3.2.82 \(\int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx\) [182]

Optimal. Leaf size=250 \[ -\frac {b f}{2 d (f h-e i)^2 (h+i x)}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {b f^2 \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \]

[Out]

-1/2*b*f/d/(-e*i+f*h)^2/(i*x+h)-1/2*b*f^2*ln(f*x+e)/d/(-e*i+f*h)^3+1/2*(a+b*ln(c*(f*x+e)))/d/(-e*i+f*h)/(i*x+h
)^2-f*i*(f*x+e)*(a+b*ln(c*(f*x+e)))/d/(-e*i+f*h)^3/(i*x+h)+3/2*b*f^2*ln(i*x+h)/d/(-e*i+f*h)^3-f^2*(a+b*ln(c*(f
*x+e)))*ln(1+(-e*i+f*h)/i/(f*x+e))/d/(-e*i+f*h)^3+b*f^2*polylog(2,(e*i-f*h)/i/(f*x+e))/d/(-e*i+f*h)^3

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Rubi [A]
time = 0.37, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2458, 12, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} \frac {b f^2 \text {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {f^2 \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (h+i x)^2 (f h-e i)}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {b f}{2 d (h+i x) (f h-e i)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(e + f*x)])/((d*e + d*f*x)*(h + i*x)^3),x]

[Out]

-1/2*(b*f)/(d*(f*h - e*i)^2*(h + i*x)) - (b*f^2*Log[e + f*x])/(2*d*(f*h - e*i)^3) + (a + b*Log[c*(e + f*x)])/(
2*d*(f*h - e*i)*(h + i*x)^2) - (f*i*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*(f*h - e*i)^3*(h + i*x)) + (3*b*f^2
*Log[h + i*x])/(2*d*(f*h - e*i)^3) - (f^2*(a + b*Log[c*(e + f*x)])*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h
- e*i)^3) + (b*f^2*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rubi steps

\begin {align*} \int \frac {a+b \log (c (e+f x))}{(h+182 x)^3 (d e+d f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{d x \left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )^3} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )^3} \, dx,x,e+f x\right )}{d f}\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (182 e-f h)}+\frac {182 \text {Subst}\left (\int \frac {a+b \log (c x)}{\left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )^3} \, dx,x,e+f x\right )}{d f (182 e-f h)}\\ &=-\frac {a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}-\frac {182 \text {Subst}\left (\int \frac {a+b \log (c x)}{\left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (182 e-f h)^2}+\frac {f \text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )} \, dx,x,e+f x\right )}{d (182 e-f h)^2}+\frac {b \text {Subst}\left (\int \frac {1}{x \left (\frac {-182 e+f h}{f}+\frac {182 x}{f}\right )^2} \, dx,x,e+f x\right )}{2 d (182 e-f h)}\\ &=-\frac {a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}+\frac {182 f (e+f x) (a+b \log (c (e+f x)))}{d (182 e-f h)^3 (h+182 x)}+\frac {(182 f) \text {Subst}\left (\int \frac {a+b \log (c x)}{\frac {-182 e+f h}{f}+\frac {182 x}{f}} \, dx,x,e+f x\right )}{d (182 e-f h)^3}-\frac {(182 b f) \text {Subst}\left (\int \frac {1}{\frac {-182 e+f h}{f}+\frac {182 x}{f}} \, dx,x,e+f x\right )}{d (182 e-f h)^3}-\frac {f^2 \text {Subst}\left (\int \frac {a+b \log (c x)}{x} \, dx,x,e+f x\right )}{d (182 e-f h)^3}+\frac {b \text {Subst}\left (\int \left (\frac {182 f^2}{(182 e-f h) (182 e-f h-182 x)^2}+\frac {182 f^2}{(182 e-f h)^2 (182 e-f h-182 x)}+\frac {f^2}{(182 e-f h)^2 x}\right ) \, dx,x,e+f x\right )}{2 d (182 e-f h)}\\ &=-\frac {b f}{2 d (182 e-f h)^2 (h+182 x)}-\frac {3 b f^2 \log (h+182 x)}{2 d (182 e-f h)^3}+\frac {b f^2 \log (e+f x)}{2 d (182 e-f h)^3}-\frac {a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}+\frac {182 f (e+f x) (a+b \log (c (e+f x)))}{d (182 e-f h)^3 (h+182 x)}+\frac {f^2 \log \left (-\frac {f (h+182 x)}{182 e-f h}\right ) (a+b \log (c (e+f x)))}{d (182 e-f h)^3}-\frac {f^2 (a+b \log (c (e+f x)))^2}{2 b d (182 e-f h)^3}-\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {182 x}{-182 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (182 e-f h)^3}\\ &=-\frac {b f}{2 d (182 e-f h)^2 (h+182 x)}-\frac {3 b f^2 \log (h+182 x)}{2 d (182 e-f h)^3}+\frac {b f^2 \log (e+f x)}{2 d (182 e-f h)^3}-\frac {a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}+\frac {182 f (e+f x) (a+b \log (c (e+f x)))}{d (182 e-f h)^3 (h+182 x)}+\frac {f^2 \log \left (-\frac {f (h+182 x)}{182 e-f h}\right ) (a+b \log (c (e+f x)))}{d (182 e-f h)^3}-\frac {f^2 (a+b \log (c (e+f x)))^2}{2 b d (182 e-f h)^3}+\frac {b f^2 \text {Li}_2\left (\frac {182 (e+f x)}{182 e-f h}\right )}{d (182 e-f h)^3}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 226, normalized size = 0.90 \begin {gather*} \frac {\frac {(f h-e i)^2 (a+b \log (c (e+f x)))}{(h+i x)^2}+\frac {2 f (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac {f^2 (a+b \log (c (e+f x)))^2}{b}-2 b f^2 (\log (e+f x)-\log (h+i x))-\frac {b f (f h-e i+f (h+i x) \log (e+f x)-f (h+i x) \log (h+i x))}{h+i x}-2 f^2 (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )-2 b f^2 \text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )}{2 d (f h-e i)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(e + f*x)])/((d*e + d*f*x)*(h + i*x)^3),x]

[Out]

(((f*h - e*i)^2*(a + b*Log[c*(e + f*x)]))/(h + i*x)^2 + (2*f*(f*h - e*i)*(a + b*Log[c*(e + f*x)]))/(h + i*x) +
(f^2*(a + b*Log[c*(e + f*x)])^2)/b - 2*b*f^2*(Log[e + f*x] - Log[h + i*x]) - (b*f*(f*h - e*i + f*(h + i*x)*Lo
g[e + f*x] - f*(h + i*x)*Log[h + i*x]))/(h + i*x) - 2*f^2*(a + b*Log[c*(e + f*x)])*Log[(f*(h + i*x))/(f*h - e*
i)] - 2*b*f^2*PolyLog[2, (i*(e + f*x))/(-(f*h) + e*i)])/(2*d*(f*h - e*i)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(704\) vs. \(2(241)=482\).
time = 1.16, size = 705, normalized size = 2.82 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(f*x+e)))/(d*f*x+d*e)/(i*x+h)^3,x,method=_RETURNVERBOSE)

[Out]

1/c/f*(-c^2*f^3/d*a/(e*i-f*h)^2/(c*e*i-h*c*f-i*(c*f*x+c*e))+c*f^3/d*a/(e*i-f*h)^3*ln(c*e*i-h*c*f-i*(c*f*x+c*e)
)-1/2*c^3*f^3/d*a/(e*i-f*h)/(c*e*i-h*c*f-i*(c*f*x+c*e))^2-c*f^3/d*a/(e*i-f*h)^3*ln(c*f*x+c*e)+c*f^3/d*b/(e*i-f
*h)^3*dilog((-c*e*i+h*c*f+i*(c*f*x+c*e))/(-c*e*i+c*f*h))+c*f^3/d*b/(e*i-f*h)^3*ln(c*f*x+c*e)*ln((-c*e*i+h*c*f+
i*(c*f*x+c*e))/(-c*e*i+c*f*h))-3/2*c*f^3/d*b/(e*i-f*h)^3*ln(c*e*i-h*c*f-i*(c*f*x+c*e))-c*f^3/d*b*i/(e*i-f*h)^3
*ln(c*f*x+c*e)*(c*f*x+c*e)/(c*e*i-h*c*f-i*(c*f*x+c*e))-1/2*c*f^3/d*b*ln(c*f*x+c*e)^2/(e*i-f*h)^3+1/2*c^2*f^3/d
*b/(e*i-f*h)^3*i/(c*e*i-h*c*f-i*(c*f*x+c*e))*e-1/2*c^2*f^4/d*b/(e*i-f*h)^3/(c*e*i-h*c*f-i*(c*f*x+c*e))*h-c^2*f
^3/d*b/(e*i-f*h)^3*i^2*ln(c*f*x+c*e)*(c*f*x+c*e)/(c*e*i-h*c*f-i*(c*f*x+c*e))^2*e+c^2*f^4/d*b/(e*i-f*h)^3*i*ln(
c*f*x+c*e)*(c*f*x+c*e)/(c*e*i-h*c*f-i*(c*f*x+c*e))^2*h+1/2*c*f^3/d*b/(e*i-f*h)^3*i^2*ln(c*f*x+c*e)*(c*f*x+c*e)
^2/(c*e*i-h*c*f-i*(c*f*x+c*e))^2)

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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. 522 vs. \(2 (240) = 480\).
time = 0.62, size = 522, normalized size = 2.09 \begin {gather*} \frac {i \, {\left (\log \left (f x + e\right ) \log \left (-\frac {f x + e}{i \, f h + e} + 1\right ) + {\rm Li}_2\left (\frac {f x + e}{i \, f h + e}\right )\right )} b f^{2}}{-i \, d f^{3} h^{3} - 3 \, d f^{2} h^{2} e + 3 i \, d f h e^{2} + d e^{3}} + \frac {{\left (2 i \, a f^{2} + {\left (2 i \, f^{2} \log \left (c\right ) - 3 i \, f^{2}\right )} b\right )} \log \left (-2 i \, h + 2 \, x\right )}{-2 i \, d f^{3} h^{3} - 6 \, d f^{2} h^{2} e + 6 i \, d f h e^{2} + 2 \, d e^{3}} + \frac {16 \, {\left (3 i \, a f^{2} h^{2} + {\left (i \, b f^{2} h^{2} - 2 \, b f^{2} h x - i \, b f^{2} x^{2}\right )} \log \left (f x + e\right )^{2} + {\left (3 i \, f^{2} h^{2} \log \left (c\right ) - i \, f^{2} h^{2}\right )} b - {\left (2 \, a f^{2} h + {\left (2 \, f^{2} h \log \left (c\right ) - f^{2} h\right )} b - {\left ({\left (2 i \, f \log \left (c\right ) - i \, f\right )} b + 2 i \, a f\right )} e\right )} x + {\left (-i \, b \log \left (c\right ) - i \, a\right )} e^{2} + {\left (4 \, a f h + {\left (4 \, f h \log \left (c\right ) - f h\right )} b\right )} e + {\left (2 i \, b f^{2} h^{2} \log \left (c\right ) + 2 i \, a f^{2} h^{2} + 4 \, b f h e + {\left (-2 i \, a f^{2} + {\left (-2 i \, f^{2} \log \left (c\right ) + 3 i \, f^{2}\right )} b\right )} x^{2} - 2 \, {\left (2 \, a f^{2} h - i \, b f e + 2 \, {\left (f^{2} h \log \left (c\right ) - f^{2} h\right )} b\right )} x - i \, b e^{2}\right )} \log \left (f x + e\right )\right )}}{32 i \, d f^{3} h^{5} + 96 \, d f^{2} h^{4} e - 96 i \, d f h^{3} e^{2} - 32 \, d h^{2} e^{3} - 32 \, {\left (i \, d f^{3} h^{3} + 3 \, d f^{2} h^{2} e - 3 i \, d f h e^{2} - d e^{3}\right )} x^{2} - 64 \, {\left (d f^{3} h^{4} - 3 i \, d f^{2} h^{3} e - 3 \, d f h^{2} e^{2} + i \, d h e^{3}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="maxima")

[Out]

I*(log(f*x + e)*log(-(f*x + e)/(I*f*h + e) + 1) + dilog((f*x + e)/(I*f*h + e)))*b*f^2/(-I*d*f^3*h^3 - 3*d*f^2*
h^2*e + 3*I*d*f*h*e^2 + d*e^3) + (2*I*a*f^2 + (2*I*f^2*log(c) - 3*I*f^2)*b)*log(-2*I*h + 2*x)/(-2*I*d*f^3*h^3
- 6*d*f^2*h^2*e + 6*I*d*f*h*e^2 + 2*d*e^3) + 16*(3*I*a*f^2*h^2 + (I*b*f^2*h^2 - 2*b*f^2*h*x - I*b*f^2*x^2)*log
(f*x + e)^2 + (3*I*f^2*h^2*log(c) - I*f^2*h^2)*b - (2*a*f^2*h + (2*f^2*h*log(c) - f^2*h)*b - ((2*I*f*log(c) -
I*f)*b + 2*I*a*f)*e)*x + (-I*b*log(c) - I*a)*e^2 + (4*a*f*h + (4*f*h*log(c) - f*h)*b)*e + (2*I*b*f^2*h^2*log(c
) + 2*I*a*f^2*h^2 + 4*b*f*h*e + (-2*I*a*f^2 + (-2*I*f^2*log(c) + 3*I*f^2)*b)*x^2 - 2*(2*a*f^2*h - I*b*f*e + 2*
(f^2*h*log(c) - f^2*h)*b)*x - I*b*e^2)*log(f*x + e))/(32*I*d*f^3*h^5 + 96*d*f^2*h^4*e - 96*I*d*f*h^3*e^2 - 32*
d*h^2*e^3 - 32*(I*d*f^3*h^3 + 3*d*f^2*h^2*e - 3*I*d*f*h*e^2 - d*e^3)*x^2 - 64*(d*f^3*h^4 - 3*I*d*f^2*h^3*e - 3
*d*f*h^2*e^2 + I*d*h*e^3)*x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="fricas")

[Out]

integral((I*b*log(c*f*x + c*e) + I*a)/(I*d*f*h^3*x - 3*d*f*h^2*x^2 - 3*I*d*f*h*x^3 + d*f*x^4 + (I*d*h^3 - 3*d*
h^2*x - 3*I*d*h*x^2 + d*x^3)*e), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(f*x+e)))/(d*f*x+d*e)/(i*x+h)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="giac")

[Out]

integrate((b*log((f*x + e)*c) + a)/((d*f*x + d*e)*(h + I*x)^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,\left (e+f\,x\right )\right )}{{\left (h+i\,x\right )}^3\,\left (d\,e+d\,f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(e + f*x)))/((h + i*x)^3*(d*e + d*f*x)),x)

[Out]

int((a + b*log(c*(e + f*x)))/((h + i*x)^3*(d*e + d*f*x)), x)

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