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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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])

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]

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*}

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Mathematica [A]  time = 0.06, 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*(-1/6*(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]))/b^4 + (c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(4*d)

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fricas [B]  time = 0.80, size = 322, normalized size = 2.16 \[ \frac {6 \, A b^{4} d^{4} i^{3} x^{4} - 6 \, B b^{4} c^{4} i^{3} \log \left (d x + c\right ) + 2 \, {\left ({\left (12 \, A - B\right )} b^{4} c d^{3} + B a b^{3} d^{4}\right )} i^{3} x^{3} + 3 \, {\left (3 \, {\left (4 \, A - B\right )} b^{4} c^{2} d^{2} + 4 \, B a b^{3} c d^{3} - B a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 6 \, {\left ({\left (4 \, A - 3 \, B\right )} b^{4} c^{3} d + 6 \, B a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} i^{3} x + 6 \, {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} i^{3} \log \left (b x + a\right ) + 6 \, {\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{24 \, b^{4} d} \]

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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giac [B]  time = 1.09, size = 3969, normalized size = 26.64 \[ \text {result too large to display} \]

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]

-1/24*(6*B*b^9*c^5*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 30*B*a*b^8*c^4*d*i*e^5*log(-b*e + (b*x*e + a*
e)*d/(d*x + c)) + 60*B*a^2*b^7*c^3*d^2*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 60*B*a^3*b^6*c^2*d^3*i*e^
5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 30*B*a^4*b^5*c*d^4*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 6*B
*a^5*b^4*d^5*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 24*(b*x*e + a*e)*B*b^8*c^5*d*i*e^4*log(-b*e + (b*x*
e + a*e)*d/(d*x + c))/(d*x + c) + 120*(b*x*e + a*e)*B*a*b^7*c^4*d^2*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c) - 240*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 240*
(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 120*(b*x*e + a*e)*B*a^
4*b^4*c*d^5*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a^5*b^3*d^6*i*e^4*log(-
b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 36*(b*x*e + a*e)^2*B*b^7*c^5*d^2*i*e^3*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^2 - 180*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^2 + 360*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 360*
(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 180*(b*x*e + a*e)^
2*B*a^4*b^3*c*d^6*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 36*(b*x*e + a*e)^2*B*a^5*b^2*d^7*i
*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e + a*e)^3*B*b^6*c^5*d^3*i*e^2*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 120*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*
x + c))/(d*x + c)^3 - 240*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^3 + 240*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 120*(b
*x*e + a*e)^3*B*a^4*b^2*c*d^7*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 24*(b*x*e + a*e)^3*B*a
^5*b*d^8*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 6*(b*x*e + a*e)^4*B*b^5*c^5*d^4*i*e*log(-b*
e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 30*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*i*e*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^4 - 60*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 30*(b*x*
e + a*e)^4*B*a^4*b*c*d^8*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 6*(b*x*e + a*e)^4*B*a^5*d^9*i
*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 24*(b*x*e + a*e)*B*b^8*c^5*d*i*e^4*log((b*x*e + a*e)/(d
*x + c))/(d*x + c) - 120*(b*x*e + a*e)*B*a*b^7*c^4*d^2*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 240*(b*x
*e + a*e)*B*a^2*b^6*c^3*d^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 240*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4
*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 120*(b*x*e + a*e)*B*a^4*b^4*c*d^5*i*e^4*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c) - 24*(b*x*e + a*e)*B*a^5*b^3*d^6*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 36*(b*x*e + a
*e)^2*B*b^7*c^5*d^2*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 180*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*i*e^3
*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 360*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*i*e^3*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^2 + 360*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 18
0*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 36*(b*x*e + a*e)^2*B*a^5*b^
2*d^7*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 24*(b*x*e + a*e)^3*B*b^6*c^5*d^3*i*e^2*log((b*x*e + a*e
)/(d*x + c))/(d*x + c)^3 - 120*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3
+ 240*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 240*(b*x*e + a*e)^3*B
*a^3*b^3*c^2*d^6*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 120*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*i*e^2*lo
g((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 24*(b*x*e + a*e)^3*B*a^5*b*d^8*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*
x + c)^3 - 6*(b*x*e + a*e)^4*B*b^5*c^5*d^4*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 30*(b*x*e + a*e)^4*B
*a*b^4*c^4*d^5*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*i*e*log((b*
x*e + a*e)/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x
+ c)^4 - 30*(b*x*e + a*e)^4*B*a^4*b*c*d^8*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 6*(b*x*e + a*e)^4*B*
a^5*d^9*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 6*A*b^9*c^5*i*e^5 - 11*B*b^9*c^5*i*e^5 - 30*A*a*b^8*c^4
*d*i*e^5 + 55*B*a*b^8*c^4*d*i*e^5 + 60*A*a^2*b^7*c^3*d^2*i*e^5 - 110*B*a^2*b^7*c^3*d^2*i*e^5 - 60*A*a^3*b^6*c^
2*d^3*i*e^5 + 110*B*a^3*b^6*c^2*d^3*i*e^5 + 30*A*a^4*b^5*c*d^4*i*e^5 - 55*B*a^4*b^5*c*d^4*i*e^5 - 6*A*a^5*b^4*
d^5*i*e^5 + 11*B*a^5*b^4*d^5*i*e^5 + 26*(b*x*e + a*e)*B*b^8*c^5*d*i*e^4/(d*x + c) - 130*(b*x*e + a*e)*B*a*b^7*
c^4*d^2*i*e^4/(d*x + c) + 260*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*i*e^4/(d*x + c) - 260*(b*x*e + a*e)*B*a^3*b^5*c^
2*d^4*i*e^4/(d*x + c) + 130*(b*x*e + a*e)*B*a^4*b^4*c*d^5*i*e^4/(d*x + c) - 26*(b*x*e + a*e)*B*a^5*b^3*d^6*i*e
^4/(d*x + c) - 21*(b*x*e + a*e)^2*B*b^7*c^5*d^2*i*e^3/(d*x + c)^2 + 105*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*i*e^3/
(d*x + c)^2 - 210*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*i*e^3/(d*x + c)^2 + 210*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*
i*e^3/(d*x + c)^2 - 105*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*i*e^3/(d*x + c)^2 + 21*(b*x*e + a*e)^2*B*a^5*b^2*d^7*i
*e^3/(d*x + c)^2 + 6*(b*x*e + a*e)^3*B*b^6*c^5*d^3*i*e^2/(d*x + c)^3 - 30*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*i*e^
2/(d*x + c)^3 + 60*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*i*e^2/(d*x + c)^3 - 60*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*
i*e^2/(d*x + c)^3 + 30*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*i*e^2/(d*x + c)^3 - 6*(b*x*e + a*e)^3*B*a^5*b*d^8*i*e^2
/(d*x + c)^3)*(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^8*d*e^4 - 4*(b*x*e +
a*e)*b^7*d^2*e^3/(d*x + c) + 6*(b*x*e + a*e)^2*b^6*d^3*e^2/(d*x + c)^2 - 4*(b*x*e + a*e)^3*b^5*d^4*e/(d*x + c)
^3 + (b*x*e + a*e)^4*b^4*d^5/(d*x + c)^4)

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maple [B]  time = 0.14, size = 2172, normalized size = 14.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.15, size = 439, normalized size = 2.95 \[ \frac {1}{4} \, A d^{3} i^{3} x^{4} + A c d^{2} i^{3} x^{3} + \frac {3}{2} \, A c^{2} d i^{3} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B c^{3} i^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B c^{2} d i^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B c d^{2} i^{3} + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B d^{3} i^{3} + A c^{3} i^{3} x \]

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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mupad [B]  time = 4.80, size = 566, normalized size = 3.80 \[ x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {c\,d\,i^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{b}+\frac {A\,a\,c\,d^2\,i^3}{b}\right )}{4\,b\,d}+\frac {c^2\,i^3\,\left (12\,A\,a\,d+8\,A\,b\,c+3\,B\,a\,d-3\,B\,b\,c\right )}{2\,b}-\frac {a\,c\,\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {c\,d\,i^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{2\,b}+\frac {A\,a\,c\,d^2\,i^3}{2\,b}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,c^3\,i^3\,x+\frac {3\,B\,c^2\,d\,i^3\,x^2}{2}+B\,c\,d^2\,i^3\,x^3+\frac {B\,d^3\,i^3\,x^4}{4}\right )+x^3\,\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,d^3\,i^3-4\,B\,a^3\,b\,c\,d^2\,i^3+6\,B\,a^2\,b^2\,c^2\,d\,i^3-4\,B\,a\,b^3\,c^3\,i^3\right )}{4\,b^4}+\frac {A\,d^3\,i^3\,x^4}{4}-\frac {B\,c^4\,i^3\,\ln \left (c+d\,x\right )}{4\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(((4*a*d + 4*b*c)*((((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d - B*b*c))/(4*b) - (A*d^2*i^3*(4*a*d + 4*b*c))/(4*b
))*(4*a*d + 4*b*c))/(4*b*d) - (c*d*i^3*(4*A*a*d + 6*A*b*c + B*a*d - B*b*c))/b + (A*a*c*d^2*i^3)/b))/(4*b*d) +
(c^2*i^3*(12*A*a*d + 8*A*b*c + 3*B*a*d - 3*B*b*c))/(2*b) - (a*c*((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d - B*b*c)
)/(4*b) - (A*d^2*i^3*(4*a*d + 4*b*c))/(4*b)))/(b*d)) - x^2*((((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d - B*b*c))/(
4*b) - (A*d^2*i^3*(4*a*d + 4*b*c))/(4*b))*(4*a*d + 4*b*c))/(8*b*d) - (c*d*i^3*(4*A*a*d + 6*A*b*c + B*a*d - B*b
*c))/(2*b) + (A*a*c*d^2*i^3)/(2*b)) + log((e*(a + b*x))/(c + d*x))*((B*d^3*i^3*x^4)/4 + B*c^3*i^3*x + (3*B*c^2
*d*i^3*x^2)/2 + B*c*d^2*i^3*x^3) + x^3*((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d - B*b*c))/(12*b) - (A*d^2*i^3*(4*
a*d + 4*b*c))/(12*b)) - (log(a + b*x)*(B*a^4*d^3*i^3 - 4*B*a*b^3*c^3*i^3 + 6*B*a^2*b^2*c^2*d*i^3 - 4*B*a^3*b*c
*d^2*i^3))/(4*b^4) + (A*d^3*i^3*x^4)/4 - (B*c^4*i^3*log(c + d*x))/(4*d)

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sympy [B]  time = 4.12, size = 706, normalized size = 4.74 \[ \frac {A d^{3} i^{3} x^{4}}{4} - \frac {B a i^{3} \left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} i^{3} - 4 B a^{3} b c^{2} d^{2} i^{3} + 6 B a^{2} b^{2} c^{3} d i^{3} + \frac {B a^{2} d i^{3} \left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{b} - 5 B a b^{3} c^{4} i^{3} - B a c i^{3} \left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{B a^{4} d^{4} i^{3} - 4 B a^{3} b c d^{3} i^{3} + 6 B a^{2} b^{2} c^{2} d^{2} i^{3} - 4 B a b^{3} c^{3} d i^{3} - B b^{4} c^{4} i^{3}} \right )}}{4 b^{4}} - \frac {B c^{4} i^{3} \log {\left (x + \frac {B a^{4} c d^{3} i^{3} - 4 B a^{3} b c^{2} d^{2} i^{3} + 6 B a^{2} b^{2} c^{3} d i^{3} - 4 B a b^{3} c^{4} i^{3} - \frac {B b^{4} c^{5} i^{3}}{d}}{B a^{4} d^{4} i^{3} - 4 B a^{3} b c d^{3} i^{3} + 6 B a^{2} b^{2} c^{2} d^{2} i^{3} - 4 B a b^{3} c^{3} d i^{3} - B b^{4} c^{4} i^{3}} \right )}}{4 d} + x^{3} \left (A c d^{2} i^{3} + \frac {B a d^{3} i^{3}}{12 b} - \frac {B c d^{2} i^{3}}{12}\right ) + x^{2} \left (\frac {3 A c^{2} d i^{3}}{2} - \frac {B a^{2} d^{3} i^{3}}{8 b^{2}} + \frac {B a c d^{2} i^{3}}{2 b} - \frac {3 B c^{2} d i^{3}}{8}\right ) + x \left (A c^{3} i^{3} + \frac {B a^{3} d^{3} i^{3}}{4 b^{3}} - \frac {B a^{2} c d^{2} i^{3}}{b^{2}} + \frac {3 B a c^{2} d i^{3}}{2 b} - \frac {3 B c^{3} i^{3}}{4}\right ) + \left (B c^{3} i^{3} x + \frac {3 B c^{2} d i^{3} x^{2}}{2} + B c d^{2} i^{3} x^{3} + \frac {B d^{3} i^{3} x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

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

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