3.1.23 \(\int (c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [23]
Optimal. Leaf size=149 \[ -\frac {B (b c-a d)^3 i^3 x}{4 b^3}-\frac {B (b c-a d)^2 i^3 (c+d x)^2}{8 b^2 d}-\frac {B (b c-a d) i^3 (c+d x)^3}{12 b d}-\frac {B (b c-a d)^4 i^3 \log (a+b x)}{4 b^4 d}+\frac {i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d} \]
[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 = 0.07, 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 = {2548, 21, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
-1/4*(B*(b*c - a*d)^3*i^3*x)/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 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])
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 = 0.04, size = 120, normalized size = 0.81 \begin {gather*} \frac {i^3 \left (-\frac {B (b c-a d) \left (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)\right )}{6 b^4}+(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{4 d} \end {gather*}
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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2665\) vs.
\(2(139)=278\).
time = 0.54, size = 2666, normalized size = 17.89
| | |
method |
result |
size |
| | |
risch |
\(\frac {i^{3} \left (d x +c \right )^{4} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 d}+\frac {i^{3} d^{3} A \,x^{4}}{4}+i^{3} d^{2} A c \,x^{3}+\frac {i^{3} d^{3} B a \,x^{3}}{12 b}-\frac {i^{3} d^{2} B c \,x^{3}}{12}+\frac {3 i^{3} d A \,c^{2} x^{2}}{2}-\frac {i^{3} d^{3} B \,a^{2} x^{2}}{8 b^{2}}+\frac {i^{3} d^{2} B a c \,x^{2}}{2 b}-\frac {3 i^{3} d B \,c^{2} x^{2}}{8}+i^{3} A \,c^{3} x -\frac {i^{3} d^{3} B \ln \left (b x +a \right ) a^{4}}{4 b^{4}}+\frac {i^{3} d^{2} B \ln \left (b x +a \right ) a^{3} c}{b^{3}}-\frac {3 i^{3} d B \ln \left (b x +a \right ) a^{2} c^{2}}{2 b^{2}}+\frac {i^{3} B \ln \left (b x +a \right ) a \,c^{3}}{b}-\frac {i^{3} B \ln \left (b x +a \right ) c^{4}}{4 d}+\frac {i^{3} d^{3} B \,a^{3} x}{4 b^{3}}-\frac {i^{3} d^{2} B \,a^{2} c x}{b^{2}}+\frac {3 i^{3} d B a \,c^{2} x}{2 b}-\frac {3 i^{3} B \,c^{3} x}{4}\) |
\(314\) |
derivativedivides |
\(\text {Expression too large to display}\) |
\(2666\) |
default |
\(\text {Expression too large to display}\) |
\(2666\) |
| | |
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|
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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,method=_RETURNVERBOSE)
[Out]
-1/d^2*e*(a*d-b*c)*(-3/8*B*d^3*e*i^3/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^2*c-B*d^7*i^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))^3/b^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^3+3/4*B*d^3/e*i
^3/b^3*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^2*c-3/4*B*d^2/e*i^3/b^2*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*
d)*a*c^2+1/4*B*d^2*e^2*i^3*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a*c^2-3/4*B*d^7/e*i^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))^4/b^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^2*c-9/2*B*d^5*e*i^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))^2/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^2
*c+3/4*B*d^6/e*i^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))^4/b^2/(b*e-(b*e/d+(a*d-b*c)*e
/d/(d*x+c))*d)^4*a*c^2-3*B*d^3*e^2*i^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/(b*e-(b
*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a*c^2-1/4*B*d*i^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c^3-1/12*B*d*e^2*i^3*
b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*c^3+1/12*B*d^4*e^2*i^3/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a
^3-1/8*B*d*e*i^3*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c^3+3/4*B*d^2*i^3/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+
c))*d)*a*c^2+3/8*B*d^2*e*i^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a*c^2-1/4*B*d^3*e^2*i^3/(b*e-(b*e/d+(a*d-
b*c)*e/d/(d*x+c))*d)^3*a^2*c+1/4*B*d/e*i^3/b*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c^3+1/8*B*d^4*e*i^3/b^2/(
b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^3+3/2*B*d^6*e*i^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))^2/b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^3-1/4*B*d^5/e*i^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))^4/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*c^3+B*d^2*e^2*i^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^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*c^3-3/2*B*d^3*e*i^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))^2*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*c^3+1/4
*B*d^8/e*i^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))^4/b^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*
x+c))*d)^4*a^3-B*d^5*e^2*i^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/(b*e-(b*e/d+(a*d-
b*c)*e/d/(d*x+c))*d)^4*a^3-3*B*d^5*i^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))^3/b/(b*e-
(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a*c^2+3*B*d^4*e^2*i^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^2*c+3*B*d^6*i^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))^3/b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a^2*c+9/2*B*d^4*e*i^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))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*a*c^2-1/4*B*d^4/e*i^3/b^4*ln(
b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^3+B*d^4*i^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
))^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4*c^3-3/4*B*d^3*i^3/b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^2*c
-1/4*A*d*e^3*i^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^4+1/4*B*d
^4*i^3/b^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a^3)
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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. 426 vs. \(2 (125) = 250\).
time = 0.29, size = 426, normalized size = 2.86 \begin {gather*} -\frac {1}{4} i \, A d^{3} x^{4} - i \, A c d^{2} x^{3} - \frac {3}{2} i \, A c^{2} d 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 c^{3} - \frac {3}{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 c^{2} d - \frac {1}{2} 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 c d^{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 d^{3} - i \, A c^{3} x \end {gather*}
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*I*A*d^3*x^4 - I*A*c*d^2*x^3 - 3/2*I*A*c^2*d*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*c^3 - 3/2*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*c^2*d - 1/2*I*(2*x^3*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*l
og(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 - 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*d^3 - I*
A*c^3*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. 302 vs. \(2 (125) = 250\).
time = 0.43, size = 302, normalized size = 2.03 \begin {gather*} \frac {-6 i \, A b^{4} d^{4} x^{4} + 6 i \, B b^{4} c^{4} \log \left (\frac {d x + c}{d}\right ) - 2 \, {\left ({\left (12 i \, A - i \, B\right )} b^{4} c d^{3} + i \, B a b^{3} d^{4}\right )} x^{3} - 3 \, {\left (3 \, {\left (4 i \, A - i \, B\right )} b^{4} c^{2} d^{2} + 4 i \, B a b^{3} c d^{3} - i \, B a^{2} b^{2} d^{4}\right )} x^{2} - 6 \, {\left ({\left (4 i \, A - 3 i \, B\right )} b^{4} c^{3} d + 6 i \, B a b^{3} c^{2} d^{2} - 4 i \, B a^{2} b^{2} c d^{3} + i \, B a^{3} b d^{4}\right )} x - 6 \, {\left (i \, B b^{4} d^{4} x^{4} + 4 i \, B b^{4} c d^{3} x^{3} + 6 i \, B b^{4} c^{2} d^{2} x^{2} + 4 i \, B b^{4} c^{3} d x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) - 6 \, {\left (4 i \, B a b^{3} c^{3} d - 6 i \, B a^{2} b^{2} c^{2} d^{2} + 4 i \, B a^{3} b c d^{3} - i \, B a^{4} d^{4}\right )} \log \left (\frac {b x + a}{b}\right )}{24 \, b^{4} d} \end {gather*}
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*I*A*b^4*d^4*x^4 + 6*I*B*b^4*c^4*log((d*x + c)/d) - 2*((12*I*A - I*B)*b^4*c*d^3 + I*B*a*b^3*d^4)*x^3 -
3*(3*(4*I*A - I*B)*b^4*c^2*d^2 + 4*I*B*a*b^3*c*d^3 - I*B*a^2*b^2*d^4)*x^2 - 6*((4*I*A - 3*I*B)*b^4*c^3*d + 6*
I*B*a*b^3*c^2*d^2 - 4*I*B*a^2*b^2*c*d^3 + I*B*a^3*b*d^4)*x - 6*(I*B*b^4*d^4*x^4 + 4*I*B*b^4*c*d^3*x^3 + 6*I*B*
b^4*c^2*d^2*x^2 + 4*I*B*b^4*c^3*d*x)*log((b*x + a)*e/(d*x + c)) - 6*(4*I*B*a*b^3*c^3*d - 6*I*B*a^2*b^2*c^2*d^2
+ 4*I*B*a^3*b*c*d^3 - I*B*a^4*d^4)*log((b*x + a)/b))/(b^4*d)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 706 vs.
\(2 (128) = 256\).
time = 2.42, size = 706, normalized size = 4.74 \begin {gather*} \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} \cdot \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 )} \end {gather*}
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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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 3885 vs. \(2 (125) = 250\).
time = 4.34, size = 3885, normalized size = 26.07 \begin {gather*} \text {Too large to display} \end {gather*}
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*I*B*b^9*c^5*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 30*I*B*a*b^8*c^4*d*e^5*log(-b*e + (b*x*e + a*
e)*d/(d*x + c)) - 60*I*B*a^2*b^7*c^3*d^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 60*I*B*a^3*b^6*c^2*d^3*e^
5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 30*I*B*a^4*b^5*c*d^4*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 6*I
*B*a^5*b^4*d^5*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 24*I*(b*x*e + a*e)*B*b^8*c^5*d*e^4*log(-b*e + (b*x*
e + a*e)*d/(d*x + c))/(d*x + c) - 120*I*(b*x*e + a*e)*B*a*b^7*c^4*d^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c) + 240*I*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 240*
I*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 120*I*(b*x*e + a*e)*B*
a^4*b^4*c*d^5*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 24*I*(b*x*e + a*e)*B*a^5*b^3*d^6*e^4*log(-
b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 36*I*(b*x*e + a*e)^2*B*b^7*c^5*d^2*e^3*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^2 + 180*I*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^2 - 360*I*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 360*
I*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 180*I*(b*x*e + a*e
)^2*B*a^4*b^3*c*d^6*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 36*I*(b*x*e + a*e)^2*B*a^5*b^2*d^7
*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 24*I*(b*x*e + a*e)^3*B*b^6*c^5*d^3*e^2*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 120*I*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*e^2*log(-b*e + (b*x*e + a*e)*d/(d*
x + c))/(d*x + c)^3 + 240*I*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^3 - 240*I*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 120*I*
(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 24*I*(b*x*e + a*e)^3*B
*a^5*b*d^8*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 6*I*(b*x*e + a*e)^4*B*b^5*c^5*d^4*e*log(-b*
e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 30*I*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*e*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^4 - 60*I*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^4 + 60*I*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 30*I*(b*
x*e + a*e)^4*B*a^4*b*c*d^8*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 6*I*(b*x*e + a*e)^4*B*a^5*d^9
*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 24*I*(b*x*e + a*e)*B*b^8*c^5*d*e^4*log((b*x*e + a*e)/(d
*x + c))/(d*x + c) + 120*I*(b*x*e + a*e)*B*a*b^7*c^4*d^2*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 240*I*(b
*x*e + a*e)*B*a^2*b^6*c^3*d^3*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 240*I*(b*x*e + a*e)*B*a^3*b^5*c^2*d
^4*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 120*I*(b*x*e + a*e)*B*a^4*b^4*c*d^5*e^4*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c) + 24*I*(b*x*e + a*e)*B*a^5*b^3*d^6*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 36*I*(b*x*e +
a*e)^2*B*b^7*c^5*d^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 180*I*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*e^3
*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 360*I*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*e^3*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^2 - 360*I*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 18
0*I*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 36*I*(b*x*e + a*e)^2*B*a^5*
b^2*d^7*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 24*I*(b*x*e + a*e)^3*B*b^6*c^5*d^3*e^2*log((b*x*e + a*e
)/(d*x + c))/(d*x + c)^3 + 120*I*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3
- 240*I*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 240*I*(b*x*e + a*e)^3
*B*a^3*b^3*c^2*d^6*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 120*I*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*e^2*lo
g((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 24*I*(b*x*e + a*e)^3*B*a^5*b*d^8*e^2*log((b*x*e + a*e)/(d*x + c))/(d*
x + c)^3 + 6*I*(b*x*e + a*e)^4*B*b^5*c^5*d^4*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 30*I*(b*x*e + a*e)^4
*B*a*b^4*c^4*d^5*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 60*I*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*e*log((b*
x*e + a*e)/(d*x + c))/(d*x + c)^4 - 60*I*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*e*log((b*x*e + a*e)/(d*x + c))/(d*x
+ c)^4 + 30*I*(b*x*e + a*e)^4*B*a^4*b*c*d^8*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 6*I*(b*x*e + a*e)^4*
B*a^5*d^9*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 6*I*A*b^9*c^5*e^5 + 11*I*B*b^9*c^5*e^5 + 30*I*A*a*b^8*c
^4*d*e^5 - 55*I*B*a*b^8*c^4*d*e^5 - 60*I*A*a^2*b^7*c^3*d^2*e^5 + 110*I*B*a^2*b^7*c^3*d^2*e^5 + 60*I*A*a^3*b^6*
c^2*d^3*e^5 - 110*I*B*a^3*b^6*c^2*d^3*e^5 - 30*I*A*a^4*b^5*c*d^4*e^5 + 55*I*B*a^4*b^5*c*d^4*e^5 + 6*I*A*a^5*b^
4*d^5*e^5 - 11*I*B*a^5*b^4*d^5*e^5 - 26*I*(b*x*e + a*e)*B*b^8*c^5*d*e^4/(d*x + c) + 130*I*(b*x*e + a*e)*B*a*b^
7*c^4*d^2*e^4/(d*x + c) - 260*I*(b*x*e + a*e)*B...
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Mupad [B]
time = 4.80, size = 566, normalized size = 3.80 \begin {gather*} 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} \end {gather*}
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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