3.2.1 \(\int \frac {f+g x}{(a+b \log (c (d+e x)^n))^3} \, dx\) [101]
Optimal. Leaf size=261 \[ \frac {e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac {2 e^{-\frac {2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
[Out]
1/2*(-d*g+e*f)*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^2/exp(a/b/n)/n^3/((c*(e*x+d)^n)^(1/n))+2*g*(e*x+d)^
2*Ei(2*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^2/exp(2*a/b/n)/n^3/((c*(e*x+d)^n)^(2/n))-1/2*(e*x+d)*(g*x+f)/b/e/n/(a+
b*ln(c*(e*x+d)^n))^2+1/2*(-d*g+e*f)*(e*x+d)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))-(e*x+d)*(g*x+f)/b^2/e/n^2/(a+b*l
n(c*(e*x+d)^n))
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Rubi [A]
time = 0.26, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2447, 2446,
2436, 2337, 2209, 2437, 2347, 2334} \begin {gather*} \frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac {2 g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}+\frac {(d+e x) (e f-d g)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f + g*x)/(a + b*Log[c*(d + e*x)^n])^3,x]
[Out]
((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(2*b^3*e^2*E^(a/(b*n))*n^3*(c*(d + e*x
)^n)^n^(-1)) + (2*g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^3*e^2*E^((2*a)/(b*n))*
n^3*(c*(d + e*x)^n)^(2/n)) - ((d + e*x)*(f + g*x))/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) + ((e*f - d*g)*(d +
e*x))/(2*b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])) - ((d + e*x)*(f + g*x))/(b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])
)
Rule 2209
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Rule 2334
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]
Rule 2337
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]
Rule 2347
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]
Rule 2436
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
Rule 2437
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]
Rule 2446
Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]
Rule 2447
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(d
+ e*x)*(f + g*x)^q*((a + b*Log[c*(d + e*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[q*((e*f - d*g)/(b*e*n*(p + 1))), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]
Rubi steps
\begin {align*} \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {\int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{b n}-\frac {(e f-d g) \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{2 b e n}\\ &=-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 n^2}-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}-\frac {(e f-d g) \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{2 b e^2 n}\\ &=-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b^2 n^2}-\frac {(e f-d g) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^2 n^2}-\frac {(e f-d g) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^2 n^2}\\ &=-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(2 g) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}+\frac {(2 (e f-d g)) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}-\frac {\left ((e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^2 n^3}-\frac {\left ((e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^2 n^3}\\ &=-\frac {3 e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(2 g) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^2 n^2}+\frac {(2 (e f-d g)) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^2 n^2}\\ &=-\frac {3 e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (2 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^2 n^3}+\frac {\left (2 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^2 n^3}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac {2 e^{-\frac {2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 256, normalized size = 0.98 \begin {gather*} -\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-4 g (d+e x) \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b e^{\frac {2 a}{b n}} n \left (c (d+e x)^n\right )^{2/n} \left (b e n (f+g x)+a (e f+d g+2 e g x)+b (d g+e (f+2 g x)) \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e^2 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)/(a + b*Log[c*(d + e*x)^n])^3,x]
[Out]
-1/2*((d + e*x)*(-(E^(a/(b*n))*(e*f - d*g)*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*
n)]*(a + b*Log[c*(d + e*x)^n])^2) - 4*g*(d + e*x)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*L
og[c*(d + e*x)^n])^2 + b*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n)*(b*e*n*(f + g*x) + a*(e*f + d*g + 2*e*g*x) +
b*(d*g + e*(f + 2*g*x))*Log[c*(d + e*x)^n])))/(b^3*e^2*E^((2*a)/(b*n))*n^3*(c*(d + e*x)^n)^(2/n)*(a + b*Log[c*
(d + e*x)^n])^2)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.80, size = 3114, normalized size = 11.93
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(3114\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)
[Out]
-(2*b*d*e*f*n+6*a*d*e*g*x+I*Pi*b*d^2*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-3*I*Pi*b*d*e*g*x*csgn(I*c)*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)+2*a*d^2*g+2*a*d*e*f+4*a*e^2*g*x^2+2*a*e^2*f*x+2*ln(c)*b*d*e*f+I*Pi*b*d*e*f*csgn(I*c)
*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*e^2*f*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*e^2*f*x*csgn(I*(e*x+d)^n)*csgn(I*
c*(e*x+d)^n)^2-3*I*Pi*b*d*e*g*x*csgn(I*c*(e*x+d)^n)^3-2*I*Pi*b*e^2*g*x^2*csgn(I*c*(e*x+d)^n)^3-I*Pi*b*e^2*f*x*
csgn(I*c*(e*x+d)^n)^3-I*Pi*b*d*e*f*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*I*Pi*b*e^2*g*x^2*csgn(I*(
e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-2*I*Pi*b*e^2*g*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+3*I*Pi*b*d*
e*g*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*Pi*b*d*e*g*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+2*b*e^2*g*n*x^2
+2*b*e^2*f*n*x-I*Pi*b*e^2*f*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*b*e^2*f*x*ln((e*x+d)^n)+2*b*d*
e*f*ln((e*x+d)^n)+4*b*e^2*g*x^2*ln((e*x+d)^n)+I*Pi*b*d^2*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+2*b*d*e*g*n
*x+6*b*d*e*g*x*ln((e*x+d)^n)+4*ln(c)*b*e^2*g*x^2+2*ln(c)*b*e^2*f*x+I*Pi*b*d*e*f*csgn(I*(e*x+d)^n)*csgn(I*c*(e*
x+d)^n)^2-I*Pi*b*d^2*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*I*Pi*b*e^2*g*x^2*csgn(I*c)*csgn(I*c*(
e*x+d)^n)^2+2*ln(c)*b*d^2*g-I*Pi*b*d*e*f*csgn(I*c*(e*x+d)^n)^3+2*b*d^2*g*ln((e*x+d)^n)-I*Pi*b*d^2*g*csgn(I*c*(
e*x+d)^n)^3+6*ln(c)*b*d*e*g*x)/b^2/e^2/n^2/(2*a+2*b*ln(c)+2*b*ln((e*x+d)^n)-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b
*Pi*csgn(I*c*(e*x+d)^n)^3)^2-1/2/b^3/n^3*f*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x
+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)
^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e
*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c
*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*x-1/2/b^3/n^3/e*f*c^(-1/n)*((e*x+d)^n)^(-1/n
)*exp(-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b
*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*P
i*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)
^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d-2
/b^3/n^3*g*c^(-2/n)*((e*x+d)^n)^(-2/n)*exp(-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*cs
gn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)
/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*
x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d
)^n)-n*ln(e*x+d))+2*a)/b/n)*x^2-4/b^3/n^3/e*g*c^(-2/n)*((e*x+d)^n)^(-2/n)*exp(-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^
2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+
d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e
*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d*x-2/b^3/n^3/e^2*g*c^(-2/n)*((e*x+d)^n)^(-2/n)
*exp(-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(
I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csg
n(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d^2+1/2/b^
3/n^3/e*d*g*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*
Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3
+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(
I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln
((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*x+1/2/b^3/n^3/e^2*d^2*g*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(
I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csg
n(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d
)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2
-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")
[Out]
-1/2*(b*d^2*g*log(c) + a*d^2*g + ((g*n + 2*g*log(c))*b + 2*a*g)*x^2*e^2 + (((f*n + f*log(c))*b + a*f)*e^2 + (3
*a*d*g + (d*g*n + 3*d*g*log(c))*b)*e)*x + (a*d*f + (d*f*n + d*f*log(c))*b)*e + (2*b*g*x^2*e^2 + b*d^2*g + b*d*
f*e + (3*b*d*g*e + b*f*e^2)*x)*log((x*e + d)^n))/(b^4*n^2*e^2*log((x*e + d)^n)^2 + 2*(b^4*n^2*log(c) + a*b^3*n
^2)*e^2*log((x*e + d)^n) + (b^4*n^2*log(c)^2 + 2*a*b^3*n^2*log(c) + a^2*b^2*n^2)*e^2) + integrate(1/2*(4*g*x*e
+ 3*d*g + f*e)/(b^3*n^2*e*log((x*e + d)^n) + (b^3*n^2*log(c) + a*b^2*n^2)*e), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs.
\(2 (264) = 528\).
time = 0.40, size = 594, normalized size = 2.28 \begin {gather*} -\frac {{\left ({\left (a^{2} d g - a^{2} f e + {\left (b^{2} d g n^{2} - b^{2} f n^{2} e\right )} \log \left (x e + d\right )^{2} + {\left (b^{2} d g - b^{2} f e\right )} \log \left (c\right )^{2} + 2 \, {\left (a b d g n - a b f n e + {\left (b^{2} d g n - b^{2} f n e\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 2 \, {\left (a b d g - a b f e\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (x e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) + {\left (a b d^{2} g n + {\left ({\left (b^{2} g n^{2} + 2 \, a b g n\right )} x^{2} + {\left (b^{2} f n^{2} + a b f n\right )} x\right )} e^{2} + {\left (b^{2} d f n^{2} + a b d f n + {\left (b^{2} d g n^{2} + 3 \, a b d g n\right )} x\right )} e + {\left (b^{2} d^{2} g n^{2} + {\left (2 \, b^{2} g n^{2} x^{2} + b^{2} f n^{2} x\right )} e^{2} + {\left (3 \, b^{2} d g n^{2} x + b^{2} d f n^{2}\right )} e\right )} \log \left (x e + d\right ) + {\left (b^{2} d^{2} g n + {\left (2 \, b^{2} g n x^{2} + b^{2} f n x\right )} e^{2} + {\left (3 \, b^{2} d g n x + b^{2} d f n\right )} e\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 4 \, {\left (b^{2} g n^{2} \log \left (x e + d\right )^{2} + b^{2} g \log \left (c\right )^{2} + 2 \, a b g \log \left (c\right ) + a^{2} g + 2 \, {\left (b^{2} g n \log \left (c\right ) + a b g n\right )} \log \left (x e + d\right )\right )} \operatorname {log\_integral}\left ({\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} n^{5} e^{2} \log \left (x e + d\right )^{2} + b^{5} n^{3} e^{2} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e^{2} \log \left (c\right ) + a^{2} b^{3} n^{3} e^{2} + 2 \, {\left (b^{5} n^{4} e^{2} \log \left (c\right ) + a b^{4} n^{4} e^{2}\right )} \log \left (x e + d\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")
[Out]
-1/2*((a^2*d*g - a^2*f*e + (b^2*d*g*n^2 - b^2*f*n^2*e)*log(x*e + d)^2 + (b^2*d*g - b^2*f*e)*log(c)^2 + 2*(a*b*
d*g*n - a*b*f*n*e + (b^2*d*g*n - b^2*f*n*e)*log(c))*log(x*e + d) + 2*(a*b*d*g - a*b*f*e)*log(c))*e^((b*log(c)
+ a)/(b*n))*log_integral((x*e + d)*e^((b*log(c) + a)/(b*n))) + (a*b*d^2*g*n + ((b^2*g*n^2 + 2*a*b*g*n)*x^2 + (
b^2*f*n^2 + a*b*f*n)*x)*e^2 + (b^2*d*f*n^2 + a*b*d*f*n + (b^2*d*g*n^2 + 3*a*b*d*g*n)*x)*e + (b^2*d^2*g*n^2 + (
2*b^2*g*n^2*x^2 + b^2*f*n^2*x)*e^2 + (3*b^2*d*g*n^2*x + b^2*d*f*n^2)*e)*log(x*e + d) + (b^2*d^2*g*n + (2*b^2*g
*n*x^2 + b^2*f*n*x)*e^2 + (3*b^2*d*g*n*x + b^2*d*f*n)*e)*log(c))*e^(2*(b*log(c) + a)/(b*n)) - 4*(b^2*g*n^2*log
(x*e + d)^2 + b^2*g*log(c)^2 + 2*a*b*g*log(c) + a^2*g + 2*(b^2*g*n*log(c) + a*b*g*n)*log(x*e + d))*log_integra
l((x^2*e^2 + 2*d*x*e + d^2)*e^(2*(b*log(c) + a)/(b*n))))*e^(-2*(b*log(c) + a)/(b*n))/(b^5*n^5*e^2*log(x*e + d)
^2 + b^5*n^3*e^2*log(c)^2 + 2*a*b^4*n^3*e^2*log(c) + a^2*b^3*n^3*e^2 + 2*(b^5*n^4*e^2*log(c) + a*b^4*n^4*e^2)*
log(x*e + d))
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f + g x}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(a+b*ln(c*(e*x+d)**n))**3,x)
[Out]
Integral((f + g*x)/(a + b*log(c*(d + e*x)**n))**3, x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4114 vs.
\(2 (264) = 528\).
time = 5.28, size = 4114, normalized size = 15.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")
[Out]
-(x*e + d)^2*b^2*g*n^2*e*log(x*e + d)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^
4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e + d)*b^2*
d*g*n^2*e*log(x*e + d)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x
*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - 1/2*b^2*d*g*n^2*Ei(log(c)/n + a/(
b*n) + log(x*e + d))*e^(-a/(b*n) + 1)*log(x*e + d)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)
*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1
/n)) - 1/2*(x*e + d)^2*b^2*g*n^2*e/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n
^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e + d)*b^2*d*g
*n^2*e/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^
3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - 1/2*(x*e + d)*b^2*f*n^2*e^2*log(x*e + d)/(b^5*n^5
*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2
+ 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*b^2*f*n^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n)
+ 2)*log(x*e + d)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e
+ d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 2*b^2*g*n^2*Ei(2*log(c)/n
+ 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)*log(x*e + d)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*l
og(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^
3*e^3)*c^(2/n)) - (x*e + d)^2*b^2*g*n*e*log(c)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c)
+ 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e
+ d)*b^2*d*g*n*e*log(c)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(
x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - b^2*d*g*n*Ei(log(c)/n + a/(b*n)
+ log(x*e + d))*e^(-a/(b*n) + 1)*log(x*e + d)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)
*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1
/n)) - 1/2*(x*e + d)*b^2*f*n^2*e^2/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n
^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - (x*e + d)^2*a*b*g*n*e
/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*
log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e + d)*a*b*d*g*n*e/(b^5*n^5*e^3*log(x*e + d)^2 +
2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log
(c) + a^2*b^3*n^3*e^3) - a*b*d*g*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 1)*log(x*e + d)/((b^5*n
^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^
2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) - 1/2*(x*e + d)*b^2*f*n*e^2*log(c)/(b^5*n^5*e^3*log(x*e
+ d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^
3*e^3*log(c) + a^2*b^3*n^3*e^3) + b^2*f*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 2)*log(x*e + d)*
log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n
^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 4*b^2*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2
*log(x*e + d))*e^(-2*a/(b*n) + 1)*log(x*e + d)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d
)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(
2/n)) - 1/2*b^2*d*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 1)*log(c)^2/((b^5*n^5*e^3*log(x*e + d)
^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3
*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) - 1/2*(x*e + d)*a*b*f*n*e^2/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*lo
g(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3
*e^3) + a*b*f*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 2)*log(x*e + d)/((b^5*n^5*e^3*log(x*e + d)
^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3
*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 4*a*b*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)
*log(x*e + d)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d)
+ b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(2/n)) - a*b*d*g*Ei(log(c)/n + a/(b*n) +
log(x*e + d))*e^(-a/(b*n) + 1)*log(c)/((b^5*n^5...
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {f+g\,x}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)/(a + b*log(c*(d + e*x)^n))^3,x)
[Out]
int((f + g*x)/(a + b*log(c*(d + e*x)^n))^3, x)
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