3.1.7 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^3} \, dx\) [7]

Optimal. Leaf size=85 \[ -\frac {B i (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) g^3 (a+b x)^2} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2562, 2341} \begin {gather*} -\frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^3,x]

[Out]

-1/4*(B*i*(c + d*x)^2)/((b*c - a*d)*g^3*(a + b*x)^2) - (i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2
*(b*c - a*d)*g^3*(a + b*x)^2)

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2562

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

Rubi steps

\begin {align*} \int \frac {(7 c+7 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^3 (a+b x)^3}+\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(7 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(7 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {(7 B d) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {(7 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {(7 B d (b c-a d)) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (7 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {(7 B d (b c-a d)) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (7 B (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^2 g^3}\\ &=-\frac {7 B (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {7 B d}{2 b^2 g^3 (a+b x)}-\frac {7 B d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {7 B d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(85)=170\).
time = 0.11, size = 208, normalized size = 2.45 \begin {gather*} \frac {i \left (-\frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 (a+b x)^2}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a+b x)}-\frac {B d \left (\frac {1}{a+b x}+\frac {d \log (a+b x)}{b c-a d}-\frac {d \log (c+d x)}{b c-a d}\right )}{b^2}-\frac {B \left (\frac {b c-a d}{(a+b x)^2}-\frac {2 d}{a+b x}-\frac {2 d^2 \log (a+b x)}{b c-a d}+\frac {2 d^2 \log (c+d x)}{b c-a d}\right )}{4 b^2}\right )}{g^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^3,x]

[Out]

(i*(-1/2*((b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*(a + b*x)^2) - (d*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(b^2*(a + b*x)) - (B*d*((a + b*x)^(-1) + (d*Log[a + b*x])/(b*c - a*d) - (d*Log[c + d*x])/(b*c - a*d
)))/b^2 - (B*((b*c - a*d)/(a + b*x)^2 - (2*d)/(a + b*x) - (2*d^2*Log[a + b*x])/(b*c - a*d) + (2*d^2*Log[c + d*
x])/(b*c - a*d)))/(4*b^2)))/g^3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(81)=162\).
time = 0.46, size = 177, normalized size = 2.08

method	result	size
norman	\(\frac {\frac {B c d i x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a d -c b \right )}-\frac {2 A a d i +2 A b c i +B a d i +B b c i}{4 g \,b^{2}}-\frac {\left (2 A d i +B d i \right ) x}{2 g b}+\frac {B i \,c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a d -c b \right )}+\frac {B \,d^{2} i \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a d -c b \right ) g}}{g^{2} \left (b x +a \right )^{2}}\)	\(170\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i \,d^{2} e A}{2 \left (a d -c b \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}\right )}{d^{2}}\)	\(177\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i \,d^{2} e A}{2 \left (a d -c b \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}\right )}{d^{2}}\)	\(177\)
risch	\(-\frac {B i \left (2 b d x +a d +c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (b x +a \right )^{2} b^{2} g^{3}}-\frac {i \left (2 B \ln \left (d x +c \right ) b^{2} d^{2} x^{2}-2 B \ln \left (-b x -a \right ) b^{2} d^{2} x^{2}+4 B \ln \left (d x +c \right ) a b \,d^{2} x -4 B \ln \left (-b x -a \right ) a b \,d^{2} x +4 A a b \,d^{2} x -4 A \,b^{2} c d x +2 B \ln \left (d x +c \right ) a^{2} d^{2}-2 B \,a^{2} \ln \left (-b x -a \right ) d^{2}+2 B a b \,d^{2} x -2 B \,b^{2} c d x +2 A \,a^{2} d^{2}-2 A \,b^{2} c^{2}+B \,a^{2} d^{2}-B \,b^{2} c^{2}\right )}{4 \left (b x +a \right )^{2} b^{2} g^{3} \left (a d -c b \right )}\)	\(249\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x,method=_RETURNVERBOSE)

[Out]

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

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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. 570 vs. \(2 (80) = 160\).
time = 0.28, size = 570, normalized size = 6.71 \begin {gather*} -\frac {1}{4} i \, B d {\left (\frac {2 \, {\left (2 \, b x + a\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} + \frac {3 \, a b c - a^{2} d + 2 \, {\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac {1}{4} i \, B c {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {i \, {\left (2 \, b x + a\right )} A d}{2 \, {\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac {i \, A c}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algorithm="maxima")

[Out]

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

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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. 177 vs. \(2 (80) = 160\).
time = 0.39, size = 177, normalized size = 2.08 \begin {gather*} -\frac {{\left (2 i \, A + i \, B\right )} b^{2} c^{2} + {\left (-2 i \, A - i \, B\right )} a^{2} d^{2} - 2 \, {\left ({\left (-2 i \, A - i \, B\right )} b^{2} c d + {\left (2 i \, A + i \, B\right )} a b d^{2}\right )} x - 2 \, {\left (-i \, B b^{2} d^{2} x^{2} - 2 i \, B b^{2} c d x - i \, B b^{2} c^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (73) = 146\).
time = 2.83, size = 384, normalized size = 4.52 \begin {gather*} - \frac {B d^{2} i \log {\left (x + \frac {- \frac {B a^{2} d^{4} i}{a d - b c} + \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i - \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \log {\left (x + \frac {\frac {B a^{2} d^{4} i}{a d - b c} - \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i + \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {- 2 A a d i - 2 A b c i - B a d i - B b c i + x \left (- 4 A b d i - 2 B b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- B a d i - B b c i - 2 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**3,x)

[Out]

-B*d**2*i*log(x + (-B*a**2*d**4*i/(a*d - b*c) + 2*B*a*b*c*d**3*i/(a*d - b*c) + B*a*d**3*i - B*b**2*c**2*d**2*i
/(a*d - b*c) + B*b*c*d**2*i)/(2*B*b*d**3*i))/(2*b**2*g**3*(a*d - b*c)) + B*d**2*i*log(x + (B*a**2*d**4*i/(a*d
- b*c) - 2*B*a*b*c*d**3*i/(a*d - b*c) + B*a*d**3*i + B*b**2*c**2*d**2*i/(a*d - b*c) + B*b*c*d**2*i)/(2*B*b*d**
3*i))/(2*b**2*g**3*(a*d - b*c)) + (-2*A*a*d*i - 2*A*b*c*i - B*a*d*i - B*b*c*i + x*(-4*A*b*d*i - 2*B*b*d*i))/(4
*a**2*b**2*g**3 + 8*a*b**3*g**3*x + 4*b**4*g**3*x**2) + (-B*a*d*i - B*b*c*i - 2*B*b*d*i*x)*log(e*(a + b*x)/(c
+ d*x))/(2*a**2*b**2*g**3 + 4*a*b**3*g**3*x + 2*b**4*g**3*x**2)

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Giac [A]
time = 2.85, size = 115, normalized size = 1.35 \begin {gather*} \frac {{\left (-2 i \, B e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) - 2 i \, A e^{3} - i \, B e^{3}\right )} {\left (d x + c\right )}^{2} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{4 \, {\left (b x e + a e\right )}^{2} g^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.58, size = 197, normalized size = 2.32 \begin {gather*} -\frac {x\,\left (2\,A\,b\,d\,i+B\,b\,d\,i\right )+A\,a\,d\,i+A\,b\,c\,i+\frac {B\,a\,d\,i}{2}+\frac {B\,b\,c\,i}{2}}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{2\,b^2\,g^3}+\frac {B\,a\,d\,i}{2\,b^3\,g^3}+\frac {B\,d\,i\,x}{b^2\,g^3}\right )}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^3,x)

[Out]

- (x*(2*A*b*d*i + B*b*d*i) + A*a*d*i + A*b*c*i + (B*a*d*i)/2 + (B*b*c*i)/2)/(2*a^2*b^2*g^3 + 2*b^4*g^3*x^2 + 4
*a*b^3*g^3*x) - (log((e*(a + b*x))/(c + d*x))*((B*c*i)/(2*b^2*g^3) + (B*a*d*i)/(2*b^3*g^3) + (B*d*i*x)/(b^2*g^
3)))/(2*a*x + b*x^2 + a^2/b) - (B*d^2*i*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*1i)/(b^2*g^3*(a*d - b*c))

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