3.4.87 \(\int x^2 (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [387]

Optimal. Leaf size=558 \[ -\frac {a g i^2 m x}{3 j^2}-\frac {b d^2 f n x}{3 e^2}+\frac {4 b d^2 g m n x}{9 e^2}+\frac {4 b g i^2 m n x}{9 j^2}+\frac {b d g i m n x}{3 e j}-\frac {5 b d g m n x^2}{36 e}-\frac {5 b g i m n x^2}{36 j}+\frac {2}{27} b g m n x^3-\frac {b d^3 g m n \log (d+e x)}{9 e^3}-\frac {b d^2 g i m n \log (d+e x)}{6 e^2 j}-\frac {b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\frac {g i m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b g i^3 m n \log (i+j x)}{9 j^3}-\frac {b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac {g i^3 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{3 j^3}-\frac {b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i^3 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{3 j^3}+\frac {b d^3 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{3 e^3} \]

[Out]

-1/3*a*g*i^2*m*x/j^2-1/3*b*d^2*f*n*x/e^2+4/9*b*d^2*g*m*n*x/e^2+4/9*b*g*i^2*m*n*x/j^2+1/3*b*d*g*i*m*n*x/e/j-5/3
6*b*d*g*m*n*x^2/e-5/36*b*g*i*m*n*x^2/j+2/27*b*g*m*n*x^3-1/9*b*d^3*g*m*n*ln(e*x+d)/e^3-1/6*b*d^2*g*i*m*n*ln(e*x
+d)/e^2/j-1/3*b*g*i^2*m*(e*x+d)*ln(c*(e*x+d)^n)/e/j^2+1/6*g*i*m*x^2*(a+b*ln(c*(e*x+d)^n))/j-1/9*g*m*x^3*(a+b*l
n(c*(e*x+d)^n))-1/9*b*g*i^3*m*n*ln(j*x+i)/j^3-1/6*b*d*g*i^2*m*n*ln(j*x+i)/e/j^2+1/3*g*i^3*m*(a+b*ln(c*(e*x+d)^
n))*ln(e*(j*x+i)/(-d*j+e*i))/j^3-1/3*b*d^2*g*n*(j*x+i)*ln(h*(j*x+i)^m)/e^2/j+1/6*b*d*n*x^2*(f+g*ln(h*(j*x+i)^m
))/e-1/9*b*n*x^3*(f+g*ln(h*(j*x+i)^m))+1/3*b*d^3*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^3+1/3*x^3
*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))+1/3*b*g*i^3*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^3+1/3*b*d^3*g*
m*n*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^3

________________________________________________________________________________________

Rubi [A]
time = 0.43, antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2489, 45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \begin {gather*} \frac {b d^3 g m n \text {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{3 e^3}+\frac {b g i^3 m n \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{3 j^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {g i^3 m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 j^3}+\frac {g i m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {a g i^2 m x}{3 j^2}-\frac {b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3}-\frac {b d^3 g m n \log (d+e x)}{9 e^3}-\frac {b d^2 f n x}{3 e^2}-\frac {b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}-\frac {b d^2 g i m n \log (d+e x)}{6 e^2 j}+\frac {4 b d^2 g m n x}{9 e^2}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac {b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac {b d g i m n x}{3 e j}-\frac {5 b d g m n x^2}{36 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^3 m n \log (i+j x)}{9 j^3}+\frac {4 b g i^2 m n x}{9 j^2}-\frac {5 b g i m n x^2}{36 j}+\frac {2}{27} b g m n x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a*g*i^2*m*x)/j^2 - (b*d^2*f*n*x)/(3*e^2) + (4*b*d^2*g*m*n*x)/(9*e^2) + (4*b*g*i^2*m*n*x)/(9*j^2) + (b*d*
g*i*m*n*x)/(3*e*j) - (5*b*d*g*m*n*x^2)/(36*e) - (5*b*g*i*m*n*x^2)/(36*j) + (2*b*g*m*n*x^3)/27 - (b*d^3*g*m*n*L
og[d + e*x])/(9*e^3) - (b*d^2*g*i*m*n*Log[d + e*x])/(6*e^2*j) - (b*g*i^2*m*(d + e*x)*Log[c*(d + e*x)^n])/(3*e*
j^2) + (g*i*m*x^2*(a + b*Log[c*(d + e*x)^n]))/(6*j) - (g*m*x^3*(a + b*Log[c*(d + e*x)^n]))/9 - (b*g*i^3*m*n*Lo
g[i + j*x])/(9*j^3) - (b*d*g*i^2*m*n*Log[i + j*x])/(6*e*j^2) + (g*i^3*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i +
j*x))/(e*i - d*j)])/(3*j^3) - (b*d^2*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(3*e^2*j) + (b*d*n*x^2*(f + g*Log[h*(i
+ j*x)^m]))/(6*e) - (b*n*x^3*(f + g*Log[h*(i + j*x)^m]))/9 + (b*d^3*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f +
g*Log[h*(i + j*x)^m]))/(3*e^3) + (x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/3 + (b*g*i^3*m*n*
PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) + (b*d^3*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(3*e^3
)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, 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 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 2440

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

Rule 2441

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

Rule 2442

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2489

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

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac {1}{3} (g j m) \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{387+j x} \, dx-\frac {1}{3} (b e n) \int \frac {x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac {1}{3} (g j m) \int \left (\frac {149769 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac {387 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac {57960603 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3 (387+j x)}\right ) \, dx-\frac {1}{3} (b e n) \int \left (\frac {d^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{e^3}-\frac {d x \left (f+g \log \left (h (387+j x)^m\right )\right )}{e^2}+\frac {x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{e}-\frac {d^3 \left (f+g \log \left (h (387+j x)^m\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac {1}{3} (g m) \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-\frac {(49923 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j^2}+\frac {(19320201 g m) \int \frac {a+b \log \left (c (d+e x)^n\right )}{387+j x} \, dx}{j^2}+\frac {(129 g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j}-\frac {1}{3} (b n) \int x^2 \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx-\frac {\left (b d^2 n\right ) \int \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx}{3 e^2}+\frac {\left (b d^3 n\right ) \int \frac {f+g \log \left (h (387+j x)^m\right )}{d+e x} \, dx}{3 e^2}+\frac {(b d n) \int x \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx}{3 e}\\ &=-\frac {49923 a g m x}{j^2}-\frac {b d^2 f n x}{3 e^2}+\frac {129 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {19320201 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (387+j x)}{387 e-d j}\right )}{j^3}+\frac {b d n x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{387 e-d j}\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac {(49923 b g m) \int \log \left (c (d+e x)^n\right ) \, dx}{j^2}-\frac {\left (b d^2 g n\right ) \int \log \left (h (387+j x)^m\right ) \, dx}{3 e^2}+\frac {1}{9} (b e g m n) \int \frac {x^3}{d+e x} \, dx-\frac {(19320201 b e g m n) \int \frac {\log \left (\frac {e (387+j x)}{387 e-d j}\right )}{d+e x} \, dx}{j^3}-\frac {(129 b e g m n) \int \frac {x^2}{d+e x} \, dx}{2 j}+\frac {1}{9} (b g j m n) \int \frac {x^3}{387+j x} \, dx-\frac {\left (b d^3 g j m n\right ) \int \frac {\log \left (\frac {j (d+e x)}{-387 e+d j}\right )}{387+j x} \, dx}{3 e^3}-\frac {(b d g j m n) \int \frac {x^2}{387+j x} \, dx}{6 e}\\ &=-\frac {49923 a g m x}{j^2}-\frac {b d^2 f n x}{3 e^2}+\frac {129 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {19320201 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (387+j x)}{387 e-d j}\right )}{j^3}+\frac {b d n x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{387 e-d j}\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac {(49923 b g m) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j^2}-\frac {\left (b d^2 g n\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,387+j x\right )}{3 e^2 j}-\frac {\left (b d^3 g m n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-387 e+d j}\right )}{x} \, dx,x,387+j x\right )}{3 e^3}+\frac {1}{9} (b e g m n) \int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx-\frac {(19320201 b g m n) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{387 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^3}-\frac {(129 b e g m n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{2 j}+\frac {1}{9} (b g j m n) \int \left (\frac {149769}{j^3}-\frac {387 x}{j^2}+\frac {x^2}{j}-\frac {57960603}{j^3 (387+j x)}\right ) \, dx-\frac {(b d g j m n) \int \left (-\frac {387}{j^2}+\frac {x}{j}+\frac {149769}{j^2 (387+j x)}\right ) \, dx}{6 e}\\ &=-\frac {49923 a g m x}{j^2}-\frac {b d^2 f n x}{3 e^2}+\frac {4 b d^2 g m n x}{9 e^2}+\frac {66564 b g m n x}{j^2}+\frac {129 b d g m n x}{e j}-\frac {5 b d g m n x^2}{36 e}-\frac {215 b g m n x^2}{4 j}+\frac {2}{27} b g m n x^3-\frac {b d^3 g m n \log (d+e x)}{9 e^3}-\frac {129 b d^2 g m n \log (d+e x)}{2 e^2 j}-\frac {49923 b g m (d+e x) \log \left (c (d+e x)^n\right )}{e j^2}+\frac {129 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {6440067 b g m n \log (387+j x)}{j^3}-\frac {49923 b d g m n \log (387+j x)}{2 e j^2}+\frac {19320201 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (387+j x)}{387 e-d j}\right )}{j^3}-\frac {b d^2 g n (387+j x) \log \left (h (387+j x)^m\right )}{3 e^2 j}+\frac {b d n x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{387 e-d j}\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac {19320201 b g m n \text {Li}_2\left (-\frac {j (d+e x)}{387 e-d j}\right )}{j^3}+\frac {b d^3 g m n \text {Li}_2\left (\frac {e (387+j x)}{387 e-d j}\right )}{3 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.52, size = 492, normalized size = 0.88 \begin {gather*} \frac {6 b n \log (d+e x) \left (-6 e^3 g i^3 m \log (i+j x)+6 g \left (e^3 i^3-d^3 j^3\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (-6 e^2 g i^2 m-3 d e g i j m+2 d^2 j^2 (3 f-g m)+6 d^2 g j^2 \log \left (h (i+j x)^m\right )\right )\right )+e \left (6 g i m \left (6 a e^2 i^2-b \left (2 e^2 i^2+3 d e i j+6 d^2 j^2\right ) n\right ) \log (i+j x)+6 b e^2 \log \left (c (d+e x)^n\right ) \left (6 f j^3 x^3+g j m x \left (-6 i^2+3 i j x-2 j^2 x^2\right )+6 g i^3 m \log (i+j x)+6 g j^3 x^3 \log \left (h (i+j x)^m\right )\right )+j \left (6 a e^2 x \left (6 f j^2 x^2+g m \left (-6 i^2+3 i j x-2 j^2 x^2\right )\right )+b n \left (12 d^2 j^2 (-3 f+4 g m) x+3 d e \left (6 f j^2 x^2+g m \left (12 i^2+12 i j x-5 j^2 x^2\right )\right )+e^2 x \left (-12 f j^2 x^2+g m \left (48 i^2-15 i j x+8 j^2 x^2\right )\right )\right )-6 g j^2 x \left (-6 a e^2 x^2+b n \left (6 d^2-3 d e x+2 e^2 x^2\right )\right ) \log \left (h (i+j x)^m\right )\right )\right )+36 b g \left (e^3 i^3-d^3 j^3\right ) m n \text {Li}_2\left (\frac {j (d+e x)}{-e i+d j}\right )}{108 e^3 j^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b*n*Log[d + e*x]*(-6*e^3*g*i^3*m*Log[i + j*x] + 6*g*(e^3*i^3 - d^3*j^3)*m*Log[(e*(i + j*x))/(e*i - d*j)] +
d*j*(-6*e^2*g*i^2*m - 3*d*e*g*i*j*m + 2*d^2*j^2*(3*f - g*m) + 6*d^2*g*j^2*Log[h*(i + j*x)^m])) + e*(6*g*i*m*(6
*a*e^2*i^2 - b*(2*e^2*i^2 + 3*d*e*i*j + 6*d^2*j^2)*n)*Log[i + j*x] + 6*b*e^2*Log[c*(d + e*x)^n]*(6*f*j^3*x^3 +
g*j*m*x*(-6*i^2 + 3*i*j*x - 2*j^2*x^2) + 6*g*i^3*m*Log[i + j*x] + 6*g*j^3*x^3*Log[h*(i + j*x)^m]) + j*(6*a*e^
2*x*(6*f*j^2*x^2 + g*m*(-6*i^2 + 3*i*j*x - 2*j^2*x^2)) + b*n*(12*d^2*j^2*(-3*f + 4*g*m)*x + 3*d*e*(6*f*j^2*x^2
+ g*m*(12*i^2 + 12*i*j*x - 5*j^2*x^2)) + e^2*x*(-12*f*j^2*x^2 + g*m*(48*i^2 - 15*i*j*x + 8*j^2*x^2))) - 6*g*j
^2*x*(-6*a*e^2*x^2 + b*n*(6*d^2 - 3*d*e*x + 2*e^2*x^2))*Log[h*(i + j*x)^m])) + 36*b*g*(e^3*i^3 - d^3*j^3)*m*n*
PolyLog[2, (j*(d + e*x))/(-(e*i) + d*j)])/(108*e^3*j^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.59, size = 3680, normalized size = 6.59

method	result	size
risch	\(\text {Expression too large to display}\)	\(3680\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m)),x,method=_RETURNVERBOSE)

[Out]

-1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*csgn(I*h*(j*x+i)^m)^3-1/9*n*b*f*x^3+1/3*x^3*a*f+49/108/e^3*b*d^3*g*m*
n-1/3/e^3*b*d^3*g*m*n*dilog(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))+1/6*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g*m/j^3
*i^3*ln(j*x+i)+1/3*ln(h)*x^3*a*g-1/3/e^3*b*d^3*g*m*n*ln(e*x+d)*ln(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))-1/6*I*Pi*b*f
*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*csgn(I*(j*x+i)^m)
*csgn(I*h*(j*x+i)^m)^2+1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h*(j*x+i)^m)^3-1/6*I*Pi*b*f*x^
3*csgn(I*c*(e*x+d)^n)^3+1/6/e*b*d*f*n*x^2+2/9/e^2/j*g*i*m*b*d^2*n-1/9*x^3*a*g*m+1/3*a*g*x^3*ln((j*x+i)^m)+1/12
*I/j*Pi*x^2*b*g*i*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/18*I*Pi*x^3*b*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(
j*x+i)^m)+1/18*I*Pi*x^3*b*g*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/6*I*ln(h)*Pi*x^3*b*g*csgn(I*c)
*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/6*I/e^2*Pi*x*b*d^2*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m
)-1/9*n*b*g*ln((j*x+i)^m)*x^3+1/3*b*ln(c)*g*x^3*ln((j*x+i)^m)+1/6/j*x^2*a*g*i*m-1/9/j^3*g*i^3*m*ln((e*x+d)*j-d
*j+e*i)*b*n+1/3/e^3*ln(e*x+d)*b*d^3*f*n+1/6*I*Pi*b*f*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/6*I*Pi*b*f*x^3*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/9*ln(h)*x^3*b*g*n-1/9*ln(c)*x^3*b*g*m+1/3*ln(h)*ln(c)*x^3*b*g+1/9/e/j^2*
g*i^2*m*b*d*n+(1/3*x^3*b*g*ln((j*x+i)^m)+1/18*b*(-3*I*Pi*g*j^3*x^3*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i
)^m)+3*I*Pi*g*j^3*x^3*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+3*I*Pi*g*j^3*x^3*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2
-3*I*Pi*g*j^3*x^3*csgn(I*h*(j*x+i)^m)^3+6*ln(h)*g*j^3*x^3-2*g*j^3*m*x^3+6*f*j^3*x^3+3*g*i*j^2*m*x^2+6*g*i^3*m*
ln(j*x+i)-6*g*i^2*j*m*x)/j^3)*ln((e*x+d)^n)+1/3*ln(c)*b*f*x^3+1/3*a*g*m/j^3*i^3*ln(j*x+i)+1/12*b*Pi^2*csgn(I*c
)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^3*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/6*I/e^3*ln(e*x+d)*Pi*b
*d^3*g*n*csgn(I*h*(j*x+i)^m)^3-1/3/j^3*b*g*i^3*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/3/j^3*b*g*i^3*m*n*ln
(j*x+i)*ln(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/18*I*Pi*x^3*b*g*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/12*b*P
i^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^3*g*csgn(I*h*(j*x+i)^m)^3-1/12*b*Pi^2*csgn(I*c)*csgn(I*c
*(e*x+d)^n)^2*x^3*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/18*I*Pi*x^3*b*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^
2-1/6*I/j^2*Pi*x*b*g*i^2*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/12*I/j*Pi*x^2*b*g*i*m*csgn(I*(e*x+d)^n)*c
sgn(I*c*(e*x+d)^n)^2-1/6*I*ln(c)*Pi*x^3*b*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/6*I*b*Pi*csgn(I*
c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g*x^3*ln((j*x+i)^m)+1/6*I*Pi*x^3*a*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+
i)^m)^2-1/6*I*ln(h)*Pi*x^3*b*g*csgn(I*c*(e*x+d)^n)^3-1/6*I*ln(c)*Pi*x^3*b*g*csgn(I*h*(j*x+i)^m)^3+1/18*I*Pi*x^
3*b*g*m*csgn(I*c*(e*x+d)^n)^3-1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g*x^3*ln((j*x+i)^m)+1/12*b*Pi^2*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h*(j*x+i)^m)^3+1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*csgn(I*h)*csgn(I*
h*(j*x+i)^m)^2-1/9*b*d^3*g*m*n*ln(e*x+d)/e^3-1/6*b*d^2*g*i*m*n*ln(e*x+d)/e^2/j+1/6*I*b*Pi*csgn(I*(e*x+d)^n)*cs
gn(I*c*(e*x+d)^n)^2*g*m/j^3*i^3*ln(j*x+i)+1/12*I/e*Pi*x^2*b*d*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/12
*I/e*Pi*x^2*b*d*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/12*I/e*Pi*x^2*b*d*g*n*csgn(I*h)*csgn(I*h
*(j*x+i)^m)^2+1/6*I*Pi*x^3*a*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/6*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c
*(e*x+d)^n)*g*m/j^3*i^3*ln(j*x+i)+1/6*I/j^2*Pi*x*b*g*i^2*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/1
2*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/12*b*Pi^2*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/12*b*Pi^2*csgn(I*
c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^3*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+2/27*b*g*m*n*x^3-1/18*I*Pi*x^3*
b*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g*m/j^3*i^3*ln(j*x+i)+1/6*I/j^2*Pi*x*b*
g*i^2*m*csgn(I*c*(e*x+d)^n)^3+1/6*I/e^2*Pi*x*b*d^2*g*n*csgn(I*h*(j*x+i)^m)^3+1/3/e^3*ln(e*x+d)*ln(h)*b*d^3*g*n
-1/3/e^2*ln(h)*x*b*d^2*g*n+1/6/j*ln(c)*x^2*b*g*i*m-1/3/j^2*ln(c)*x*b*g*i^2*m+1/6/e*ln(h)*x^2*b*d*g*n+1/3*b*ln(
c)*g*m/j^3*i^3*ln(j*x+i)-1/3/e^2/j*g*i*m*ln((e*x+d)*j-d*j+e*i)*b*d^2*n-1/3/e/j^2*ln(e*x+d)*b*d*g*i^2*m*n-1/6/e
/j^2*g*i^2*m*ln((e*x+d)*j-d*j+e*i)*b*d*n+1/6*I*ln(c)*Pi*x^3*b*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/12*b
*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/12*b*Pi^2*csgn(I*(e*x+d)
^n)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)^2*x^3*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/12*I/j*Pi*x^2*b*g*i*m*csgn(I*c*(e*x+d)^n)^3-1/12*I/e*Pi*
x^2*b*d*g*n*csgn(I*h*(j*x+i)^m)^3-1/6*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i
)^m)+1/18*I*Pi*x^3*b*g*n*csgn(I*h*(j*x+i)^m)^3-1/3*a*g*i^2*m*x/j^2-1/3*b*d^2*f*n*x/e^2+1/3/e^3*n*b*g*ln((j*x+i
)^m)*d^3*ln(e*x+d)+1/6/e*n*b*g*ln((j*x+i)^m)*d*...

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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(x^2*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

1/3*b*f*x^3*log((x*e + d)^n*c) + 1/3*a*g*x^3*log((j*x + I)^m*h) + 1/3*a*f*x^3 - 1/18*a*g*j*m*((2*j^2*x^3 - 3*I
*j*x^2 - 6*x)/j^3 + 6*I*log(j*x + I)/j^4) + 1/18*(6*d^3*e^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x
)*e^(-3))*b*f*n*e + 1/18*b*g*((6*I*m*n*e^3*log(j*x + I)*log(x*e + d) + (3*d*j^3*n*x^2*e^2 - 6*d^2*j^3*n*x*e +
6*d^3*j^3*n*log(x*e + d) + 6*j^3*x^3*e^3*log((x*e + d)^n) - 2*(j^3*n - 3*j^3*log(c))*x^3*e^3)*log((j*x + I)^m)
+ (3*I*j^2*m*x^2*e^3 - 2*(j^3*m - 3*j^3*log(h))*x^3*e^3 + 6*j*m*x*e^3 - 6*I*m*e^3*log(j*x + I))*log((x*e + d)
^n))*e^(-3)/j^3 - 18*integrate(-1/18*(2*(2*j^3*m*n - 3*j^3*n*log(h) - 3*(j^3*m - 3*j^3*log(h))*log(c))*x^4*e^4
+ ((-I*j^2*m*n - 6*I*j^2*n*log(h) + 18*I*j^2*log(c)*log(h))*e^4 - (d*j^3*m*n + 6*(j^3*m - 3*j^3*log(h))*d*log
(c))*e^3)*x^3 + 3*(d^2*j^3*m*n*e^2 + 6*I*d*j^2*e^3*log(c)*log(h) - j*m*n*e^4)*x^2 + 6*(d^3*j^3*m*n*e - I*m*n*e
^4)*x - 6*(d^4*j^3*m*n + I*d*m*n*e^3 + (d^3*j^3*m*n*e + I*m*n*e^4)*x)*log(x*e + d))/(j^3*x^2*e^4 + I*d*j^2*e^3
+ (d*j^3*e^3 + I*j^2*e^4)*x), 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(x^2*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")

[Out]

1/18*(18*j^3*integral(1/18*(18*a*d*f*j^3*x^2 + (-3*I*b*g*j^2*m*n*x^2 - 6*b*g*j*m*n*x + 2*(9*a*f*j^3 + (b*g*j^3
*m - 3*b*f*j^3)*n)*x^3)*e + 6*(3*a*d*g*j^3*m*x^2 - (-I*b*g*m*n + (b*g*j^3*m*n - 3*a*g*j^3*m)*x^3)*e + 3*(b*g*j
^3*m*x^3*e + b*d*g*j^3*m*x^2)*log(c))*log(j*x + I) + 18*(b*f*j^3*x^3*e + b*d*f*j^3*x^2)*log(c) + 6*(3*a*d*g*j^
3*x^2 - (b*g*j^3*n - 3*a*g*j^3)*x^3*e + 3*(b*g*j^3*x^3*e + b*d*g*j^3*x^2)*log(c))*log(h))/(j^3*x*e + d*j^3), x
) + (6*b*g*j^3*n*x^3*log(h) + 3*I*b*g*j^2*m*n*x^2 + 6*b*g*j*m*n*x - 2*(b*g*j^3*m - 3*b*f*j^3)*n*x^3 + 6*(b*g*j
^3*m*n*x^3 - I*b*g*m*n)*log(j*x + I))*log(x*e + d))/j^3

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*(e*x+d)**n))*(f+g*ln(h*(j*x+i)**m)),x)

[Out]

Timed out

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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(x^2*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)*(g*log((j*x + I)^m*h) + f)*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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