3.2.80 \(\int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)} \, dx\) [180]

Optimal. Leaf size=87 \[ -\frac {(a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}+\frac {b \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2458, 12, 2379, 2438} \begin {gather*} \frac {b \text {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}-\frac {\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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 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+180 x) (d e+d f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{d x \left (\frac {-180 e+f h}{f}+\frac {180 x}{f}\right )} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {-180 e+f h}{f}+\frac {180 x}{f}\right )} \, dx,x,e+f x\right )}{d f}\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x} \, dx,x,e+f x\right )}{d (180 e-f h)}+\frac {180 \text {Subst}\left (\int \frac {a+b \log (c x)}{\frac {-180 e+f h}{f}+\frac {180 x}{f}} \, dx,x,e+f x\right )}{d f (180 e-f h)}\\ &=\frac {\log \left (-\frac {f (h+180 x)}{180 e-f h}\right ) (a+b \log (c (e+f x)))}{d (180 e-f h)}-\frac {(a+b \log (c (e+f x)))^2}{2 b d (180 e-f h)}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {180 x}{-180 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (180 e-f h)}\\ &=\frac {\log \left (-\frac {f (h+180 x)}{180 e-f h}\right ) (a+b \log (c (e+f x)))}{d (180 e-f h)}-\frac {(a+b \log (c (e+f x)))^2}{2 b d (180 e-f h)}+\frac {b \text {Li}_2\left (\frac {180 (e+f x)}{180 e-f h}\right )}{d (180 e-f h)}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 91, normalized size = 1.05 \begin {gather*} \frac {(a+b \log (c (e+f x))) \left (a+b \log (c (e+f x))-2 b \log \left (\frac {f (h+i x)}{f h-e i}\right )\right )-2 b^2 \text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )}{2 b d (f h-e i)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(86)=172\).
time = 1.08, size = 215, normalized size = 2.47

method	result	size
risch	\(-\frac {a \ln \left (f x +e \right )}{d \left (e i -f h \right )}+\frac {a \ln \left (i x +h \right )}{d \left (e i -f h \right )}-\frac {b \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )}+\frac {b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}\)	\(179\)
derivativedivides	\(\frac {\frac {c f a \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )}-\frac {c f a \ln \left (c f x +c e \right )}{d \left (e i -f h \right )}+\frac {c f b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {c f b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {c f b \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )}}{c f}\)	\(215\)
default	\(\frac {\frac {c f a \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )}-\frac {c f a \ln \left (c f x +c e \right )}{d \left (e i -f h \right )}+\frac {c f b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {c f b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {c f b \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )}}{c f}\)	\(215\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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

[Out]

a*(log(f*x + e)/(d*f*h - I*d*e) - log(h + I*x)/(d*f*h - I*d*e)) - b*integrate((I*log(f*x + e) + I*log(c))/(d*f
*x^2 - I*d*h*e + (-I*d*f*h + d*e)*x), 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),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{e h + e i x + f h x + f i x^{2}}\, dx + \int \frac {b \log {\left (c e + c f x \right )}}{e h + e i x + f h x + f i x^{2}}\, dx}{d} \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),x)

[Out]

(Integral(a/(e*h + e*i*x + f*h*x + f*i*x**2), x) + Integral(b*log(c*e + c*f*x)/(e*h + e*i*x + f*h*x + f*i*x**2
), x))/d

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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),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,\left (e+f\,x\right )\right )}{\left (h+i\,x\right )\,\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)*(d*e + d*f*x)),x)

[Out]

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

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