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3.1.2 \(\int (a g+b g x)^2 (c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [2]

Optimal. Leaf size=180 \[ \frac {B (b c-a d)^3 g^2 i x}{12 b d^2}-\frac {B (b c-a d)^2 g^2 i (a+b x)^2}{24 b^2 d}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2}-\frac {B (b c-a d)^4 g^2 i \log (c+d x)}{12 b^2 d^3} \]

[Out]

1/12*B*(-a*d+b*c)^3*g^2*i*x/b/d^2-1/24*B*(-a*d+b*c)^2*g^2*i*(b*x+a)^2/b^2/d+1/4*g^2*i*(b*x+a)^3*(d*x+c)*(A+B*l
n(e*(b*x+a)/(d*x+c)))/b+1/12*(-a*d+b*c)*g^2*i*(b*x+a)^3*(A-B+B*ln(e*(b*x+a)/(d*x+c)))/b^2-1/12*B*(-a*d+b*c)^4*
g^2*i*ln(d*x+c)/b^2/d^3

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Rubi [A]
time = 0.10, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2560, 2548, 21, 45} \begin {gather*} \frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}-\frac {B g^2 i (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}-\frac {B g^2 i (a+b x)^2 (b c-a d)^2}{24 b^2 d}+\frac {B g^2 i x (b c-a d)^3}{12 b d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(B*(b*c - a*d)^3*g^2*i*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*(a + b*x)^2)/(24*b^2*d) + (g^2*i*(a + b*x)^3*(c
+ d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A - B + B*Log[(e*(a + b*x
))/(c + d*x)]))/(12*b^2) - (B*(b*c - a*d)^4*g^2*i*Log[c + d*x])/(12*b^2*d^3)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
 n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
 d*x, a + b*x])

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 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rule 2560

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
)*((h_.) + (i_.)*(x_)), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(h + i*x)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)]
)/(g*(m + 2))), x] + Dist[i*((b*c - a*d)/(b*d*(m + 2))), Int[(f + g*x)^m*(A - B*n + B*Log[e*((a + b*x)^n/(c +
d*x)^n)]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[
b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IGtQ[m, -2]

Rubi steps

\begin {align*} \int (2 c+2 d x) (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac {2 (b c-a d) (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}+\frac {2 d (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}\right ) \, dx\\ &=\frac {(2 (b c-a d)) \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b}+\frac {(2 d) \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b g}\\ &=\frac {2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac {(B d) \int \frac {(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{2 b^2 g^2}-\frac {(2 B (b c-a d)) \int \frac {(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{3 b^2 g}\\ &=\frac {2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac {\left (B d (b c-a d) g^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{2 b^2}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b^2}\\ &=\frac {2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac {\left (B d (b c-a d) g^2\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{2 b^2}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac {B (b c-a d)^3 g^2 x}{6 b d^2}-\frac {B (b c-a d)^2 g^2 (a+b x)^2}{12 b^2 d}-\frac {B (b c-a d) g^2 (a+b x)^3}{6 b^2}+\frac {2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac {B (b c-a d)^4 g^2 \log (c+d x)}{6 b^2 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 217, normalized size = 1.21 \begin {gather*} \frac {g^2 i \left (8 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 d (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )}{d^3}-\frac {B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^3}\right )}{24 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(g^2*i*(8*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*d*(a + b*x)^4*(A + B*Log[(e*(a + b*
x))/(c + d*x)]) + (4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]))/d
^3 - (B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a
*d)^3*Log[c + d*x]))/d^3))/(24*b^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4929\) vs. \(2(170)=340\).
time = 0.59, size = 4930, normalized size = 27.39

method result size
risch \(\frac {i \,g^{2} B x \left (3 b^{2} d \,x^{3}+8 a b d \,x^{2}+4 b^{2} c \,x^{2}+6 x \,a^{2} d +12 a b c x +12 c \,a^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12}+\frac {i \,g^{2} b^{2} d A \,x^{4}}{4}+\frac {2 i \,g^{2} b d A a \,x^{3}}{3}+\frac {i \,g^{2} b^{2} A c \,x^{3}}{3}+\frac {i \,g^{2} b d B a \,x^{3}}{12}-\frac {i \,g^{2} b^{2} B c \,x^{3}}{12}+\frac {i \,g^{2} d A \,a^{2} x^{2}}{2}+i \,g^{2} b A a c \,x^{2}+\frac {5 i \,g^{2} d B \,a^{2} x^{2}}{24}-\frac {i \,g^{2} b B a c \,x^{2}}{6}-\frac {i \,g^{2} b^{2} B \,c^{2} x^{2}}{24 d}+i \,g^{2} A \,a^{2} c x -\frac {i \,g^{2} B \ln \left (-d x -c \right ) a^{2} c^{2}}{2 d}+\frac {i \,g^{2} b B \ln \left (-d x -c \right ) a \,c^{3}}{3 d^{2}}-\frac {i \,g^{2} b^{2} B \ln \left (-d x -c \right ) c^{4}}{12 d^{3}}-\frac {i \,g^{2} d B \ln \left (b x +a \right ) a^{4}}{12 b^{2}}+\frac {i \,g^{2} B \ln \left (b x +a \right ) a^{3} c}{3 b}+\frac {i \,g^{2} d B \,a^{3} x}{12 b}+\frac {i \,g^{2} B \,a^{2} c x}{6}-\frac {i \,g^{2} b B a \,c^{2} x}{3 d}+\frac {i \,g^{2} b^{2} B \,c^{3} x}{12 d^{2}}\) \(396\)
derivativedivides \(\text {Expression too large to display}\) \(4930\)
default \(\text {Expression too large to display}\) \(4930\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURNVERBOSE)

[Out]

-1/d^2*e*(a*d-b*c)*(B*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(b*e-(b*e/d+(a*d
-b*c)*e/d/(d*x+c))*d)^2*c^3-5/24*B*d^2*e*g^2*i/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^3-B*d^3*e^2*g^2*i*b*l
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^3+1/4*B
*d^6/e*g^2*i/b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*
x+c))*d)^4*a^3-1/4*B*d^3/e*g^2*i*b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4/(b*e-(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*d)^4*c^3+3/4*B*d^4/e*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c
))^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a*c^2+3*B*d^4*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b
*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^2*c-2*B*d^4*g^2*i/b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c
))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^3+2*B*d*g^2*i*b^2*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*c^3+3*B*d^2*g^2*i*ln(
b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^2*c+B*e^2
*g^2*i*b^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)
^4*c^3-B*d^5*g^2*i/b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)*e/d
/(d*x+c))*d)^4*a^3+B*d^2*g^2*i*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*d)^4*c^3-1/12*B/d*e^2*g^2*i*b^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*c^3-5/8*B*e*g^
2*i/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a*b^2*c^2+1/4*B*e^2*g^2*i*b^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d
)^3*a*c^2-1/12*B*d^2/e*g^2*i/b^2*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^3+1/12*B/d/e*g^2*i*b*ln(b*e-(b*e/d+
(a*d-b*c)*e/d/(d*x+c))*d)*c^3+5/24*B/d*e*g^2*i/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*b^3*c^3+1/12*B*d^2*e^2*
g^2*i*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^3+3/2*B*d^2/e*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(
a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a*c^2-1/2*B*d/e*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(
d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c^3-3/2*B*d*e*g^2*i*ln(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*b^3*c^3-1/4*B/
e*g^2*i*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a*c^2+1/4*B*g^2*i*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a*c^
2+1/12*B*d^2*g^2*i/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^3-1/12*B/d*g^2*i*b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d
*x+c))*d)*c^3-A*d^2*e*g^2*i*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(-2/3*b*e/d^3/(b*e-(b*e/d+(a*d-b*c)*
e/d/(d*x+c))*d)^3+1/2/d^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2+1/4*b^2*e^2/d^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d
*x+c))*d)^4)-1/4*B*d*g^2*i/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^2*c-3*B*d*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x
+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a*c^2-6*B*d^2*g^2*i*b*ln(b*e/d+(a
*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a*c^2-3*B*d^3*g^2
*i*b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a
*c^2-2/3*B*d^2/e*g^2*i*b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)
*e/d/(d*x+c))*d)^3*c^3+5/8*B*d*e*g^2*i/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^2*b*c+2*B*d^3*e*g^2*i*ln(b*e/
d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^3-1/4*B*d*e^2
*g^2*i*b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^2*c-3*B*d*e^2*g^2*i*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(
b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a*c^2-3/2*B*d^3/e*g^2*i*ln(b*e/d+(a*d-b*c
)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^2*c-2*B*d^4/e*g^2*i
/b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^2
*c-3/4*B*d^5/e*g^2*i/b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4/(b*e-(b*e/d+(a*d-b*c)*e
/d/(d*x+c))*d)^4*a^2*c-9/2*B*d^3*e*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-
(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^2*b*c+9/2*B*d^2*e*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*
e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a*b^2*c^2-6*B*d^2*e*g^2*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c
))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^2*b*c+6*B*d*e*g^2*i*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a*b^2*c^2+2*B*d^3/e*g^2
*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a*c
^2+2/3*B*d^5/e*g^2*i/b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/(b*e-(b*e/d+(a*d-b*c)
*e/d/(d*x+c))*d)^3*a^3+1/2*B*d^4/e*g^2*i*ln(b*e...

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (167) = 334\).
time = 0.30, size = 675, normalized size = 3.75 \begin {gather*} \frac {1}{4} i \, A b^{2} d g^{2} x^{4} + \frac {1}{3} i \, A b^{2} c g^{2} x^{3} + \frac {2}{3} i \, A a b d g^{2} x^{3} + i \, A a b c g^{2} x^{2} + \frac {1}{2} i \, A a^{2} d g^{2} x^{2} + i \, {\left (x \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{2} c g^{2} + i \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a b c g^{2} + \frac {1}{6} i \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} c g^{2} + \frac {1}{2} i \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{2} d g^{2} + \frac {1}{3} i \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b d g^{2} + \frac {1}{24} i \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{2} d g^{2} + i \, A a^{2} c g^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/4*I*A*b^2*d*g^2*x^4 + 1/3*I*A*b^2*c*g^2*x^3 + 2/3*I*A*a*b*d*g^2*x^3 + I*A*a*b*c*g^2*x^2 + 1/2*I*A*a^2*d*g^2*
x^2 + I*(x*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a^2*c*g^2 + I*(x^2*lo
g(b*x*e/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a*b*
c*g^2 + 1/6*I*(2*x^3*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 -
((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*b^2*c*g^2 + 1/2*I*(x^2*log(b*x*e/(d*x + c) +
a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^2*d*g^2 + 1/3*I*(2*x^3
*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)
*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*b*d*g^2 + 1/24*I*(6*x^4*log(b*x*e/(d*x + c) + a*e/(d*x + c)) -
6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x
^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^2*d*g^2 + I*A*a^2*c*g^2*x

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (167) = 334\).
time = 0.40, size = 375, normalized size = 2.08 \begin {gather*} \frac {6 i \, A b^{4} d^{4} g^{2} x^{4} - 2 \, {\left ({\left (-4 i \, A + i \, B\right )} b^{4} c d^{3} + {\left (-8 i \, A - i \, B\right )} a b^{3} d^{4}\right )} g^{2} x^{3} + {\left (-i \, B b^{4} c^{2} d^{2} - 4 \, {\left (-6 i \, A + i \, B\right )} a b^{3} c d^{3} + {\left (12 i \, A + 5 i \, B\right )} a^{2} b^{2} d^{4}\right )} g^{2} x^{2} - 2 \, {\left (-i \, B b^{4} c^{3} d + 4 i \, B a b^{3} c^{2} d^{2} + 2 \, {\left (-6 i \, A - i \, B\right )} a^{2} b^{2} c d^{3} - i \, B a^{3} b d^{4}\right )} g^{2} x - 2 \, {\left (-4 i \, B a^{3} b c d^{3} + i \, B a^{4} d^{4}\right )} g^{2} \log \left (\frac {b x + a}{b}\right ) - 2 \, {\left (i \, B b^{4} c^{4} - 4 i \, B a b^{3} c^{3} d + 6 i \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} \log \left (\frac {d x + c}{d}\right ) - 2 \, {\left (-3 i \, B b^{4} d^{4} g^{2} x^{4} - 12 i \, B a^{2} b^{2} c d^{3} g^{2} x + 4 \, {\left (-i \, B b^{4} c d^{3} - 2 i \, B a b^{3} d^{4}\right )} g^{2} x^{3} + 6 \, {\left (-2 i \, B a b^{3} c d^{3} - i \, B a^{2} b^{2} d^{4}\right )} g^{2} x^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{24 \, b^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/24*(6*I*A*b^4*d^4*g^2*x^4 - 2*((-4*I*A + I*B)*b^4*c*d^3 + (-8*I*A - I*B)*a*b^3*d^4)*g^2*x^3 + (-I*B*b^4*c^2*
d^2 - 4*(-6*I*A + I*B)*a*b^3*c*d^3 + (12*I*A + 5*I*B)*a^2*b^2*d^4)*g^2*x^2 - 2*(-I*B*b^4*c^3*d + 4*I*B*a*b^3*c
^2*d^2 + 2*(-6*I*A - I*B)*a^2*b^2*c*d^3 - I*B*a^3*b*d^4)*g^2*x - 2*(-4*I*B*a^3*b*c*d^3 + I*B*a^4*d^4)*g^2*log(
(b*x + a)/b) - 2*(I*B*b^4*c^4 - 4*I*B*a*b^3*c^3*d + 6*I*B*a^2*b^2*c^2*d^2)*g^2*log((d*x + c)/d) - 2*(-3*I*B*b^
4*d^4*g^2*x^4 - 12*I*B*a^2*b^2*c*d^3*g^2*x + 4*(-I*B*b^4*c*d^3 - 2*I*B*a*b^3*d^4)*g^2*x^3 + 6*(-2*I*B*a*b^3*c*
d^3 - I*B*a^2*b^2*d^4)*g^2*x^2)*log((b*x + a)*e/(d*x + c)))/(b^2*d^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (163) = 326\).
time = 2.82, size = 850, normalized size = 4.72 \begin {gather*} \frac {A b^{2} d g^{2} i x^{4}}{4} - \frac {B a^{3} g^{2} i \left (a d - 4 b c\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{2} i + \frac {B a^{4} d^{3} g^{2} i \left (a d - 4 b c\right )}{b} - 10 B a^{3} b c^{2} d^{2} g^{2} i - B a^{3} c d^{2} g^{2} i \left (a d - 4 b c\right ) + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 b^{2}} - \frac {B c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{2} i - 10 B a^{3} b c^{2} d^{2} g^{2} i + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i + B a b c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) - \frac {B b^{2} c^{3} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 d^{3}} + x^{3} \cdot \left (\frac {2 A a b d g^{2} i}{3} + \frac {A b^{2} c g^{2} i}{3} + \frac {B a b d g^{2} i}{12} - \frac {B b^{2} c g^{2} i}{12}\right ) + x^{2} \left (\frac {A a^{2} d g^{2} i}{2} + A a b c g^{2} i + \frac {5 B a^{2} d g^{2} i}{24} - \frac {B a b c g^{2} i}{6} - \frac {B b^{2} c^{2} g^{2} i}{24 d}\right ) + x \left (A a^{2} c g^{2} i + \frac {B a^{3} d g^{2} i}{12 b} + \frac {B a^{2} c g^{2} i}{6} - \frac {B a b c^{2} g^{2} i}{3 d} + \frac {B b^{2} c^{3} g^{2} i}{12 d^{2}}\right ) + \left (B a^{2} c g^{2} i x + \frac {B a^{2} d g^{2} i x^{2}}{2} + B a b c g^{2} i x^{2} + \frac {2 B a b d g^{2} i x^{3}}{3} + \frac {B b^{2} c g^{2} i x^{3}}{3} + \frac {B b^{2} d g^{2} i x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**2*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**2*d*g**2*i*x**4/4 - B*a**3*g**2*i*(a*d - 4*b*c)*log(x + (B*a**4*c*d**3*g**2*i + B*a**4*d**3*g**2*i*(a*d -
 4*b*c)/b - 10*B*a**3*b*c**2*d**2*g**2*i - B*a**3*c*d**2*g**2*i*(a*d - 4*b*c) + 4*B*a**2*b**2*c**3*d*g**2*i -
B*a*b**3*c**4*g**2*i)/(B*a**4*d**4*g**2*i - 4*B*a**3*b*c*d**3*g**2*i - 6*B*a**2*b**2*c**2*d**2*g**2*i + 4*B*a*
b**3*c**3*d*g**2*i - B*b**4*c**4*g**2*i))/(12*b**2) - B*c**2*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)*log(
x + (B*a**4*c*d**3*g**2*i - 10*B*a**3*b*c**2*d**2*g**2*i + 4*B*a**2*b**2*c**3*d*g**2*i - B*a*b**3*c**4*g**2*i
+ B*a*b*c**2*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2) - B*b**2*c**3*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2
*c**2)/d)/(B*a**4*d**4*g**2*i - 4*B*a**3*b*c*d**3*g**2*i - 6*B*a**2*b**2*c**2*d**2*g**2*i + 4*B*a*b**3*c**3*d*
g**2*i - B*b**4*c**4*g**2*i))/(12*d**3) + x**3*(2*A*a*b*d*g**2*i/3 + A*b**2*c*g**2*i/3 + B*a*b*d*g**2*i/12 - B
*b**2*c*g**2*i/12) + x**2*(A*a**2*d*g**2*i/2 + A*a*b*c*g**2*i + 5*B*a**2*d*g**2*i/24 - B*a*b*c*g**2*i/6 - B*b*
*2*c**2*g**2*i/(24*d)) + x*(A*a**2*c*g**2*i + B*a**3*d*g**2*i/(12*b) + B*a**2*c*g**2*i/6 - B*a*b*c**2*g**2*i/(
3*d) + B*b**2*c**3*g**2*i/(12*d**2)) + (B*a**2*c*g**2*i*x + B*a**2*d*g**2*i*x**2/2 + B*a*b*c*g**2*i*x**2 + 2*B
*a*b*d*g**2*i*x**3/3 + B*b**2*c*g**2*i*x**3/3 + B*b**2*d*g**2*i*x**4/4)*log(e*(a + b*x)/(c + d*x))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3911 vs. \(2 (167) = 334\).
time = 2.78, size = 3911, normalized size = 21.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

1/24*(2*I*B*b^9*c^5*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 10*I*B*a*b^8*c^4*d*g^2*e^5*log(-b*e + (b*x
*e + a*e)*d/(d*x + c)) + 20*I*B*a^2*b^7*c^3*d^2*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 20*I*B*a^3*b^6
*c^2*d^3*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 10*I*B*a^4*b^5*c*d^4*g^2*e^5*log(-b*e + (b*x*e + a*e)
*d/(d*x + c)) - 2*I*B*a^5*b^4*d^5*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 8*I*(b*x*e + a*e)*B*b^8*c^5*
d*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 40*I*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^2*e^4*log(-b*
e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 80*I*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^2*e^4*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c) + 80*I*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))
/(d*x + c) - 40*I*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 8*I*
(b*x*e + a*e)*B*a^5*b^3*d^6*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 12*I*(b*x*e + a*e)^2*B*b
^7*c^5*d^2*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 60*I*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^
2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 120*I*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^2*e^3*log(
-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 120*I*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g^2*e^3*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 60*I*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/
(d*x + c))/(d*x + c)^2 - 12*I*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
 + c)^2 - 8*I*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 40*I*(
b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 80*I*(b*x*e + a*e)^
3*B*a^2*b^4*c^3*d^5*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 80*I*(b*x*e + a*e)^3*B*a^3*b^3
*c^2*d^6*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 40*I*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^2*
e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 8*I*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*e^2*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 2*I*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x
+ c))/(d*x + c)^4 - 10*I*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^4 + 20*I*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 20*I*(b*
x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 10*I*(b*x*e + a*e)^4*
B*a^4*b*c*d^8*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 2*I*(b*x*e + a*e)^4*B*a^5*d^9*g^2*e*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 8*I*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*e^2*log((b*x*e + a*e)/
(d*x + c))/(d*x + c)^3 - 40*I*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3
 + 80*I*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 80*I*(b*x*e + a*e
)^3*B*a^3*b^3*c^2*d^6*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 40*I*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*
g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 8*I*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*e^2*log((b*x*e + a*e)/(
d*x + c))/(d*x + c)^3 - 2*I*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 10*
I*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 20*I*(b*x*e + a*e)^4*B*a^2*
b^3*c^3*d^6*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 20*I*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^2*e*log(
(b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 10*I*(b*x*e + a*e)^4*B*a^4*b*c*d^8*g^2*e*log((b*x*e + a*e)/(d*x + c))/(
d*x + c)^4 + 2*I*(b*x*e + a*e)^4*B*a^5*d^9*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 2*I*A*b^9*c^5*g^2*
e^5 + I*B*b^9*c^5*g^2*e^5 - 10*I*A*a*b^8*c^4*d*g^2*e^5 - 5*I*B*a*b^8*c^4*d*g^2*e^5 + 20*I*A*a^2*b^7*c^3*d^2*g^
2*e^5 + 10*I*B*a^2*b^7*c^3*d^2*g^2*e^5 - 20*I*A*a^3*b^6*c^2*d^3*g^2*e^5 - 10*I*B*a^3*b^6*c^2*d^3*g^2*e^5 + 10*
I*A*a^4*b^5*c*d^4*g^2*e^5 + 5*I*B*a^4*b^5*c*d^4*g^2*e^5 - 2*I*A*a^5*b^4*d^5*g^2*e^5 - I*B*a^5*b^4*d^5*g^2*e^5
- 8*I*(b*x*e + a*e)*A*b^8*c^5*d*g^2*e^4/(d*x + c) - 2*I*(b*x*e + a*e)*B*b^8*c^5*d*g^2*e^4/(d*x + c) + 40*I*(b*
x*e + a*e)*A*a*b^7*c^4*d^2*g^2*e^4/(d*x + c) + 10*I*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^2*e^4/(d*x + c) - 80*I*(b*
x*e + a*e)*A*a^2*b^6*c^3*d^3*g^2*e^4/(d*x + c) - 20*I*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^2*e^4/(d*x + c) + 80*I
*(b*x*e + a*e)*A*a^3*b^5*c^2*d^4*g^2*e^4/(d*x + c) + 20*I*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^2*e^4/(d*x + c) -
40*I*(b*x*e + a*e)*A*a^4*b^4*c*d^5*g^2*e^4/(d*x + c) - 10*I*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^2*e^4/(d*x + c) +
8*I*(b*x*e + a*e)*A*a^5*b^3*d^6*g^2*e^4/(d*x + c) + 2*I*(b*x*e + a*e)*B*a^5*b^3*d^6*g^2*e^4/(d*x + c) + 12*I*(
b*x*e + a*e)^2*A*b^7*c^5*d^2*g^2*e^3/(d*x + c)^2 - I*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g^2*e^3/(d*x + c)^2 - 60*I*
(b*x*e + a*e)^2*A*a*b^6*c^4*d^3*g^2*e^3/(d*x + c)^2 + 5*I*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^2*e^3/(d*x + c)^2
+ 120*I*(b*x*e + a*e)^2*A*a^2*b^5*c^3*d^4*g^2*e...

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Mupad [B]
time = 5.11, size = 638, normalized size = 3.54 \begin {gather*} x^3\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )-x^2\,\left (\frac {\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{24\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{6\,d}+\frac {A\,a\,b\,c\,g^2\,i}{2}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^2\,c\,g^2\,i\,x+\frac {B\,a\,g^2\,i\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {B\,b\,g^2\,i\,x^3\,\left (2\,a\,d+b\,c\right )}{3}+\frac {B\,b^2\,d\,g^2\,i\,x^4}{4}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{12\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{3\,d}+A\,a\,b\,c\,g^2\,i\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {a\,g^2\,i\,\left (2\,A\,a^2\,d^2+6\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+12\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{2\,b\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,i\,a^2\,c^2\,d^2\,g^2-4\,B\,i\,a\,b\,c^3\,d\,g^2+B\,i\,b^2\,c^4\,g^2\right )}{12\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,d\,g^2\,i-4\,B\,a^3\,b\,c\,g^2\,i\right )}{12\,b^2}+\frac {A\,b^2\,d\,g^2\,i\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))),x)

[Out]

x^3*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/12 - (A*b*g^2*i*(12*a*d + 12*b*c))/36) - x^2*((((b*g^2*i*(
12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(24*b*d) - (g^2*
i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d - B*a*b*c*d))/(6*d) + (A*a*b*c*g^2*i)/2)
 + log((e*(a + b*x))/(c + d*x))*(B*a^2*c*g^2*i*x + (B*a*g^2*i*x^2*(a*d + 2*b*c))/2 + (B*b*g^2*i*x^3*(2*a*d + b
*c))/3 + (B*b^2*d*g^2*i*x^4)/4) + x*(((12*a*d + 12*b*c)*((((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 -
(A*b*g^2*i*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(12*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^
2 - B*b^2*c^2 + 18*A*a*b*c*d - B*a*b*c*d))/(3*d) + A*a*b*c*g^2*i))/(12*b*d) - (a*c*((b*g^2*i*(12*A*a*d + 8*A*b
*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12))/(b*d) + (a*g^2*i*(2*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^
2*d^2 - 2*B*b^2*c^2 + 12*A*a*b*c*d + B*a*b*c*d))/(2*b*d)) - (log(c + d*x)*(B*b^2*c^4*g^2*i + 6*B*a^2*c^2*d^2*g
^2*i - 4*B*a*b*c^3*d*g^2*i))/(12*d^3) - (log(a + b*x)*(B*a^4*d*g^2*i - 4*B*a^3*b*c*g^2*i))/(12*b^2) + (A*b^2*d
*g^2*i*x^4)/4

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