Integrand size = 18, antiderivative size = 133 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3} \] Output:
1/6*b*p/e/(-a*e+b*d)/(e*x+d)^2+1/3*b^2*p/e/(-a*e+b*d)^2/(e*x+d)+1/3*b^3*p* ln(b*x+a)/e/(-a*e+b*d)^3-1/3*ln(c*(b*x+a)^p)/e/(e*x+d)^3-1/3*b^3*p*ln(e*x+ d)/e/(-a*e+b*d)^3
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.79 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {-2 \log \left (c (a+b x)^p\right )+\frac {b p (d+e x) \left ((b d-a e) (3 b d-a e+2 b e x)+2 b^2 (d+e x)^2 \log (a+b x)-2 b^2 (d+e x)^2 \log (d+e x)\right )}{(b d-a e)^3}}{6 e (d+e x)^3} \] Input:
Integrate[Log[c*(a + b*x)^p]/(d + e*x)^4,x]
Output:
(-2*Log[c*(a + b*x)^p] + (b*p*(d + e*x)*((b*d - a*e)*(3*b*d - a*e + 2*b*e* x) + 2*b^2*(d + e*x)^2*Log[a + b*x] - 2*b^2*(d + e*x)^2*Log[d + e*x]))/(b* d - a*e)^3)/(6*e*(d + e*x)^3)
Time = 0.49 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2842, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 2842 |
| \(\displaystyle \frac {b p \int \frac {1}{(a+b x) (d+e x)^3}dx}{3 e}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {b p \int \left (\frac {b^3}{(b d-a e)^3 (a+b x)}-\frac {e b^2}{(b d-a e)^3 (d+e x)}-\frac {e b}{(b d-a e)^2 (d+e x)^2}-\frac {e}{(b d-a e) (d+e x)^3}\right )dx}{3 e}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b p \left (\frac {b^2 \log (a+b x)}{(b d-a e)^3}-\frac {b^2 \log (d+e x)}{(b d-a e)^3}+\frac {b}{(d+e x) (b d-a e)^2}+\frac {1}{2 (d+e x)^2 (b d-a e)}\right )}{3 e}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}\) |
Input:
Int[Log[c*(a + b*x)^p]/(d + e*x)^4,x]
Output:
-1/3*Log[c*(a + b*x)^p]/(e*(d + e*x)^3) + (b*p*(1/(2*(b*d - a*e)*(d + e*x) ^2) + b/((b*d - a*e)^2*(d + e*x)) + (b^2*Log[a + b*x])/(b*d - a*e)^3 - (b^ 2*Log[d + e*x])/(b*d - a*e)^3))/(3*e)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
Time = 2.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.83
| method | result | size |
| parts | \(-\frac {\ln \left (c \left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {p b \left (-\frac {b^{2} \ln \left (b x +a \right )}{\left (e a -b d \right )^{3}}-\frac {1}{2 \left (e a -b d \right ) \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{\left (e a -b d \right )^{3}}+\frac {b}{\left (e a -b d \right )^{2} \left (e x +d \right )}\right )}{3 e}\) | \(111\) |
| parallelrisch | \(-\frac {3 b^{4} d^{3} e^{2} p +2 \ln \left (c \left (b x +a \right )^{p}\right ) a^{3} b \,e^{5}-2 \ln \left (c \left (b x +a \right )^{p}\right ) b^{4} d^{3} e^{2}-2 x^{2} a \,b^{3} e^{5} p +2 x^{2} b^{4} d \,e^{4} p +x \,a^{2} b^{2} e^{5} p +5 x \,b^{4} d^{2} e^{3} p -6 \ln \left (c \left (b x +a \right )^{p}\right ) a^{2} b^{2} d \,e^{4}+6 \ln \left (c \left (b x +a \right )^{p}\right ) a \,b^{3} d^{2} e^{3}+2 \ln \left (b x +a \right ) x^{3} b^{4} e^{5} p -2 \ln \left (e x +d \right ) x^{3} b^{4} e^{5} p +2 \ln \left (b x +a \right ) b^{4} d^{3} e^{2} p -2 \ln \left (e x +d \right ) b^{4} d^{3} e^{2} p +a^{2} b^{2} d \,e^{4} p -4 a \,b^{3} d^{2} e^{3} p -6 x a \,b^{3} d \,e^{4} p +6 \ln \left (b x +a \right ) x^{2} b^{4} d \,e^{4} p -6 \ln \left (e x +d \right ) x^{2} b^{4} d \,e^{4} p +6 \ln \left (b x +a \right ) x \,b^{4} d^{2} e^{3} p -6 \ln \left (e x +d \right ) x \,b^{4} d^{2} e^{3} p}{6 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{3} b \,e^{3}}\) | \(387\) |
| risch | \(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {6 a \,b^{2} d \,e^{2} p x -a^{2} b \,e^{3} p x -2 \ln \left (b x +a \right ) b^{3} d^{3} p +2 a \,b^{2} e^{3} p \,x^{2}-2 b^{3} d \,e^{2} p \,x^{2}+2 \ln \left (-e x -d \right ) b^{3} d^{3} p -a^{2} b d p \,e^{2}+4 a \,b^{2} d^{2} p e -5 b^{3} d^{2} e p x +6 \ln \left (c \right ) a^{2} b d \,e^{2}-6 \ln \left (c \right ) a \,b^{2} d^{2} e -3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-2 \ln \left (b x +a \right ) b^{3} e^{3} p \,x^{3}+2 \ln \left (-e x -d \right ) b^{3} e^{3} p \,x^{3}+2 \ln \left (c \right ) b^{3} d^{3}-2 \ln \left (c \right ) a^{3} e^{3}+i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+3 i \pi a \,b^{2} d^{2} e \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-6 \ln \left (b x +a \right ) b^{3} d \,e^{2} p \,x^{2}+6 \ln \left (-e x -d \right ) b^{3} d \,e^{2} p \,x^{2}-6 \ln \left (b x +a \right ) b^{3} d^{2} e p x +6 \ln \left (-e x -d \right ) b^{3} d^{2} e p x -3 i \pi a \,b^{2} d^{2} e \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 i \pi a \,b^{2} d^{2} e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 b^{3} d^{3} p +3 i \pi a \,b^{2} d^{2} e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{6 \left (e x +d \right )^{3} \left (a^{2} e^{2}-2 d e a b +d^{2} b^{2}\right ) \left (e a -b d \right ) e}\) | \(873\) |
Input:
int(ln(c*(b*x+a)^p)/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
-1/3*ln(c*(b*x+a)^p)/e/(e*x+d)^3+1/3*p*b/e*(-b^2/(a*e-b*d)^3*ln(b*x+a)-1/2 /(a*e-b*d)/(e*x+d)^2+b^2/(a*e-b*d)^3*ln(e*x+d)+b/(a*e-b*d)^2/(e*x+d))
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (123) = 246\).
Time = 0.09 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.33 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {2 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} p x^{2} + {\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} p x + {\left (3 \, b^{3} d^{3} - 4 \, a b^{2} d^{2} e + a^{2} b d e^{2}\right )} p + 2 \, {\left (b^{3} e^{3} p x^{3} + 3 \, b^{3} d e^{2} p x^{2} + 3 \, b^{3} d^{2} e p x + {\left (3 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} p\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{3} e^{3} p x^{3} + 3 \, b^{3} d e^{2} p x^{2} + 3 \, b^{3} d^{2} e p x + b^{3} d^{3} p\right )} \log \left (e x + d\right ) - 2 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (c\right )}{6 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}} \] Input:
integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="fricas")
Output:
1/6*(2*(b^3*d*e^2 - a*b^2*e^3)*p*x^2 + (5*b^3*d^2*e - 6*a*b^2*d*e^2 + a^2* b*e^3)*p*x + (3*b^3*d^3 - 4*a*b^2*d^2*e + a^2*b*d*e^2)*p + 2*(b^3*e^3*p*x^ 3 + 3*b^3*d*e^2*p*x^2 + 3*b^3*d^2*e*p*x + (3*a*b^2*d^2*e - 3*a^2*b*d*e^2 + a^3*e^3)*p)*log(b*x + a) - 2*(b^3*e^3*p*x^3 + 3*b^3*d*e^2*p*x^2 + 3*b^3*d ^2*e*p*x + b^3*d^3*p)*log(e*x + d) - 2*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b* d*e^2 - a^3*e^3)*log(c))/(b^3*d^6*e - 3*a*b^2*d^5*e^2 + 3*a^2*b*d^4*e^3 - a^3*d^3*e^4 + (b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6 - a^3*e^7)*x^ 3 + 3*(b^3*d^4*e^3 - 3*a*b^2*d^3*e^4 + 3*a^2*b*d^2*e^5 - a^3*d*e^6)*x^2 + 3*(b^3*d^5*e^2 - 3*a*b^2*d^4*e^3 + 3*a^2*b*d^3*e^4 - a^3*d^2*e^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 4571 vs. \(2 (109) = 218\).
Time = 18.40 (sec) , antiderivative size = 4571, normalized size of antiderivative = 34.37 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:
integrate(ln(c*(b*x+a)**p)/(e*x+d)**4,x)
Output:
Piecewise(((a*log(c*(a + b*x)**p)/b - p*x + x*log(c*(a + b*x)**p))/d**4, E q(e, 0)), (-p/(9*d**3*e + 27*d**2*e**2*x + 27*d*e**3*x**2 + 9*e**4*x**3) - 3*log(c*(b*d/e + b*x)**p)/(9*d**3*e + 27*d**2*e**2*x + 27*d*e**3*x**2 + 9 *e**4*x**3), Eq(a, b*d/e)), (-2*a**3*e**3*log(c*(a + b*x)**p)/(6*a**3*d**3 *e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18* a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a **2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b** 2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5 *e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) - a**2*b*d*e**2* p/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e **7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e* *5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e** 3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 6*a**2*b*d*e**2*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5* x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2 *b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b **2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b* *2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e** 3*x**2 - 6*b**3*d**3*e**4*x**3) - a**2*b*e**3*p*x/(6*a**3*d**3*e**4 + 1...
Time = 0.04 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.74 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {{\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {2 \, b^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b e x + 3 \, b d - a e}{b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x}\right )} b p}{6 \, e} - \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{3 \, {\left (e x + d\right )}^{3} e} \] Input:
integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="maxima")
Output:
1/6*(2*b^2*log(b*x + a)/(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3 ) - 2*b^2*log(e*x + d)/(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3) + (2*b*e*x + 3*b*d - a*e)/(b^2*d^4 - 2*a*b*d^3*e + a^2*d^2*e^2 + (b^2*d^2 *e^2 - 2*a*b*d*e^3 + a^2*e^4)*x^2 + 2*(b^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e ^3)*x))*b*p/e - 1/3*log((b*x + a)^p*c)/((e*x + d)^3*e)
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (123) = 246\).
Time = 0.12 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.74 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^{3} p \log \left (b x + a\right )}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {b^{3} p \log \left (e x + d\right )}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {p \log \left (b x + a\right )}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {2 \, b^{2} e^{2} p x^{2} + 5 \, b^{2} d e p x - a b e^{2} p x + 3 \, b^{2} d^{2} p - a b d e p - 2 \, b^{2} d^{2} \log \left (c\right ) + 4 \, a b d e \log \left (c\right ) - 2 \, a^{2} e^{2} \log \left (c\right )}{6 \, {\left (b^{2} d^{2} e^{4} x^{3} - 2 \, a b d e^{5} x^{3} + a^{2} e^{6} x^{3} + 3 \, b^{2} d^{3} e^{3} x^{2} - 6 \, a b d^{2} e^{4} x^{2} + 3 \, a^{2} d e^{5} x^{2} + 3 \, b^{2} d^{4} e^{2} x - 6 \, a b d^{3} e^{3} x + 3 \, a^{2} d^{2} e^{4} x + b^{2} d^{5} e - 2 \, a b d^{4} e^{2} + a^{2} d^{3} e^{3}\right )}} \] Input:
integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="giac")
Output:
1/3*b^3*p*log(b*x + a)/(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3* e^4) - 1/3*b^3*p*log(e*x + d)/(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4) - 1/3*p*log(b*x + a)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^ 3*e) + 1/6*(2*b^2*e^2*p*x^2 + 5*b^2*d*e*p*x - a*b*e^2*p*x + 3*b^2*d^2*p - a*b*d*e*p - 2*b^2*d^2*log(c) + 4*a*b*d*e*log(c) - 2*a^2*e^2*log(c))/(b^2*d ^2*e^4*x^3 - 2*a*b*d*e^5*x^3 + a^2*e^6*x^3 + 3*b^2*d^3*e^3*x^2 - 6*a*b*d^2 *e^4*x^2 + 3*a^2*d*e^5*x^2 + 3*b^2*d^4*e^2*x - 6*a*b*d^3*e^3*x + 3*a^2*d^2 *e^4*x + b^2*d^5*e - 2*a*b*d^4*e^2 + a^2*d^3*e^3)
Time = 26.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^2\,p\,x}{3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}-\frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{3\,e\,{\left (d+e\,x\right )}^3}-\frac {a\,b\,p}{6\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^2\,d\,p}{2\,e\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^3\,p\,\mathrm {atan}\left (\frac {a\,e\,1{}\mathrm {i}+b\,d\,1{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}\right )\,2{}\mathrm {i}}{3\,e\,{\left (a\,e-b\,d\right )}^3} \] Input:
int(log(c*(a + b*x)^p)/(d + e*x)^4,x)
Output:
(b^2*p*x)/(3*(a*e - b*d)^2*(d + e*x)^2) - log(c*(a + b*x)^p)/(3*e*(d + e*x )^3) + (b^3*p*atan((a*e*1i + b*d*1i + b*e*x*2i)/(a*e - b*d))*2i)/(3*e*(a*e - b*d)^3) - (a*b*p)/(6*(a*e - b*d)^2*(d + e*x)^2) + (b^2*d*p)/(2*e*(a*e - b*d)^2*(d + e*x)^2)
Time = 0.17 (sec) , antiderivative size = 898, normalized size of antiderivative = 6.75 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
int(log(c*(b*x+a)^p)/(e*x+d)^4,x)
Output:
( - 6*log(a + b*x)*a**3*d**3*e**3*p - 18*log(a + b*x)*a**3*d**2*e**4*p*x - 18*log(a + b*x)*a**3*d*e**5*p*x**2 - 6*log(a + b*x)*a**3*e**6*p*x**3 + 18 *log(a + b*x)*a**2*b*d**4*e**2*p + 54*log(a + b*x)*a**2*b*d**3*e**3*p*x + 54*log(a + b*x)*a**2*b*d**2*e**4*p*x**2 + 18*log(a + b*x)*a**2*b*d*e**5*p* x**3 - 18*log(a + b*x)*a*b**2*d**5*e*p - 54*log(a + b*x)*a*b**2*d**4*e**2* p*x - 54*log(a + b*x)*a*b**2*d**3*e**3*p*x**2 - 18*log(a + b*x)*a*b**2*d** 2*e**4*p*x**3 + 6*log(d + e*x)*b**3*d**6*p + 18*log(d + e*x)*b**3*d**5*e*p *x + 18*log(d + e*x)*b**3*d**4*e**2*p*x**2 + 6*log(d + e*x)*b**3*d**3*e**3 *p*x**3 + 18*log((a + b*x)**p*c)*a**3*d**2*e**4*x + 18*log((a + b*x)**p*c) *a**3*d*e**5*x**2 + 6*log((a + b*x)**p*c)*a**3*e**6*x**3 - 54*log((a + b*x )**p*c)*a**2*b*d**3*e**3*x - 54*log((a + b*x)**p*c)*a**2*b*d**2*e**4*x**2 - 18*log((a + b*x)**p*c)*a**2*b*d*e**5*x**3 + 54*log((a + b*x)**p*c)*a*b** 2*d**4*e**2*x + 54*log((a + b*x)**p*c)*a*b**2*d**3*e**3*x**2 + 18*log((a + b*x)**p*c)*a*b**2*d**2*e**4*x**3 - 18*log((a + b*x)**p*c)*b**3*d**5*e*x - 18*log((a + b*x)**p*c)*b**3*d**4*e**2*x**2 - 6*log((a + b*x)**p*c)*b**3*d **3*e**3*x**3 - 3*a**2*b*d**4*e**2*p - 3*a**2*b*d**3*e**3*p*x + 10*a*b**2* d**5*e*p + 12*a*b**2*d**4*e**2*p*x - 2*a*b**2*d**2*e**4*p*x**3 - 7*b**3*d* *6*p - 9*b**3*d**5*e*p*x + 2*b**3*d**3*e**3*p*x**3)/(18*d**3*e*(a**3*d**3* e**3 + 3*a**3*d**2*e**4*x + 3*a**3*d*e**5*x**2 + a**3*e**6*x**3 - 3*a**2*b *d**4*e**2 - 9*a**2*b*d**3*e**3*x - 9*a**2*b*d**2*e**4*x**2 - 3*a**2*b*...