3.2.83 \(\int \frac {\log (c (a+b x)^p)}{(d+e x)^4} \, dx\) [183]

Optimal. Leaf size=133 \[ \frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2442, 46} \begin {gather*} \frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3}+\frac {b^2 p}{3 e (d+e x) (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {b p}{6 e (d+e x)^2 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^p]/(d + e*x)^4,x]

[Out]

(b*p)/(6*e*(b*d - a*e)*(d + e*x)^2) + (b^2*p)/(3*e*(b*d - a*e)^2*(d + e*x)) + (b^3*p*Log[a + b*x])/(3*e*(b*d -
a*e)^3) - Log[c*(a + b*x)^p]/(3*e*(d + e*x)^3) - (b^3*p*Log[d + e*x])/(3*e*(b*d - a*e)^3)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2442

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] - Dist[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]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 105, normalized size = 0.79 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x)^p]/(d + e*x)^4,x]

[Out]

(-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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.72, size = 873, normalized size = 6.56

method	result	size
risch	\(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {-i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 a \,b^{2} e^{3} p \,x^{2}-2 b^{3} d \,e^{2} p \,x^{2}-a^{2} b \,e^{3} p x -5 b^{3} d^{2} e p x -a^{2} b d p \,e^{2}+4 a \,b^{2} d^{2} p e -3 b^{3} d^{3} p -3 i \pi a \,b^{2} d^{2} e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 i \pi a \,b^{2} d^{2} e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {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 +2 \ln \left (-e x -d \right ) b^{3} d^{3} p -2 \ln \left (b x +a \right ) b^{3} d^{3} p +6 a \,b^{2} d \,e^{2} p x -i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{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} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+3 i \pi a \,b^{2} d^{2} e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-6 \ln \left (c \right ) a \,b^{2} d^{2} e +3 i \pi a \,b^{2} d^{2} e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-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}+6 \ln \left (c \right ) a^{2} b d \,e^{2}}{6 \left (e x +d \right )^{3} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (a e -b d \right ) e}\)	\(873\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x+a)^p)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3/e/(e*x+d)^3*ln((b*x+a)^p)+1/6*(-3*I*Pi*a^2*b*d*e^2*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)*csgn(I*c)+3*I*Pi
*a*b^2*d^2*e*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)*csgn(I*c)+3*I*Pi*a^2*b*d*e^2*csgn(I*(b*x+a)^p)*csgn(I*c*(b*
x+a)^p)^2+2*a*b^2*e^3*p*x^2-2*b^3*d*e^2*p*x^2-a^2*b*e^3*p*x-5*b^3*d^2*e*p*x-a^2*b*d*p*e^2+4*a*b^2*d^2*p*e-3*b^
3*d^3*p-3*I*Pi*a*b^2*d^2*e*csgn(I*c*(b*x+a)^p)^2*csgn(I*c)+3*I*Pi*a^2*b*d*e^2*csgn(I*c*(b*x+a)^p)^2*csgn(I*c)-
3*I*Pi*a*b^2*d^2*e*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)^2-6*ln(b*x+a)*b^3*d*e^2*p*x^2+6*ln(-e*x-d)*b^3*d*e^2*
p*x^2-6*ln(b*x+a)*b^3*d^2*e*p*x+6*ln(-e*x-d)*b^3*d^2*e*p*x-I*Pi*a^3*e^3*csgn(I*c*(b*x+a)^p)^2*csgn(I*c)+2*ln(-
e*x-d)*b^3*d^3*p+I*Pi*b^3*d^3*csgn(I*c*(b*x+a)^p)^2*csgn(I*c)-I*Pi*a^3*e^3*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^
p)^2-2*ln(b*x+a)*b^3*d^3*p+3*I*Pi*a*b^2*d^2*e*csgn(I*c*(b*x+a)^p)^3-I*Pi*b^3*d^3*csgn(I*(b*x+a)^p)*csgn(I*c*(b
*x+a)^p)*csgn(I*c)+6*a*b^2*d*e^2*p*x+2*ln(c)*b^3*d^3-2*ln(c)*a^3*e^3+I*Pi*a^3*e^3*csgn(I*(b*x+a)^p)*csgn(I*c*(
b*x+a)^p)*csgn(I*c)-3*I*Pi*a^2*b*d*e^2*csgn(I*c*(b*x+a)^p)^3+I*Pi*b^3*d^3*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p
)^2-6*ln(c)*a*b^2*d^2*e-2*ln(b*x+a)*b^3*e^3*p*x^3+2*ln(-e*x-d)*b^3*e^3*p*x^3+6*ln(c)*a^2*b*d*e^2-I*Pi*b^3*d^3*
csgn(I*c*(b*x+a)^p)^3+I*Pi*a^3*e^3*csgn(I*c*(b*x+a)^p)^3)/(e*x+d)^3/(a^2*e^2-2*a*b*d*e+b^2*d^2)/(a*e-b*d)/e

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Maxima [A]
time = 0.36, size = 228, normalized size = 1.71 \begin {gather*} \frac {1}{6} \, {\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 (x e + 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 x e + 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 e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (b x + a\right )}^{p} c\right )}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="maxima")

[Out]

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(x*e + 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*x*e + 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) - 1/3*e^(-1)*lo
g((b*x + a)^p*c)/(x*e + d)^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (126) = 252\).
time = 0.40, size = 431, normalized size = 3.24 \begin {gather*} \frac {3 \, b^{3} d^{3} p - {\left (2 \, a b^{2} p x^{2} - a^{2} b p x\right )} e^{3} + {\left (2 \, b^{3} d p x^{2} - 6 \, a b^{2} d p x + a^{2} b d p\right )} e^{2} + {\left (5 \, b^{3} d^{2} p x - 4 \, a b^{2} d^{2} p\right )} e + 2 \, {\left ({\left (b^{3} p x^{3} + a^{3} p\right )} e^{3} + 3 \, {\left (b^{3} d p x^{2} - a^{2} b d p\right )} e^{2} + 3 \, {\left (b^{3} d^{2} p x + a b^{2} d^{2} p\right )} e\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{3} p x^{3} e^{3} + 3 \, b^{3} d p x^{2} e^{2} + 3 \, b^{3} d^{2} p x e + b^{3} d^{3} p\right )} \log \left (x e + 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 - a^{3} x^{3} e^{7} + 3 \, {\left (a^{2} b d x^{3} - a^{3} d x^{2}\right )} e^{6} - 3 \, {\left (a b^{2} d^{2} x^{3} - 3 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} e^{5} + {\left (b^{3} d^{3} x^{3} - 9 \, a b^{2} d^{3} x^{2} + 9 \, a^{2} b d^{3} x - a^{3} d^{3}\right )} e^{4} + 3 \, {\left (b^{3} d^{4} x^{2} - 3 \, a b^{2} d^{4} x + a^{2} b d^{4}\right )} e^{3} + 3 \, {\left (b^{3} d^{5} x - a b^{2} d^{5}\right )} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(3*b^3*d^3*p - (2*a*b^2*p*x^2 - a^2*b*p*x)*e^3 + (2*b^3*d*p*x^2 - 6*a*b^2*d*p*x + a^2*b*d*p)*e^2 + (5*b^3*
d^2*p*x - 4*a*b^2*d^2*p)*e + 2*((b^3*p*x^3 + a^3*p)*e^3 + 3*(b^3*d*p*x^2 - a^2*b*d*p)*e^2 + 3*(b^3*d^2*p*x + a
*b^2*d^2*p)*e)*log(b*x + a) - 2*(b^3*p*x^3*e^3 + 3*b^3*d*p*x^2*e^2 + 3*b^3*d^2*p*x*e + b^3*d^3*p)*log(x*e + 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 - a^3*x^3*e^7 + 3*(a^2*b*d*x^3 - a
^3*d*x^2)*e^6 - 3*(a*b^2*d^2*x^3 - 3*a^2*b*d^2*x^2 + a^3*d^2*x)*e^5 + (b^3*d^3*x^3 - 9*a*b^2*d^3*x^2 + 9*a^2*b
*d^3*x - a^3*d^3)*e^4 + 3*(b^3*d^4*x^2 - 3*a*b^2*d^4*x + a^2*b*d^4)*e^3 + 3*(b^3*d^5*x - a*b^2*d^5)*e^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 4571 vs. \(2 (109) = 218\).
time = 20.01, size = 4571, normalized size = 34.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x+a)**p)/(e*x+d)**4,x)

[Out]

Piecewise((-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)), ((a*log(c*(a + b*x)**p)/b - p*x + x*log(c*(a
+ b*x)**p))/d**4, Eq(e, 0)), (-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 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 1
8*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) + 4*a*b**2*d**2*e*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*b**2*d**2*e*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) + 6*a*b**2*d*e**2*p*x/(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) + 2*a*b**2*e**3*p*x**2/(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) + 2*b**3*d**3*p*log(d
/e + x)/(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) - 3*b**3*d**3*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*b**3*d**2*
e*p*x*log(d/e + x)/(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) - 5*b**3*d**2*e*p*x/(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*b**3*d**2*e*x*log(c*(a + b*x)**p)/(6*a**3*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (126) = 252\).
time = 4.17, size = 495, normalized size = 3.72 \begin {gather*} \frac {2 \, b^{3} p x^{3} e^{3} \log \left (b x + a\right ) + 6 \, b^{3} d p x^{2} e^{2} \log \left (b x + a\right ) + 6 \, b^{3} d^{2} p x e \log \left (b x + a\right ) - 2 \, b^{3} p x^{3} e^{3} \log \left (x e + d\right ) - 6 \, b^{3} d p x^{2} e^{2} \log \left (x e + d\right ) - 6 \, b^{3} d^{2} p x e \log \left (x e + d\right ) + 2 \, b^{3} d p x^{2} e^{2} + 5 \, b^{3} d^{2} p x e + 6 \, a b^{2} d^{2} p e \log \left (b x + a\right ) - 2 \, b^{3} d^{3} p \log \left (x e + d\right ) + 3 \, b^{3} d^{3} p - 2 \, a b^{2} p x^{2} e^{3} - 6 \, a b^{2} d p x e^{2} - 4 \, a b^{2} d^{2} p e - 6 \, a^{2} b d p e^{2} \log \left (b x + a\right ) - 2 \, b^{3} d^{3} \log \left (c\right ) + 6 \, a b^{2} d^{2} e \log \left (c\right ) + a^{2} b p x e^{3} + a^{2} b d p e^{2} + 2 \, a^{3} p e^{3} \log \left (b x + a\right ) - 6 \, a^{2} b d e^{2} \log \left (c\right ) + 2 \, a^{3} e^{3} \log \left (c\right )}{6 \, {\left (b^{3} d^{3} x^{3} e^{4} + 3 \, b^{3} d^{4} x^{2} e^{3} + 3 \, b^{3} d^{5} x e^{2} + b^{3} d^{6} e - 3 \, a b^{2} d^{2} x^{3} e^{5} - 9 \, a b^{2} d^{3} x^{2} e^{4} - 9 \, a b^{2} d^{4} x e^{3} - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d x^{3} e^{6} + 9 \, a^{2} b d^{2} x^{2} e^{5} + 9 \, a^{2} b d^{3} x e^{4} + 3 \, a^{2} b d^{4} e^{3} - a^{3} x^{3} e^{7} - 3 \, a^{3} d x^{2} e^{6} - 3 \, a^{3} d^{2} x e^{5} - a^{3} d^{3} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/6*(2*b^3*p*x^3*e^3*log(b*x + a) + 6*b^3*d*p*x^2*e^2*log(b*x + a) + 6*b^3*d^2*p*x*e*log(b*x + a) - 2*b^3*p*x^
3*e^3*log(x*e + d) - 6*b^3*d*p*x^2*e^2*log(x*e + d) - 6*b^3*d^2*p*x*e*log(x*e + d) + 2*b^3*d*p*x^2*e^2 + 5*b^3
*d^2*p*x*e + 6*a*b^2*d^2*p*e*log(b*x + a) - 2*b^3*d^3*p*log(x*e + d) + 3*b^3*d^3*p - 2*a*b^2*p*x^2*e^3 - 6*a*b
^2*d*p*x*e^2 - 4*a*b^2*d^2*p*e - 6*a^2*b*d*p*e^2*log(b*x + a) - 2*b^3*d^3*log(c) + 6*a*b^2*d^2*e*log(c) + a^2*
b*p*x*e^3 + a^2*b*d*p*e^2 + 2*a^3*p*e^3*log(b*x + a) - 6*a^2*b*d*e^2*log(c) + 2*a^3*e^3*log(c))/(b^3*d^3*x^3*e
^4 + 3*b^3*d^4*x^2*e^3 + 3*b^3*d^5*x*e^2 + b^3*d^6*e - 3*a*b^2*d^2*x^3*e^5 - 9*a*b^2*d^3*x^2*e^4 - 9*a*b^2*d^4
*x*e^3 - 3*a*b^2*d^5*e^2 + 3*a^2*b*d*x^3*e^6 + 9*a^2*b*d^2*x^2*e^5 + 9*a^2*b*d^3*x*e^4 + 3*a^2*b*d^4*e^3 - a^3
*x^3*e^7 - 3*a^3*d*x^2*e^6 - 3*a^3*d^2*x*e^5 - a^3*d^3*e^4)

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Mupad [B]
time = 0.76, size = 145, normalized size = 1.09 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x)^p)/(d + e*x)^4,x)

[Out]

(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)

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