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

Optimal. Leaf size=240 \[ \frac {2 b p q (h i-g j) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac {(h i-g j) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}-\frac {2 a b j p q x}{h}-\frac {2 b^2 j p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac {2 b^2 p^2 q^2 (h i-g j) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^2}+\frac {2 b^2 j p^2 q^2 x}{h} \]

[Out]

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

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Rubi [A]  time = 0.65, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 6589, 2445} \[ \frac {2 b p q (h i-g j) \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}-\frac {2 b^2 p^2 q^2 (h i-g j) \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h^2}+\frac {(h i-g j) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}-\frac {2 a b j p q x}{h}-\frac {2 b^2 j p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}+\frac {2 b^2 j p^2 q^2 x}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 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 2396

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 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2433

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 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]]

Rule 6589

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 {(531+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{g+h x} \, dx &=\operatorname {Subst}\left (\int \frac {(531+j x) \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 )\\ &=\operatorname {Subst}\left (\int \left (\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{h}+\frac {(531 h-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{h (g+h x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {j \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(531 h-g j) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(531 h-g j) \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^2}+\operatorname {Subst}\left (\frac {j \operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(2 b f (531 h-g j) 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^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac {(531 h-g j) \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^2}-\operatorname {Subst}\left (\frac {(2 b j p q) \operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(2 b (531 h-g j) p q) \operatorname {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^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 a b j p q x}{h}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac {(531 h-g j) \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^2}+\frac {2 b (531 h-g j) 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^2}-\operatorname {Subst}\left (\frac {\left (2 b^2 j p q\right ) \operatorname {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (2 b^2 (531 h-g j) p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 a b j p q x}{h}+\frac {2 b^2 j p^2 q^2 x}{h}-\frac {2 b^2 j p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac {(531 h-g j) \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^2}+\frac {2 b (531 h-g j) 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^2}-\frac {2 b^2 (531 h-g j) p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^2}\\ \end {align*}

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Mathematica [B]  time = 0.39, size = 852, normalized size = 3.55 \[ \frac {f h j x a^2+f h i \log (g+h x) a^2-f g j \log (g+h x) a^2-2 b e h j p q a-2 b f h j p q x a+2 b e h j p q \log (e+f x) a+2 b f h j x \log \left (c \left (d (e+f x)^p\right )^q\right ) a-2 b f h i p q \log (e+f x) \log (g+h x) a+2 b f g j p q \log (e+f x) \log (g+h x) a+2 b f h i \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a-2 b f g j \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a+2 b f h i p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) a-2 b f g j p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) a-b^2 e h j p^2 q^2 \log ^2(e+f x)+b^2 f h j x \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 f h j p^2 q^2 x-2 b^2 e h j p q \log \left (c \left (d (e+f x)^p\right )^q\right )-2 b^2 f h j p q x \log \left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 e h j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 f h i p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^2 f g j p^2 q^2 \log ^2(e+f x) \log (g+h x)+b^2 f h i \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-b^2 f g j \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-2 b^2 f h i p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+2 b^2 f g j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-b^2 f h i p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+b^2 f g j p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 b^2 f h i p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-2 b^2 f g j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 b f (h i-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )+2 b^2 f (g j-h i) p^2 q^2 \text {Li}_3\left (\frac {h (e+f x)}{e h-f g}\right )}{f h^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} j x + a^{2} i + {\left (b^{2} j x + b^{2} i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, {\left (a b j x + a b i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{h x + g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (j x +i \right ) \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{2}}{h x +g}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} j {\left (\frac {x}{h} - \frac {g \log \left (h x + g\right )}{h^{2}}\right )} + \frac {a^{2} i \log \left (h x + g\right )}{h} + \int \frac {2 \, {\left (i q \log \relax (d) + i \log \relax (c)\right )} a b + {\left (i q^{2} \log \relax (d)^{2} + 2 \, i q \log \relax (c) \log \relax (d) + i \log \relax (c)^{2}\right )} b^{2} + {\left (b^{2} j x + b^{2} i\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + {\left (2 \, {\left (j q \log \relax (d) + j \log \relax (c)\right )} a b + {\left (j q^{2} \log \relax (d)^{2} + 2 \, j q \log \relax (c) \log \relax (d) + j \log \relax (c)^{2}\right )} b^{2}\right )} x + 2 \, {\left ({\left (i q \log \relax (d) + i \log \relax (c)\right )} b^{2} + a b i + {\left ({\left (j q \log \relax (d) + j \log \relax (c)\right )} b^{2} + a b j\right )} x\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x+i)*(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*(i + j*x)/(g + h*x), x)

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