3.4.88 \(\int x (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [388]
Optimal. Leaf size=397 \[ \frac {a g i m x}{2 j}+\frac {b d f n x}{2 e}-\frac {3 b d g m n x}{4 e}-\frac {3 b g i m n x}{4 j}+\frac {1}{4} b g m n x^2+\frac {b d^2 g m n \log (d+e x)}{4 e^2}+\frac {b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g i^2 m n \log (i+j x)}{4 j^2}-\frac {g i^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}+\frac {b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^2 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}-\frac {b d^2 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{2 e^2} \]
[Out]
1/2*a*g*i*m*x/j+1/2*b*d*f*n*x/e-3/4*b*d*g*m*n*x/e-3/4*b*g*i*m*n*x/j+1/4*b*g*m*n*x^2+1/4*b*d^2*g*m*n*ln(e*x+d)/
e^2+1/2*b*g*i*m*(e*x+d)*ln(c*(e*x+d)^n)/e/j-1/4*g*m*x^2*(a+b*ln(c*(e*x+d)^n))+1/4*b*g*i^2*m*n*ln(j*x+i)/j^2-1/
2*g*i^2*m*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j^2+1/2*b*d*g*n*(j*x+i)*ln(h*(j*x+i)^m)/e/j-1/4*b*n*x
^2*(f+g*ln(h*(j*x+i)^m))-1/2*b*d^2*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^2+1/2*x^2*(a+b*ln(c*(e*
x+d)^n))*(f+g*ln(h*(j*x+i)^m))-1/2*b*g*i^2*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2-1/2*b*d^2*g*m*n*polylog(2,
e*(j*x+i)/(-d*j+e*i))/e^2
________________________________________________________________________________________
Rubi [A]
time = 0.31, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2489, 45,
2463, 2436, 2332, 2442, 2441, 2440, 2438} \begin {gather*} -\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 \end {gather*}
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 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 \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) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j}+\frac {(b d g n) \text {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 ) \text {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) \text {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 = 0.34, size = 341, normalized size = 0.86 \begin {gather*} \frac {b n \log (d+e x) \left (2 e^2 g i^2 m \log (i+j x)+2 g \left (-e^2 i^2+d^2 j^2\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (-2 d f j+2 e g i m+d g j m-2 d g j \log \left (h (i+j x)^m\right )\right )\right )+e \left (g i m (-2 a e i+b (e i+2 d j) n) \log (i+j x)+j \left (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 \left (h (i+j x)^m\right )\right )+b e \log \left (c (d+e x)^n\right ) \left (-2 g i^2 m \log (i+j x)+j x \left (2 g i m+2 f j x-g j m x+2 g j x \log \left (h (i+j x)^m\right )\right )\right )\right )+2 b g \left (-e^2 i^2+d^2 j^2\right ) m n \text {Li}_2\left (\frac {j (d+e x)}{-e i+d j}\right )}{4 e^2 j^2} \end {gather*}
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)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.94, size = 3163, normalized size = 7.97
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result |
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| risch |
\(\text {Expression too large to display}\) |
\(3163\) |
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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,method=_RETURNVERBOSE)
[Out]
-5/8/e^2*b*d^2*g*m*n+1/2*ln(h)*ln(c)*x^2*b*g-1/4*ln(h)*x^2*b*g*n-1/4*ln(c)*x^2*b*g*m+(1/2*x^2*b*g*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)+1/2*x^2*a*f-1/4*n*b*f*x^2
-1/4*a*g*m*x^2+1/2*ln(c)*b*f*x^2-1/4*n*b*g*ln((j*x+i)^m)*x^2+1/2*b*ln(c)*g*x^2*ln((j*x+i)^m)+1/2*ln(h)*x^2*a*g
-1/2*m*a*g*i^2/j^2*ln(j*x+i)+1/2*a*g*x^2*ln((j*x+i)^m)-1/4/e/j*g*i*m*b*d*n+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*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*n*b*d^2*f/e^2*ln(e*x+d)+1/2/e*ln(h)
*x*b*d*g*n-1/2/e^2*ln(h)*ln(e*x+d)*b*d^2*g*n-1/2*b*ln(c)*g*m/j^2*i^2*ln(j*x+i)+1/2/j*ln(c)*x*b*g*i*m+1/8*csgn(
I*(j*x+i)^m)*csgn(I*h)*x^2*Pi^2*csgn(I*(e*x+d)^n)*g*csgn(I*h*(j*x+i)^m)*csgn(I*c*(e*x+d)^n)^2*b+1/8*Pi^2*x^2*b
*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/4*I*Pi*b*f*x^2*csgn(I*c)*
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2/j^2*b*g*i^2*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*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*I*Pi*b*f*x^2*csgn(I*c*(e*x+d)^n)^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))+1/4/j^2*g*i^2*m*ln((e*x+d)*j-d*j+e*i)*b*n+1/2/e^2*b*d^2*g*m*n*dilog
(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))-1/8*Pi^2*x^2*b*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2*csgn(I*c)*csgn(I*c*(
e*x+d)^n)^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)-1/8*I*Pi*x^2*b*g*n*csgn(I*h)*csgn
(I*h*(j*x+i)^m)^2-1/8*I*Pi*x^2*b*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/4*I*Pi*ln(c)*x^2*b*g*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*x^2*ln((j*x+i)^m)+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(h)*Pi*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+1/8*csgn(I*(j*x+i)^m)*x^2*Pi^2*g*csgn(I*h*(j*x+i)^m)
^2*csgn(I*c*(e*x+d)^n)^3*b+1/4*I*Pi*x^2*a*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/4*I*ln(h)*Pi*x^2*b*g*csg
n(I*c*(e*x+d)^n)^3-1/8*x^2*Pi^2*g*csgn(I*h*(j*x+i)^m)^3*csgn(I*c*(e*x+d)^n)^3*b-1/4*I*Pi*x^2*a*g*csgn(I*h*(j*x
+i)^m)^3-1/4*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*b*Pi*csgn(I*c)*csgn(I*(e*x
+d)^n)*csgn(I*c*(e*x+d)^n)*g*m/j^2*i^2*ln(j*x+i)+1/2/e*n*b*g*ln((j*x+i)^m)*d*x-1/2/e^2*n*b*g*ln((j*x+i)^m)*d^2
*ln(e*x+d)+1/8*Pi^2*x^2*b*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/
8*Pi^2*x^2*b*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-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*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/4*b*d
^2*g*m*n*ln(e*x+d)/e^2-1/4*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*I*b*
Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g*m/j^2*i^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*
m/j^2*i^2*ln(j*x+i)+1/4*I/e*Pi*x*b*d*g*n*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
*c*(e*x+d)^n)^2-1/4*I/e^2*ln(e*x+d)*Pi*b*d^2*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/8*csgn(I*h)*x^2*Pi^2*g*csgn
(I*h*(j*x+i)^m)^2*csgn(I*c*(e*x+d)^n)^3*b-1/4*I/j*Pi*x*b*g*i*m*csgn(I*c*(e*x+d)^n)^3+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(h)*Pi*x^2*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4*I*b*P
i*csgn(I*c*(e*x+d)^n)^3*g*m/j^2*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-1/4*I*Pi*ln(c)*x^2*b*g*csgn(I*h*(j*x+i)^m)^3+1/8*I*Pi*x^2*b*g*m*csgn(I*c*(e*x+d)^n)^3+1/8*I*Pi*x^2*b*g
*n*csgn(I*h*(j*x+i)^m)^3-1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g*x^2*ln((j*x+i)^m)-1/8*Pi^2*x^2*b*g*csgn(I*h)*csgn(
I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4*b*g*m*n*x^2+1/8*x^2*Pi^2*
csgn(I*(e*x+d)^n)*g*csgn(I*h*(j*x+i)^m)^3*csgn(I*c*(e*x+d)^n)^2*b-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*Pi^2*x^2*b*g*csgn(I*h*(j*x+i)^m)^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/8*csgn
(I*(j*x+i)^m)*x^2*Pi^2*csgn(I*(e*x+d)^n)*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*c*(e*x+d)^n)^2*b-1/8*csgn(I*(j*x+i)^m)
*csgn(I*h)*x^2*Pi^2*g*csgn(I*h*(j*x+i)^m)*csgn(I*c*(e*x+d)^n)^3*b-1/8*Pi^2*x^2*b*g*csgn(I*h)*csgn(I*h*(j*x+i)^
m)^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/8*csgn(I*h)*x^2*Pi^2*csgn(I*(e*x+d)^n)*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*c
*(e*x+d)^n)^2*b+1/4*I*Pi*x^2*a*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/8*Pi^2*x^2*b*g*csgn(I*h*(j*x+i)^m)^3*csgn(I
*c)*csgn(I*c*(e*x+d)^n)^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+1/4*I*ln(h)*Pi*x^2*b*g*
csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*I*Pi*ln(c)*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*cs
gn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/2*a*g*i*m*x/j+1/2*b*d*f*n*x/e-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)+1/4*I/e^2*ln(e*x+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*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")
[Out]
-1/4*a*g*j*m*((j*x^2 - 2*I*x)/j^2 - 2*log(j*x + I)/j^3) - 1/4*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(
-2))*b*f*n*e + 1/2*b*f*x^2*log((x*e + 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*m*n
*e^2*log(j*x + I)*log(x*e + d) - (2*d*j^2*n*x*e - 2*d^2*j^2*n*log(x*e + d) + 2*j^2*x^2*e^2*log((x*e + d)^n) -
(j^2*n - 2*j^2*log(c))*x^2*e^2)*log((j*x + I)^m) - (2*I*j*m*x*e^2 - (j^2*m - 2*j^2*log(h))*x^2*e^2 + 2*m*e^2*l
og(j*x + I))*log((x*e + d)^n))*e^(-2)/j^2 - 4*integrate(1/4*(2*(j^2*m*n - j^2*n*log(h) - (j^2*m - 2*j^2*log(h)
)*log(c))*x^3*e^3 + ((-I*j*m*n - 2*I*j*n*log(h) + 4*I*j*log(c)*log(h))*e^3 - (d*j^2*m*n + 2*(j^2*m - 2*j^2*log
(h))*d*log(c))*e^2)*x^2 - 2*(d^2*j^2*m*n*e - 2*I*d*j*e^2*log(c)*log(h) - m*n*e^3)*x + 2*(d^3*j^2*m*n + d*m*n*e
^2 + (d^2*j^2*m*n*e + m*n*e^3)*x)*log(x*e + d))/(j^2*x^2*e^3 + I*d*j*e^2 + (d*j^2*e^2 + I*j*e^3)*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*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")
[Out]
1/4*(4*j^2*integral(1/4*(4*a*d*f*j^2*x + (-2*I*b*g*j*m*n*x + (4*a*f*j^2 + (b*g*j^2*m - 2*b*f*j^2)*n)*x^2)*e +
2*(2*a*d*g*j^2*m*x - (b*g*m*n + (b*g*j^2*m*n - 2*a*g*j^2*m)*x^2)*e + 2*(b*g*j^2*m*x^2*e + b*d*g*j^2*m*x)*log(c
))*log(j*x + I) + 4*(b*f*j^2*x^2*e + b*d*f*j^2*x)*log(c) + 2*(2*a*d*g*j^2*x - (b*g*j^2*n - 2*a*g*j^2)*x^2*e +
2*(b*g*j^2*x^2*e + b*d*g*j^2*x)*log(c))*log(h))/(j^2*x*e + d*j^2), x) + (2*b*g*j^2*n*x^2*log(h) + 2*I*b*g*j*m*
n*x - (b*g*j^2*m - 2*b*f*j^2)*n*x^2 + 2*(b*g*j^2*m*n*x^2 + b*g*m*n)*log(j*x + I))*log(x*e + d))/j^2
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (h \left (i + j x\right )^{m} \right )}\right )\, dx \end {gather*}
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]
Integral(x*(a + b*log(c*(d + e*x)**n))*(f + g*log(h*(i + j*x)**m)), 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(x*(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, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\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*(a + b*log(c*(d + e*x)^n))*(f + g*log(h*(i + j*x)^m)),x)
[Out]
int(x*(a + b*log(c*(d + e*x)^n))*(f + g*log(h*(i + j*x)^m)), x)
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