3.1.43 \(\int \frac {a+b \log (c (d+e x)^n)}{(f+g x)^4} \, dx\) [43]

Optimal. Leaf size=141 \[ \frac {b e n}{6 g (e f-d g) (f+g x)^2}+\frac {b e^2 n}{3 g (e f-d g)^2 (f+g x)}+\frac {b e^3 n \log (d+e x)}{3 g (e f-d g)^3}-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}-\frac {b e^3 n \log (f+g x)}{3 g (e f-d g)^3} \]

[Out]

1/6*b*e*n/g/(-d*g+e*f)/(g*x+f)^2+1/3*b*e^2*n/g/(-d*g+e*f)^2/(g*x+f)+1/3*b*e^3*n*ln(e*x+d)/g/(-d*g+e*f)^3+1/3*(
-a-b*ln(c*(e*x+d)^n))/g/(g*x+f)^3-1/3*b*e^3*n*ln(g*x+f)/g/(-d*g+e*f)^3

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Rubi [A]
time = 0.06, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2442, 46} \begin {gather*} -\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}+\frac {b e^3 n \log (d+e x)}{3 g (e f-d g)^3}-\frac {b e^3 n \log (f+g x)}{3 g (e f-d g)^3}+\frac {b e^2 n}{3 g (f+g x) (e f-d g)^2}+\frac {b e n}{6 g (f+g x)^2 (e f-d g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])/(f + g*x)^4,x]

[Out]

(b*e*n)/(6*g*(e*f - d*g)*(f + g*x)^2) + (b*e^2*n)/(3*g*(e*f - d*g)^2*(f + g*x)) + (b*e^3*n*Log[d + e*x])/(3*g*
(e*f - d*g)^3) - (a + b*Log[c*(d + e*x)^n])/(3*g*(f + g*x)^3) - (b*e^3*n*Log[f + g*x])/(3*g*(e*f - d*g)^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 {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx &=-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}+\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)^3} \, dx}{3 g}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}+\frac {(b e n) \int \left (\frac {e^3}{(e f-d g)^3 (d+e x)}-\frac {g}{(e f-d g) (f+g x)^3}-\frac {e g}{(e f-d g)^2 (f+g x)^2}-\frac {e^2 g}{(e f-d g)^3 (f+g x)}\right ) \, dx}{3 g}\\ &=\frac {b e n}{6 g (e f-d g) (f+g x)^2}+\frac {b e^2 n}{3 g (e f-d g)^2 (f+g x)}+\frac {b e^3 n \log (d+e x)}{3 g (e f-d g)^3}-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}-\frac {b e^3 n \log (f+g x)}{3 g (e f-d g)^3}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 110, normalized size = 0.78 \begin {gather*} \frac {-2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e n (f+g x) \left ((e f-d g) (3 e f-d g+2 e g x)+2 e^2 (f+g x)^2 \log (d+e x)-2 e^2 (f+g x)^2 \log (f+g x)\right )}{(e f-d g)^3}}{6 g (f+g x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.50, size = 950, normalized size = 6.74

method	result	size
risch	\(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{3 g \left (g x +f \right )^{3}}+\frac {6 b d \,e^{2} f \,g^{2} n x +2 \ln \left (-g x -f \right ) b \,e^{3} f^{3} n -3 b \,e^{3} f^{3} n -2 \ln \left (e x +d \right ) b \,e^{3} g^{3} n \,x^{3}+2 \ln \left (-g x -f \right ) b \,e^{3} g^{3} n \,x^{3}-2 \ln \left (e x +d \right ) b \,e^{3} f^{3} n +2 a \,e^{3} f^{3}-i \pi b \,d^{3} g^{3} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,e^{3} f^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,e^{3} f^{3} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \,d^{3} g^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+6 a \,d^{2} e f \,g^{2}-6 a d \,e^{2} f^{2} g -6 \ln \left (e x +d \right ) b \,e^{3} f \,g^{2} n \,x^{2}+6 \ln \left (-g x -f \right ) b \,e^{3} f \,g^{2} n \,x^{2}-6 \ln \left (e x +d \right ) b \,e^{3} f^{2} g n x +6 \ln \left (-g x -f \right ) b \,e^{3} f^{2} g n x -5 b \,e^{3} f^{2} g n x -2 a \,d^{3} g^{3}-b \,d^{2} e f n \,g^{2}+4 b d \,e^{2} f^{2} n g -b \,d^{2} e \,g^{3} n x -2 \ln \left (c \right ) b \,d^{3} g^{3}+2 \ln \left (c \right ) b \,e^{3} f^{3}+3 i \pi b \,d^{2} e f \,g^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+3 i \pi b \,d^{2} e f \,g^{2} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-3 i \pi b d \,e^{2} f^{2} g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-3 i \pi b d \,e^{2} f^{2} g \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-3 i \pi b \,d^{2} e f \,g^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+3 i \pi b d \,e^{2} f^{2} g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b \,e^{3} f^{3} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b \,d^{3} g^{3} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b d \,e^{2} g^{3} n \,x^{2}+i \pi b \,d^{3} g^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-3 i \pi b \,d^{2} e f \,g^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+3 i \pi b d \,e^{2} f^{2} g \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,e^{3} f^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+6 \ln \left (c \right ) b \,d^{2} e f \,g^{2}-6 \ln \left (c \right ) b d \,e^{2} f^{2} g -2 b \,e^{3} f \,g^{2} n \,x^{2}}{6 \left (g x +f \right )^{3} \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (d g -e f \right ) g}\)	\(950\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (135) = 270\).
time = 0.30, size = 302, normalized size = 2.14 \begin {gather*} \frac {1}{6} \, b n {\left (\frac {2 \, e^{2} \log \left (g x + f\right )}{d^{3} g^{4} - 3 \, d^{2} f g^{3} e + 3 \, d f^{2} g^{2} e^{2} - f^{3} g e^{3}} - \frac {2 \, e^{2} \log \left (x e + d\right )}{d^{3} g^{4} - 3 \, d^{2} f g^{3} e + 3 \, d f^{2} g^{2} e^{2} - f^{3} g e^{3}} + \frac {2 \, g x e - d g + 3 \, f e}{d^{2} f^{2} g^{3} - 2 \, d f^{3} g^{2} e + f^{4} g e^{2} + {\left (d^{2} g^{5} - 2 \, d f g^{4} e + f^{2} g^{3} e^{2}\right )} x^{2} + 2 \, {\left (d^{2} f g^{4} - 2 \, d f^{2} g^{3} e + f^{3} g^{2} e^{2}\right )} x}\right )} e - \frac {b \log \left ({\left (x e + d\right )}^{n} c\right )}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} - \frac {a}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*n*(2*e^2*log(g*x + f)/(d^3*g^4 - 3*d^2*f*g^3*e + 3*d*f^2*g^2*e^2 - f^3*g*e^3) - 2*e^2*log(x*e + d)/(d^3*
g^4 - 3*d^2*f*g^3*e + 3*d*f^2*g^2*e^2 - f^3*g*e^3) + (2*g*x*e - d*g + 3*f*e)/(d^2*f^2*g^3 - 2*d*f^3*g^2*e + f^
4*g*e^2 + (d^2*g^5 - 2*d*f*g^4*e + f^2*g^3*e^2)*x^2 + 2*(d^2*f*g^4 - 2*d*f^2*g^3*e + f^3*g^2*e^2)*x))*e - 1/3*
b*log((x*e + d)^n*c)/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g) - 1/3*a/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*
x + f^3*g)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (135) = 270\).
time = 0.38, size = 475, normalized size = 3.37 \begin {gather*} -\frac {2 \, a d^{3} g^{3} - 2 \, {\left (b g^{3} n x^{3} + 3 \, b f g^{2} n x^{2} + 3 \, b f^{2} g n x + b f^{3} n\right )} e^{3} \log \left (g x + f\right ) + {\left (2 \, b f g^{2} n x^{2} + 5 \, b f^{2} g n x + 3 \, b f^{3} n - 2 \, a f^{3}\right )} e^{3} - 2 \, {\left (b d g^{3} n x^{2} + 3 \, b d f g^{2} n x + 2 \, b d f^{2} g n - 3 \, a d f^{2} g\right )} e^{2} + {\left (b d^{2} g^{3} n x + b d^{2} f g^{2} n - 6 \, a d^{2} f g^{2}\right )} e + 2 \, {\left (b d^{3} g^{3} n - 3 \, b d^{2} f g^{2} n e + 3 \, b d f^{2} g n e^{2} + {\left (b g^{3} n x^{3} + 3 \, b f g^{2} n x^{2} + 3 \, b f^{2} g n x\right )} e^{3}\right )} \log \left (x e + d\right ) + 2 \, {\left (b d^{3} g^{3} - 3 \, b d^{2} f g^{2} e + 3 \, b d f^{2} g e^{2} - b f^{3} e^{3}\right )} \log \left (c\right )}{6 \, {\left (d^{3} g^{7} x^{3} + 3 \, d^{3} f g^{6} x^{2} + 3 \, d^{3} f^{2} g^{5} x + d^{3} f^{3} g^{4} - {\left (f^{3} g^{4} x^{3} + 3 \, f^{4} g^{3} x^{2} + 3 \, f^{5} g^{2} x + f^{6} g\right )} e^{3} + 3 \, {\left (d f^{2} g^{5} x^{3} + 3 \, d f^{3} g^{4} x^{2} + 3 \, d f^{4} g^{3} x + d f^{5} g^{2}\right )} e^{2} - 3 \, {\left (d^{2} f g^{6} x^{3} + 3 \, d^{2} f^{2} g^{5} x^{2} + 3 \, d^{2} f^{3} g^{4} x + d^{2} f^{4} g^{3}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))/(g*x+f)**4,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (135) = 270\).
time = 3.83, size = 565, normalized size = 4.01 \begin {gather*} \frac {2 \, b g^{3} n x^{3} e^{3} \log \left (g x + f\right ) - 2 \, b g^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 2 \, b d g^{3} n x^{2} e^{2} - b d^{2} g^{3} n x e + 6 \, b f g^{2} n x^{2} e^{3} \log \left (g x + f\right ) - 2 \, b d^{3} g^{3} n \log \left (x e + d\right ) - 6 \, b f g^{2} n x^{2} e^{3} \log \left (x e + d\right ) + 6 \, b d^{2} f g^{2} n e \log \left (x e + d\right ) - 2 \, b f g^{2} n x^{2} e^{3} + 6 \, b d f g^{2} n x e^{2} - b d^{2} f g^{2} n e + 6 \, b f^{2} g n x e^{3} \log \left (g x + f\right ) - 6 \, b f^{2} g n x e^{3} \log \left (x e + d\right ) - 6 \, b d f^{2} g n e^{2} \log \left (x e + d\right ) - 2 \, b d^{3} g^{3} \log \left (c\right ) + 6 \, b d^{2} f g^{2} e \log \left (c\right ) - 2 \, a d^{3} g^{3} - 5 \, b f^{2} g n x e^{3} + 4 \, b d f^{2} g n e^{2} + 6 \, a d^{2} f g^{2} e + 2 \, b f^{3} n e^{3} \log \left (g x + f\right ) - 6 \, b d f^{2} g e^{2} \log \left (c\right ) - 3 \, b f^{3} n e^{3} - 6 \, a d f^{2} g e^{2} + 2 \, b f^{3} e^{3} \log \left (c\right ) + 2 \, a f^{3} e^{3}}{6 \, {\left (d^{3} g^{7} x^{3} - 3 \, d^{2} f g^{6} x^{3} e + 3 \, d^{3} f g^{6} x^{2} + 3 \, d f^{2} g^{5} x^{3} e^{2} - 9 \, d^{2} f^{2} g^{5} x^{2} e + 3 \, d^{3} f^{2} g^{5} x - f^{3} g^{4} x^{3} e^{3} + 9 \, d f^{3} g^{4} x^{2} e^{2} - 9 \, d^{2} f^{3} g^{4} x e + d^{3} f^{3} g^{4} - 3 \, f^{4} g^{3} x^{2} e^{3} + 9 \, d f^{4} g^{3} x e^{2} - 3 \, d^{2} f^{4} g^{3} e - 3 \, f^{5} g^{2} x e^{3} + 3 \, d f^{5} g^{2} e^{2} - f^{6} g e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.94, size = 283, normalized size = 2.01 \begin {gather*} \frac {2\,a\,d\,e\,f}{3\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {a\,d^2\,g}{3\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{3\,g\,{\left (f+g\,x\right )}^3}-\frac {a\,e^2\,f^2}{3\,g\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {5\,b\,e^2\,f\,n\,x}{6\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {b\,e^2\,g\,n\,x^2}{3\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,d\,e\,f\,n}{6\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {b\,e^2\,f^2\,n}{2\,g\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,d\,e\,g\,n\,x}{6\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {b\,e^3\,n\,\mathrm {atan}\left (\frac {d\,g\,1{}\mathrm {i}+e\,f\,1{}\mathrm {i}+e\,g\,x\,2{}\mathrm {i}}{d\,g-e\,f}\right )\,2{}\mathrm {i}}{3\,g\,{\left (d\,g-e\,f\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))/(f + g*x)^4,x)

[Out]

(2*a*d*e*f)/(3*(f + g*x)^3*(d*g - e*f)^2) - (a*d^2*g)/(3*(f + g*x)^3*(d*g - e*f)^2) - (b*log(c*(d + e*x)^n))/(
3*g*(f + g*x)^3) - (a*e^2*f^2)/(3*g*(f + g*x)^3*(d*g - e*f)^2) + (b*e^3*n*atan((d*g*1i + e*f*1i + e*g*x*2i)/(d
*g - e*f))*2i)/(3*g*(d*g - e*f)^3) + (5*b*e^2*f*n*x)/(6*(f + g*x)^3*(d*g - e*f)^2) + (b*e^2*g*n*x^2)/(3*(f + g
*x)^3*(d*g - e*f)^2) - (b*d*e*f*n)/(6*(f + g*x)^3*(d*g - e*f)^2) + (b*e^2*f^2*n)/(2*g*(f + g*x)^3*(d*g - e*f)^
2) - (b*d*e*g*n*x)/(6*(f + g*x)^3*(d*g - e*f)^2)

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