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

[Out]

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

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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)^3,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^3*x^3 + 3*g*h^2*x^2 + 3*g^
2*h*x + g^3), 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}}{\left (g + h x\right )^{3}}\, 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)**3,x)

[Out]

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

[Out]

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

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

[Out]

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

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