3.23 \(\int (c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=149 \[ \frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d}-\frac{B i^3 x (b c-a d)^3}{4 b^3}-\frac{B i^3 (c+d x)^2 (b c-a d)^2}{8 b^2 d}-\frac{B i^3 (b c-a d)^4 \log (a+b x)}{4 b^4 d}-\frac{B i^3 (c+d x)^3 (b c-a d)}{12 b d} \]

[Out]

-(B*(b*c - a*d)^3*i^3*x)/(4*b^3) - (B*(b*c - a*d)^2*i^3*(c + d*x)^2)/(8*b^2*d) - (B*(b*c - a*d)*i^3*(c + d*x)^
3)/(12*b*d) - (B*(b*c - a*d)^4*i^3*Log[a + b*x])/(4*b^4*d) + (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*
x)]))/(4*d)

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Rubi [A]  time = 0.0805712, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 43} \[ \frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d}-\frac{B i^3 x (b c-a d)^3}{4 b^3}-\frac{B i^3 (c+d x)^2 (b c-a d)^2}{8 b^2 d}-\frac{B i^3 (b c-a d)^4 \log (a+b x)}{4 b^4 d}-\frac{B i^3 (c+d x)^3 (b c-a d)}{12 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)^3*i^3*x)/(4*b^3) - (B*(b*c - a*d)^2*i^3*(c + d*x)^2)/(8*b^2*d) - (B*(b*c - a*d)*i^3*(c + d*x)^
3)/(12*b*d) - (B*(b*c - a*d)^4*i^3*Log[a + b*x])/(4*b^4*d) + (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*
x)]))/(4*d)

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

Rubi steps

\begin{align*} \int (23 c+23 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac{12167 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d}-\frac{B \int \frac{279841 (b c-a d) (c+d x)^3}{a+b x} \, dx}{92 d}\\ &=\frac{12167 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d}-\frac{(12167 B (b c-a d)) \int \frac{(c+d x)^3}{a+b x} \, dx}{4 d}\\ &=\frac{12167 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d}-\frac{(12167 B (b c-a d)) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{4 d}\\ &=-\frac{12167 B (b c-a d)^3 x}{4 b^3}-\frac{12167 B (b c-a d)^2 (c+d x)^2}{8 b^2 d}-\frac{12167 B (b c-a d) (c+d x)^3}{12 b d}-\frac{12167 B (b c-a d)^4 \log (a+b x)}{4 b^4 d}+\frac{12167 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0586556, size = 120, normalized size = 0.81 \[ \frac{i^3 \left ((c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{B (b c-a d) \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{6 b^4}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.171, size = 2172, normalized size = 14.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^2/d*B*i^3*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^4+1/2*e^2*d^2*B*i^3/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a
^3*c-e*d^2*B*i^3/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3*c+3/2*e*d*B*i^3/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c^2
+1/4*e^4/d*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^4*b^4-e^4*B*i^3*ln(b*e/d+(a
*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^3*b^3*a+1/2*e^2*B*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a
*c^3*b-1/3*e^3*d^2*B*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3*c-e^4*A*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*b^3*c
^3*a-1/3*e^3*B*i^3*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3*a-e^4*d^2*A*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3
*b*c+3/2*e^4*d*A*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2*b^2*c^2+1/2*e^3*d*B*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)
^3*a^2*c^2*b+1/4/d*B*i^3*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^4+1/4*e^4*d^3*A*i^3/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)^4*a^4+1/4*d^3*B*i^3/b^4*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^4-B*i^3/b*ln(d*(b*e/d+(a*d-b*c)*e/d/(
d*x+c))-b*e)*a*c^3-e*B*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^3*a-1/8*e^2*d^3*B*i^3/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*
b*c)^2*a^4+1/12*e^3*d^3*B*i^3/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^4+3/2*d*B*i^3/b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/
(d*x+c))-b*e)*a^2*c^2-d^2*B*i^3/b^3*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^3*c+1/4*e^4*d^3*B*i^3*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^4-e^4*d^2*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(
d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*b*c+2*e^4*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c
)^4*a*c^7/(d*x+c)^4-35/2*e^4*d^3*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^4*c^4
/(d*x+c)^4+1/4*e/d*B*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^4*b+3/2*e^4*d*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b
^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2*c^2-1/4*e^4/d*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^4/(d*e/(d*x+c)*a-
e/(d*x+c)*b*c)^4*c^8/(d*x+c)^4-1/4*e^4*d^7*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b^4/(d*e/(d*x+c)*a-e/(d*x+c)*
b*c)^4*a^8/(d*x+c)^4+1/4*e^4/d*A*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*b^4*c^4-3/4*e^2*d*B*i^3/(d*e/(d*x+c)*a-e/
(d*x+c)*b*c)^2*a^2*c^2+1/12*e^3/d*B*i^3*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^4+1/4*e*d^3*B*i^3/b^3/(d*e/(d*x+
c)*a-e/(d*x+c)*b*c)*a^4+2*e^4*d^6*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^
7*c/(d*x+c)^4+14*e^4*d^4*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^5*c^3/(d*x+
c)^4+14*e^4*d^2*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*c^5/(d*x+c)^4*b-7*e^
4*d^5*B*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^6*c^2/(d*x+c)^4-7*e^4*d*B*i^
3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2*c^6/(d*x+c)^4

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Maxima [B]  time = 1.18331, size = 593, normalized size = 3.98 \begin{align*} \frac{1}{4} \, A d^{3} i^{3} x^{4} + A c d^{2} i^{3} x^{3} + \frac{3}{2} \, A c^{2} d i^{3} x^{2} +{\left (x \log \left (\frac{b e x}{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 c^{3} i^{3} + \frac{3}{2} \,{\left (x^{2} \log \left (\frac{b e x}{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 c^{2} d i^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \log \left (\frac{b e x}{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 c d^{2} i^{3} + \frac{1}{24} \,{\left (6 \, x^{4} \log \left (\frac{b e x}{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 d^{3} i^{3} + A c^{3} i^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 0.838573, size = 664, normalized size = 4.46 \begin{align*} \frac{6 \, A b^{4} d^{4} i^{3} x^{4} - 6 \, B b^{4} c^{4} i^{3} \log \left (d x + c\right ) + 2 \,{\left ({\left (12 \, A - B\right )} b^{4} c d^{3} + B a b^{3} d^{4}\right )} i^{3} x^{3} + 3 \,{\left (3 \,{\left (4 \, A - B\right )} b^{4} c^{2} d^{2} + 4 \, B a b^{3} c d^{3} - B a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 6 \,{\left ({\left (4 \, A - 3 \, B\right )} b^{4} c^{3} d + 6 \, B a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} i^{3} x + 6 \,{\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} i^{3} \log \left (b x + a\right ) + 6 \,{\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{24 \, b^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 5.2841, size = 719, normalized size = 4.83 \begin{align*} \frac{A d^{3} i^{3} x^{4}}{4} - \frac{B a i^{3} \left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right ) \log{\left (x + \frac{B a^{4} c d^{3} i^{3} - 4 B a^{3} b c^{2} d^{2} i^{3} + 6 B a^{2} b^{2} c^{3} d i^{3} + \frac{B a^{2} d i^{3} \left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{b} - 5 B a b^{3} c^{4} i^{3} - B a c i^{3} \left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{B a^{4} d^{4} i^{3} - 4 B a^{3} b c d^{3} i^{3} + 6 B a^{2} b^{2} c^{2} d^{2} i^{3} - 4 B a b^{3} c^{3} d i^{3} - B b^{4} c^{4} i^{3}} \right )}}{4 b^{4}} - \frac{B c^{4} i^{3} \log{\left (x + \frac{B a^{4} c d^{3} i^{3} - 4 B a^{3} b c^{2} d^{2} i^{3} + 6 B a^{2} b^{2} c^{3} d i^{3} - 4 B a b^{3} c^{4} i^{3} - \frac{B b^{4} c^{5} i^{3}}{d}}{B a^{4} d^{4} i^{3} - 4 B a^{3} b c d^{3} i^{3} + 6 B a^{2} b^{2} c^{2} d^{2} i^{3} - 4 B a b^{3} c^{3} d i^{3} - B b^{4} c^{4} i^{3}} \right )}}{4 d} + \left (B c^{3} i^{3} x + \frac{3 B c^{2} d i^{3} x^{2}}{2} + B c d^{2} i^{3} x^{3} + \frac{B d^{3} i^{3} x^{4}}{4}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )} + \frac{x^{3} \left (12 A b c d^{2} i^{3} + B a d^{3} i^{3} - B b c d^{2} i^{3}\right )}{12 b} - \frac{x^{2} \left (- 12 A b^{2} c^{2} d i^{3} + B a^{2} d^{3} i^{3} - 4 B a b c d^{2} i^{3} + 3 B b^{2} c^{2} d i^{3}\right )}{8 b^{2}} + \frac{x \left (4 A b^{3} c^{3} i^{3} + B a^{3} d^{3} i^{3} - 4 B a^{2} b c d^{2} i^{3} + 6 B a b^{2} c^{2} d i^{3} - 3 B b^{3} c^{3} i^{3}\right )}{4 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*d**3*i**3*x**4/4 - B*a*i**3*(a*d - 2*b*c)*(a**2*d**2 - 2*a*b*c*d + 2*b**2*c**2)*log(x + (B*a**4*c*d**3*i**3
- 4*B*a**3*b*c**2*d**2*i**3 + 6*B*a**2*b**2*c**3*d*i**3 + B*a**2*d*i**3*(a*d - 2*b*c)*(a**2*d**2 - 2*a*b*c*d +
2*b**2*c**2)/b - 5*B*a*b**3*c**4*i**3 - B*a*c*i**3*(a*d - 2*b*c)*(a**2*d**2 - 2*a*b*c*d + 2*b**2*c**2))/(B*a*
*4*d**4*i**3 - 4*B*a**3*b*c*d**3*i**3 + 6*B*a**2*b**2*c**2*d**2*i**3 - 4*B*a*b**3*c**3*d*i**3 - B*b**4*c**4*i*
*3))/(4*b**4) - B*c**4*i**3*log(x + (B*a**4*c*d**3*i**3 - 4*B*a**3*b*c**2*d**2*i**3 + 6*B*a**2*b**2*c**3*d*i**
3 - 4*B*a*b**3*c**4*i**3 - B*b**4*c**5*i**3/d)/(B*a**4*d**4*i**3 - 4*B*a**3*b*c*d**3*i**3 + 6*B*a**2*b**2*c**2
*d**2*i**3 - 4*B*a*b**3*c**3*d*i**3 - B*b**4*c**4*i**3))/(4*d) + (B*c**3*i**3*x + 3*B*c**2*d*i**3*x**2/2 + B*c
*d**2*i**3*x**3 + B*d**3*i**3*x**4/4)*log(e*(a + b*x)/(c + d*x)) + x**3*(12*A*b*c*d**2*i**3 + B*a*d**3*i**3 -
B*b*c*d**2*i**3)/(12*b) - x**2*(-12*A*b**2*c**2*d*i**3 + B*a**2*d**3*i**3 - 4*B*a*b*c*d**2*i**3 + 3*B*b**2*c**
2*d*i**3)/(8*b**2) + x*(4*A*b**3*c**3*i**3 + B*a**3*d**3*i**3 - 4*B*a**2*b*c*d**2*i**3 + 6*B*a*b**2*c**2*d*i**
3 - 3*B*b**3*c**3*i**3)/(4*b**3)

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Giac [B]  time = 1.49911, size = 606, normalized size = 4.07 \begin{align*} -\frac{1}{4} \,{\left (A d^{3} i + B d^{3} i\right )} x^{4} - \frac{{\left (12 \, A b c d^{2} i + 11 \, B b c d^{2} i + B a d^{3} i\right )} x^{3}}{12 \, b} - \frac{1}{4} \,{\left (B d^{3} i x^{4} + 4 \, B c d^{2} i x^{3} + 6 \, B c^{2} d i x^{2} + 4 \, B c^{3} i x\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (12 \, A b^{2} c^{2} d i + 9 \, B b^{2} c^{2} d i + 4 \, B a b c d^{2} i - B a^{2} d^{3} i\right )} x^{2}}{8 \, b^{2}} - \frac{{\left (4 \, A b^{3} c^{3} i + B b^{3} c^{3} i + 6 \, B a b^{2} c^{2} d i - 4 \, B a^{2} b c d^{2} i + B a^{3} d^{3} i\right )} x}{4 \, b^{3}} + \frac{{\left (B b^{4} c^{4} i - 4 \, B a b^{3} c^{3} d i + 6 \, B a^{2} b^{2} c^{2} d^{2} i - 4 \, B a^{3} b c d^{3} i + B a^{4} d^{4} i\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{8 \, b^{4} d} - \frac{{\left (B b^{5} c^{5} i + 3 \, B a b^{4} c^{4} d i - 10 \, B a^{2} b^{3} c^{3} d^{2} i + 10 \, B a^{3} b^{2} c^{2} d^{3} i - 5 \, B a^{4} b c d^{4} i + B a^{5} d^{5} i\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | -b c + a d \right |}}{2 \, b d x + b c + a d +{\left | -b c + a d \right |}} \right |}\right )}{8 \, b^{4} d{\left | -b c + a d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(A*d^3*i + B*d^3*i)*x^4 - 1/12*(12*A*b*c*d^2*i + 11*B*b*c*d^2*i + B*a*d^3*i)*x^3/b - 1/4*(B*d^3*i*x^4 + 4
*B*c*d^2*i*x^3 + 6*B*c^2*d*i*x^2 + 4*B*c^3*i*x)*log((b*x + a)/(d*x + c)) - 1/8*(12*A*b^2*c^2*d*i + 9*B*b^2*c^2
*d*i + 4*B*a*b*c*d^2*i - B*a^2*d^3*i)*x^2/b^2 - 1/4*(4*A*b^3*c^3*i + B*b^3*c^3*i + 6*B*a*b^2*c^2*d*i - 4*B*a^2
*b*c*d^2*i + B*a^3*d^3*i)*x/b^3 + 1/8*(B*b^4*c^4*i - 4*B*a*b^3*c^3*d*i + 6*B*a^2*b^2*c^2*d^2*i - 4*B*a^3*b*c*d
^3*i + B*a^4*d^4*i)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^4*d) - 1/8*(B*b^5*c^5*i + 3*B*a*b^4*c^4*d*i - 1
0*B*a^2*b^3*c^3*d^2*i + 10*B*a^3*b^2*c^2*d^3*i - 5*B*a^4*b*c*d^4*i + B*a^5*d^5*i)*log(abs((2*b*d*x + b*c + a*d
- abs(-b*c + a*d))/(2*b*d*x + b*c + a*d + abs(-b*c + a*d))))/(b^4*d*abs(-b*c + a*d))