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

Optimal. Leaf size=397 \[ -\frac{b d^2 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac{b g i^2 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{2 j^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac{g i^2 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}-\frac{1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{a g i m x}{2 j}+\frac{b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac{b d^2 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2}+\frac{b d^2 g m n \log (d+e x)}{4 e^2}+\frac{b d f n x}{2 e}+\frac{b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac{3 b d g m n x}{4 e}-\frac{1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{b g i^2 m n \log (i+j x)}{4 j^2}-\frac{3 b g i m n x}{4 j}+\frac{1}{4} b g m n x^2 \]

[Out]

(a*g*i*m*x)/(2*j) + (b*d*f*n*x)/(2*e) - (3*b*d*g*m*n*x)/(4*e) - (3*b*g*i*m*n*x)/(4*j) + (b*g*m*n*x^2)/4 + (b*d
^2*g*m*n*Log[d + e*x])/(4*e^2) + (b*g*i*m*(d + e*x)*Log[c*(d + e*x)^n])/(2*e*j) - (g*m*x^2*(a + b*Log[c*(d + e
*x)^n]))/4 + (b*g*i^2*m*n*Log[i + j*x])/(4*j^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i -
d*j)])/(2*j^2) + (b*d*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(2*e*j) - (b*n*x^2*(f + g*Log[h*(i + j*x)^m]))/4 - (b
*d^2*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(2*e^2) + (x^2*(a + b*Log[c*(d + e*x)^n])
*(f + g*Log[h*(i + j*x)^m]))/2 - (b*g*i^2*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) - (b*d^2*g*m*n
*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.433456, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2439, 43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ -\frac{b d^2 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac{b g i^2 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{2 j^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac{g i^2 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}-\frac{1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{a g i m x}{2 j}+\frac{b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac{b d^2 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2}+\frac{b d^2 g m n \log (d+e x)}{4 e^2}+\frac{b d f n x}{2 e}+\frac{b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac{3 b d g m n x}{4 e}-\frac{1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{b g i^2 m n \log (i+j x)}{4 j^2}-\frac{3 b g i m n x}{4 j}+\frac{1}{4} b g m n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*g*i*m*x)/(2*j) + (b*d*f*n*x)/(2*e) - (3*b*d*g*m*n*x)/(4*e) - (3*b*g*i*m*n*x)/(4*j) + (b*g*m*n*x^2)/4 + (b*d
^2*g*m*n*Log[d + e*x])/(4*e^2) + (b*g*i*m*(d + e*x)*Log[c*(d + e*x)^n])/(2*e*j) - (g*m*x^2*(a + b*Log[c*(d + e
*x)^n]))/4 + (b*g*i^2*m*n*Log[i + j*x])/(4*j^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i -
d*j)])/(2*j^2) + (b*d*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(2*e*j) - (b*n*x^2*(f + g*Log[h*(i + j*x)^m]))/4 - (b
*d^2*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(2*e^2) + (x^2*(a + b*Log[c*(d + e*x)^n])
*(f + g*Log[h*(i + j*x)^m]))/2 - (b*g*i^2*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) - (b*d^2*g*m*n
*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*e^2)

Rule 2439

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]

Rule 43

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 2416

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 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 2295

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

Rule 2395

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 2394

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 2393

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 2391

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]

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{1}{2} (g j m) \int \frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{388+j x} \, dx-\frac{1}{2} (b e n) \int \frac{x^2 \left (f+g \log \left (h (388+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{1}{2} (g j m) \int \left (-\frac{388 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac{150544 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2 (388+j x)}\right ) \, dx-\frac{1}{2} (b e n) \int \left (-\frac{d \left (f+g \log \left (h (388+j x)^m\right )\right )}{e^2}+\frac{x \left (f+g \log \left (h (388+j x)^m\right )\right )}{e}+\frac{d^2 \left (f+g \log \left (h (388+j x)^m\right )\right )}{e^2 (d+e x)}\right ) \, dx\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{1}{2} (g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+\frac{(194 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j}-\frac{(75272 g m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{388+j x} \, dx}{j}-\frac{1}{2} (b n) \int x \left (f+g \log \left (h (388+j x)^m\right )\right ) \, dx+\frac{(b d n) \int \left (f+g \log \left (h (388+j x)^m\right )\right ) \, dx}{2 e}-\frac{\left (b d^2 n\right ) \int \frac{f+g \log \left (h (388+j x)^m\right )}{d+e x} \, dx}{2 e}\\ &=\frac{194 a g m x}{j}+\frac{b d f n x}{2 e}-\frac{1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{75272 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (388+j x)}{388 e-d j}\right )}{j^2}-\frac{1}{4} b n x^2 \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{b d^2 n \log \left (-\frac{j (d+e x)}{388 e-d j}\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )+\frac{(194 b g m) \int \log \left (c (d+e x)^n\right ) \, dx}{j}+\frac{(b d g n) \int \log \left (h (388+j x)^m\right ) \, dx}{2 e}+\frac{1}{4} (b e g m n) \int \frac{x^2}{d+e x} \, dx+\frac{(75272 b e g m n) \int \frac{\log \left (\frac{e (388+j x)}{388 e-d j}\right )}{d+e x} \, dx}{j^2}+\frac{1}{4} (b g j m n) \int \frac{x^2}{388+j x} \, dx+\frac{\left (b d^2 g j m n\right ) \int \frac{\log \left (\frac{j (d+e x)}{-388 e+d j}\right )}{388+j x} \, dx}{2 e^2}\\ &=\frac{194 a g m x}{j}+\frac{b d f n x}{2 e}-\frac{1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{75272 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (388+j x)}{388 e-d j}\right )}{j^2}-\frac{1}{4} b n x^2 \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{b d^2 n \log \left (-\frac{j (d+e x)}{388 e-d j}\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )+\frac{(194 b g m) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j}+\frac{(b d g n) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,388+j x\right )}{2 e j}+\frac{\left (b d^2 g m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-388 e+d j}\right )}{x} \, dx,x,388+j x\right )}{2 e^2}+\frac{1}{4} (b e g m n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx+\frac{(75272 b g m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{388 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2}+\frac{1}{4} (b g j m n) \int \left (-\frac{388}{j^2}+\frac{x}{j}+\frac{150544}{j^2 (388+j x)}\right ) \, dx\\ &=\frac{194 a g m x}{j}+\frac{b d f n x}{2 e}-\frac{3 b d g m n x}{4 e}-\frac{291 b g m n x}{j}+\frac{1}{4} b g m n x^2+\frac{b d^2 g m n \log (d+e x)}{4 e^2}+\frac{194 b g m (d+e x) \log \left (c (d+e x)^n\right )}{e j}-\frac{1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{37636 b g m n \log (388+j x)}{j^2}-\frac{75272 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (388+j x)}{388 e-d j}\right )}{j^2}+\frac{b d g n (388+j x) \log \left (h (388+j x)^m\right )}{2 e j}-\frac{1}{4} b n x^2 \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{b d^2 n \log \left (-\frac{j (d+e x)}{388 e-d j}\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (388+j x)^m\right )\right )-\frac{75272 b g m n \text{Li}_2\left (-\frac{j (d+e x)}{388 e-d j}\right )}{j^2}-\frac{b d^2 g m n \text{Li}_2\left (\frac{e (388+j x)}{388 e-d j}\right )}{2 e^2}\\ \end{align*}

Mathematica [A]  time = 0.637675, size = 341, normalized size = 0.86 \[ \frac{2 b g m n \left (d^2 j^2-e^2 i^2\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )+e \left (j \left (g j x (2 a e x+b n (2 d-e x)) \log \left (h (i+j x)^m\right )+a e x (2 f j x+g m (2 i-j x))-b n (d (-2 f j x+2 g i m+3 g j m x)+e x (f j x+3 g i m-g j m x))\right )+g i m \log (i+j x) (b n (2 d j+e i)-2 a e i)+b e \log \left (c (d+e x)^n\right ) \left (j x \left (2 f j x+2 g j x \log \left (h (i+j x)^m\right )+2 g i m-g j m x\right )-2 g i^2 m \log (i+j x)\right )\right )+b n \log (d+e x) \left (2 g m \left (d^2 j^2-e^2 i^2\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )+d j \left (-2 d f j-2 d g j \log \left (h (i+j x)^m\right )+d g j m+2 e g i m\right )+2 e^2 g i^2 m \log (i+j x)\right )}{4 e^2 j^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 1.639, size = 3163, normalized size = 8. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^2*g*csgn(I*h*(j*x+i)^m)^3+1/8*I*Pi*x^2*b*g*n*csgn(I*h*(j*x+i)^m)^3-1/4*x^2
*n*b*f+1/2*a*f*x^2+1/2*ln(h)*x^2*a*g+1/2*ln(c)*b*f*x^2-1/4*I*ln(c)*Pi*x^2*b*g*csgn(I*h*(j*x+i)^m)^3-1/4*I*Pi*l
n(h)*x^2*b*g*csgn(I*c*(e*x+d)^n)^3+(1/2*g*b*x^2*ln((j*x+i)^m)-1/4*b*(I*Pi*g*j^2*x^2*csgn(I*h)*csgn(I*(j*x+i)^m
)*csgn(I*h*(j*x+i)^m)-I*Pi*g*j^2*x^2*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-I*Pi*g*j^2*x^2*csgn(I*(j*x+i)^m)*csgn(I*h
*(j*x+i)^m)^2+I*Pi*g*j^2*x^2*csgn(I*h*(j*x+i)^m)^3-2*ln(h)*g*j^2*x^2+g*j^2*m*x^2+2*g*i^2*m*ln(j*x+i)-2*f*j^2*x
^2-2*g*i*j*m*x)/j^2)*ln((e*x+d)^n)-1/8*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*g*csgn(I*h*(
j*x+i)^m)^3-1/4*a*g*m*x^2-1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*g/j^2*m*i^2*ln(j*x+i)+1/2/j^2*b*g
*i^2*m*n*ln(j*x+i)*ln(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/4/e/j*g*i*m*b*d*n-5/8/e^2*b*d^2*g*m*n+1/4*I*Pi*ln(h)*x^
2*b*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*I*Pi*ln(h)*x^2*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/8*I*Pi*
x^2*b*g*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*I*Pi*b*f*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*I*Pi*b*f*x^2*cs
gn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*m*a*g*i^2/j^2*ln(j*x+i)+1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d
)^n)^2*g*x^2*ln((j*x+i)^m)+1/4*I*ln(c)*Pi*x^2*b*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/8*I*Pi*x^2*b*g*n*c
sgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/4*I*Pi*b*f*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4*b*
g*m*n*x^2+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g/j^2*m*i^2*ln(j*x+i)-1/4*I/e*Pi*x*b*d*g*n*csgn(I*h*(j*x+i)^m)^3+1/
8*I*Pi*x^2*b*g*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2*b*ln(c)*g/j^2*m*i^2*ln(j*x+i)+1/2/e*ln(h)
*x*b*d*g*n-1/2/e^2*ln(e*x+d)*ln(h)*b*d^2*g*n+1/2/j*ln(c)*x*b*g*i*m+1/2*a*g*x^2*ln((j*x+i)^m)-1/2/e^2*ln(e*x+d)
*b*d^2*f*n-1/4*I*Pi*b*f*x^2*csgn(I*c*(e*x+d)^n)^3-1/2/e^2*n*b*g*ln((j*x+i)^m)*d^2*ln(e*x+d)+1/8*I*Pi*x^2*b*g*m
*csgn(I*c*(e*x+d)^n)^3-1/4*n*b*g*ln((j*x+i)^m)*x^2+1/2*b*ln(c)*g*x^2*ln((j*x+i)^m)-1/8*b*Pi^2*csgn(I*c)*csgn(I
*c*(e*x+d)^n)^2*x^2*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/8*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^2*g*csgn(I*
(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^2*g*csgn(I*h)*csgn(I*h*(
j*x+i)^m)^2-1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^2*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1
/4*I*Pi*x^2*a*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/4*I/j*Pi*x*b*g*i*m*csgn(I*c)*csgn(I*c*(e*x+d
)^n)^2-1/8*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(
j*x+i)^m)+1/8*I*Pi*x^2*b*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/4*I/e^2*ln(e*x+d)*Pi*b*d^2*g*n*
csgn(I*h*(j*x+i)^m)^3-1/4*I*ln(c)*Pi*x^2*b*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/4*I*Pi*x^2*a*g*
csgn(I*h*(j*x+i)^m)^3+1/2/e^2*b*d^2*g*m*n*ln(e*x+d)*ln(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))+1/2/e*n*b*g*ln((j*x+i)^
m)*d*x+1/2/e^2*b*d^2*g*m*n*dilog(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))+1/4/j^2*g*i^2*m*ln((e*x+d)*j-d*j+e*i)*b*n-1/4
*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g*x^2*ln((j*x+i)^m)+1/4*I*Pi*x^2*a*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/4*I*Pi*x^
2*a*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/2*ln(c)*ln(h)*x^2*b*g-1/4*ln(c)*x^2*b*g*m-1/4*ln(h)*x^2*b*g*n-
1/8*I*Pi*x^2*b*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/4*I*ln(c)*Pi*x^2*b*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/4*
I/j*Pi*x*b*g*i*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4*I/e^2*ln(e*x+d)*Pi*b*d^2*g*n*csgn(I*h)*cs
gn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/4*I/e*Pi*x*b*d*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/8*I
*Pi*x^2*b*g*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/8*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^2*g*csgn(I*h)*csgn(I*
(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g*x^2*ln((j*x+i)^m)+1/8*b*Pi^2*csgn(
I*c*(e*x+d)^n)^3*x^2*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/8*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^2*g*csgn(I*(j*x+i)^m
)*csgn(I*h*(j*x+i)^m)^2+1/8*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^2*g*csgn(I*h*(j*x+i)^m)^3+1/8*b*Pi^2*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^2*g*csgn(I*h*(j*x+i)^m)^3+1/2*a*g*i*m*x/j+1/2*b*d*f*n*x/e+1/2/j^2*b*g*i^
2*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/4*I/e^2*ln(e*x+d)*Pi*b*d^2*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/
4*I/e^2*ln(e*x+d)*Pi*b*d^2*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n
)^2*g/j^2*m*i^2*ln(j*x+i)+1/2/e/j*g*i*m*ln((e*x+d)*j-d*j+e*i)*b*d*n+1/2/e/j*ln(e*x+d)*b*d*g*i*m*n+1/8*b*Pi^2*c
sgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/8*b*Pi^2*csgn(I*c)*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/8*b*Pi^2*csgn(I*c)*csgn(I*c*
(e*x+d)^n)^2*x^2*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+
d)^n)^2*x^2*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-3/4*b*d*g*m*n*x/e-3/4*b*g*i*m*n*x/j-1/4*I*Pi*ln(
h)*x^2*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/4*I/j*Pi*x*b*g*i*m*csgn(I*c*(e*x+d)^n)^3-1/4*I*b*
Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g*x^2*ln((j*x+i)^m)+1/4*b*d^2*g*m*n*ln(e*x+d)/e^2+1/4*I*b*P
i*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g/j^2*m*i^2*ln(j*x+i)+1/4*I/j*Pi*x*b*g*i*m*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)^2+1/4*I/e*Pi*x*b*d*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/4*I/e*Pi*x*b*d*g*n*csgn(I*(j*x+i
)^m)*csgn(I*h*(j*x+i)^m)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, b e f n{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} - \frac{1}{4} \, a g j m{\left (\frac{2 \, i^{2} \log \left (j x + i\right )}{j^{3}} + \frac{j x^{2} - 2 \, i x}{j^{2}}\right )} + \frac{1}{2} \, b f x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{1}{2} \, a g x^{2} \log \left ({\left (j x + i\right )}^{m} h\right ) + \frac{1}{2} \, a f x^{2} + \frac{1}{4} \, b g{\left (\frac{2 \, e^{2} i^{2} m n \log \left (e x + d\right ) \log \left (j x + i\right ) +{\left (2 \, e^{2} i j m x - 2 \, e^{2} i^{2} m \log \left (j x + i\right ) -{\left (j^{2} m - 2 \, j^{2} \log \left (h\right )\right )} e^{2} x^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (2 \, e^{2} j^{2} x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + 2 \, d e j^{2} n x - 2 \, d^{2} j^{2} n \log \left (e x + d\right ) -{\left (e^{2} j^{2} n - 2 \, e^{2} j^{2} \log \left (c\right )\right )} x^{2}\right )} \log \left ({\left (j x + i\right )}^{m}\right )}{e^{2} j^{2}} + 4 \, \int -\frac{2 \,{\left ({\left (j^{2} m - 2 \, j^{2} \log \left (h\right )\right )} e^{3} \log \left (c\right ) -{\left (j^{2} m n - j^{2} n \log \left (h\right )\right )} e^{3}\right )} x^{3} +{\left (d e^{2} j^{2} m n +{\left (i j m n + 2 \, i j n \log \left (h\right )\right )} e^{3} - 2 \,{\left (2 \, e^{3} i j \log \left (h\right ) -{\left (j^{2} m - 2 \, j^{2} \log \left (h\right )\right )} d e^{2}\right )} \log \left (c\right )\right )} x^{2} + 2 \,{\left (e^{3} i^{2} m n + d^{2} e j^{2} m n - 2 \, d e^{2} i j \log \left (c\right ) \log \left (h\right )\right )} x + 2 \,{\left (d e^{2} i^{2} m n - d^{3} j^{2} m n +{\left (e^{3} i^{2} m n - d^{2} e j^{2} m n\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{3} j^{2} x^{2} + d e^{2} i j +{\left (e^{3} i j + d e^{2} j^{2}\right )} x\right )}}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*e*f*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) - 1/4*a*g*j*m*(2*i^2*log(j*x + i)/j^3 + (j*x^2 - 2
*i*x)/j^2) + 1/2*b*f*x^2*log((e*x + d)^n*c) + 1/2*a*g*x^2*log((j*x + i)^m*h) + 1/2*a*f*x^2 + 1/4*b*g*((2*e^2*i
^2*m*n*log(e*x + d)*log(j*x + i) + (2*e^2*i*j*m*x - 2*e^2*i^2*m*log(j*x + i) - (j^2*m - 2*j^2*log(h))*e^2*x^2)
*log((e*x + d)^n) + (2*e^2*j^2*x^2*log((e*x + d)^n) + 2*d*e*j^2*n*x - 2*d^2*j^2*n*log(e*x + d) - (e^2*j^2*n -
2*e^2*j^2*log(c))*x^2)*log((j*x + i)^m))/(e^2*j^2) + 4*integrate(-1/4*(2*((j^2*m - 2*j^2*log(h))*e^3*log(c) -
(j^2*m*n - j^2*n*log(h))*e^3)*x^3 + (d*e^2*j^2*m*n + (i*j*m*n + 2*i*j*n*log(h))*e^3 - 2*(2*e^3*i*j*log(h) - (j
^2*m - 2*j^2*log(h))*d*e^2)*log(c))*x^2 + 2*(e^3*i^2*m*n + d^2*e*j^2*m*n - 2*d*e^2*i*j*log(c)*log(h))*x + 2*(d
*e^2*i^2*m*n - d^3*j^2*m*n + (e^3*i^2*m*n - d^2*e*j^2*m*n)*x)*log(e*x + d))/(e^3*j^2*x^2 + d*e^2*i*j + (e^3*i*
j + d*e^2*j^2)*x), x))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b f x \log \left ({\left (e x + d\right )}^{n} c\right ) + a f x +{\left (b g x \log \left ({\left (e x + d\right )}^{n} c\right ) + a g x\right )} \log \left ({\left (j x + i\right )}^{m} h\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}{\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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