3.189 \(\int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx\)

Optimal. Leaf size=273 \[ \frac {2 b f \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}+\frac {2 b f \log \left (\frac {f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}-\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^2}-\frac {f \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2}+\frac {2 b^2 f \text {Li}_2\left (-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \text {Li}_3\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2} \]

[Out]

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

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Rubi [A]  time = 0.64, antiderivative size = 300, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2411, 12, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac {2 b f \text {PolyLog}\left (2,-\frac {i (e+f x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}+\frac {2 b^2 f \text {PolyLog}\left (2,-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \text {PolyLog}\left (3,-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac {f (a+b \log (c (e+f x)))^3}{3 b d (f h-e i)^2}-\frac {f \log \left (\frac {f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2}-\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^2}+\frac {2 b f \log \left (\frac {f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2317

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

Rule 2318

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

Rule 2344

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

Rule 2347

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2391

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 2411

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]

Rule 6589

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

Rubi steps

\begin {align*} \int \frac {(a+b \log (c (e+f x)))^2}{(h+189 x)^2 (d e+d f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b \log (c x))^2}{d x \left (\frac {-189 e+f h}{f}+\frac {189 x}{f}\right )^2} \, dx,x,e+f x\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b \log (c x))^2}{x \left (\frac {-189 e+f h}{f}+\frac {189 x}{f}\right )^2} \, dx,x,e+f x\right )}{d f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b \log (c x))^2}{x \left (\frac {-189 e+f h}{f}+\frac {189 x}{f}\right )} \, dx,x,e+f x\right )}{d (189 e-f h)}+\frac {189 \operatorname {Subst}\left (\int \frac {(a+b \log (c x))^2}{\left (\frac {-189 e+f h}{f}+\frac {189 x}{f}\right )^2} \, dx,x,e+f x\right )}{d f (189 e-f h)}\\ &=-\frac {189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac {189 \operatorname {Subst}\left (\int \frac {(a+b \log (c x))^2}{\frac {-189 e+f h}{f}+\frac {189 x}{f}} \, dx,x,e+f x\right )}{d (189 e-f h)^2}+\frac {(378 b) \operatorname {Subst}\left (\int \frac {a+b \log (c x)}{\frac {-189 e+f h}{f}+\frac {189 x}{f}} \, dx,x,e+f x\right )}{d (189 e-f h)^2}+\frac {f \operatorname {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}\\ &=\frac {2 b f \log \left (-\frac {f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))}{d (189 e-f h)^2}-\frac {189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac {f \log \left (-\frac {f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2}+\frac {f \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d (189 e-f h)^2}+\frac {(2 b f) \operatorname {Subst}\left (\int \frac {(a+b \log (c x)) \log \left (1+\frac {189 x}{-189 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}-\frac {\left (2 b^2 f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {189 x}{-189 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}\\ &=\frac {2 b f \log \left (-\frac {f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))}{d (189 e-f h)^2}-\frac {189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac {f \log \left (-\frac {f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2}+\frac {f (a+b \log (c (e+f x)))^3}{3 b d (189 e-f h)^2}+\frac {2 b^2 f \text {Li}_2\left (\frac {189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}-\frac {2 b f (a+b \log (c (e+f x))) \text {Li}_2\left (\frac {189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}+\frac {\left (2 b^2 f\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {189 x}{-189 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}\\ &=\frac {2 b f \log \left (-\frac {f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))}{d (189 e-f h)^2}-\frac {189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac {f \log \left (-\frac {f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2}+\frac {f (a+b \log (c (e+f x)))^3}{3 b d (189 e-f h)^2}+\frac {2 b^2 f \text {Li}_2\left (\frac {189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}-\frac {2 b f (a+b \log (c (e+f x))) \text {Li}_2\left (\frac {189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}+\frac {2 b^2 f \text {Li}_3\left (\frac {189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}\\ \end {align*}

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Mathematica [A]  time = 0.54, size = 360, normalized size = 1.32 \[ \frac {3 a^2 f (h+i x) \log (e+f x)+3 a^2 (f h-e i)-3 a^2 f (h+i x) \log (h+i x)+3 a b \left (-2 f (h+i x) \left (\log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+\text {Li}_2\left (\frac {i (e+f x)}{e i-f h}\right )\right )+f (h+i x) \log ^2(c (e+f x))+2 (f h-e i) \log (c (e+f x))-2 f (h+i x) \log (e+f x)+2 f (h+i x) \log (h+i x)\right )+b^2 \left (-6 f (h+i x) (\log (c (e+f x))-1) \text {Li}_2\left (\frac {i (e+f x)}{e i-f h}\right )+\log (c (e+f x)) \left (f (h+i x) \log ^2(c (e+f x))-3 \log (c (e+f x)) \left (f (h+i x) \log \left (\frac {f (h+i x)}{f h-e i}\right )+i (e+f x)\right )+6 f (h+i x) \log \left (\frac {f (h+i x)}{f h-e i}\right )\right )+6 f (h+i x) \text {Li}_3\left (\frac {i (e+f x)}{e i-f h}\right )\right )}{3 d (h+i x) (f h-e i)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*(f*h - e*i) + 3*a^2*f*(h + i*x)*Log[e + f*x] - 3*a^2*f*(h + i*x)*Log[h + i*x] + 3*a*b*(-2*f*(h + i*x)*L
og[e + f*x] + 2*(f*h - e*i)*Log[c*(e + f*x)] + f*(h + i*x)*Log[c*(e + f*x)]^2 + 2*f*(h + i*x)*Log[h + i*x] - 2
*f*(h + i*x)*(Log[c*(e + f*x)]*Log[(f*(h + i*x))/(f*h - e*i)] + PolyLog[2, (i*(e + f*x))/(-(f*h) + e*i)])) + b
^2*(Log[c*(e + f*x)]*(f*(h + i*x)*Log[c*(e + f*x)]^2 + 6*f*(h + i*x)*Log[(f*(h + i*x))/(f*h - e*i)] - 3*Log[c*
(e + f*x)]*(i*(e + f*x) + f*(h + i*x)*Log[(f*(h + i*x))/(f*h - e*i)])) - 6*f*(h + i*x)*(-1 + Log[c*(e + f*x)])
*PolyLog[2, (i*(e + f*x))/(-(f*h) + e*i)] + 6*f*(h + i*x)*PolyLog[3, (i*(e + f*x))/(-(f*h) + e*i)]))/(3*d*(f*h
- e*i)^2*(h + i*x))

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (c f x + c e\right )^{2} + 2 \, a b \log \left (c f x + c e\right ) + a^{2}}{d f i^{2} x^{3} + d e h^{2} + {\left (2 \, d f h i + d e i^{2}\right )} x^{2} + {\left (d f h^{2} + 2 \, d e h i\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (f x +e \right ) c \right )+a \right )^{2}}{\left (d f x +d e \right ) \left (i x +h \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e)/(i*x+h)^2,x)

[Out]

int((a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e)/(i*x+h)^2,x)

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maxima [B]  time = 0.90, size = 622, normalized size = 2.28 \[ a^{2} {\left (\frac {f \log \left (f x + e\right )}{d f^{2} h^{2} - 2 \, d e f h i + d e^{2} i^{2}} - \frac {f \log \left (i x + h\right )}{d f^{2} h^{2} - 2 \, d e f h i + d e^{2} i^{2}} + \frac {1}{d f h^{2} - d e h i + {\left (d f h i - d e i^{2}\right )} x}\right )} - \frac {{\left (\log \left (f x + e\right )^{2} \log \left (\frac {f i x + e i}{f h - e i} + 1\right ) + 2 \, {\rm Li}_2\left (-\frac {f i x + e i}{f h - e i}\right ) \log \left (f x + e\right ) - 2 \, {\rm Li}_{3}(-\frac {f i x + e i}{f h - e i})\right )} b^{2} f}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} + \frac {3 \, {\left (f h - e i\right )} b^{2} \log \relax (c)^{2} + {\left (b^{2} f i x + b^{2} f h\right )} \log \left (f x + e\right )^{3} + 6 \, {\left (f h - e i\right )} a b \log \relax (c) + 3 \, {\left (a b f h + {\left (f h \log \relax (c) - e i\right )} b^{2} + {\left (a b f i + {\left (f i \log \relax (c) - f i\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )^{2} + 3 \, {\left (2 \, {\left (f h \log \relax (c) - e i\right )} a b + {\left (f h \log \relax (c)^{2} - 2 \, e i \log \relax (c)\right )} b^{2} + {\left (2 \, {\left (f i \log \relax (c) - f i\right )} a b + {\left (f i \log \relax (c)^{2} - 2 \, f i \log \relax (c)\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )}{3 \, {\left ({\left (f^{2} h^{2} i - 2 \, e f h i^{2} + e^{2} i^{3}\right )} d x + {\left (f^{2} h^{3} - 2 \, e f h^{2} i + e^{2} h i^{2}\right )} d\right )}} - \frac {2 \, {\left ({\left (f \log \relax (c) - f\right )} b^{2} + a b f\right )} {\left (\log \left (f x + e\right ) \log \left (\frac {f i x + e i}{f h - e i} + 1\right ) + {\rm Li}_2\left (-\frac {f i x + e i}{f h - e i}\right )\right )}}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} - \frac {{\left (2 \, {\left (f \log \relax (c) - f\right )} a b + {\left (f \log \relax (c)^{2} - 2 \, f \log \relax (c)\right )} b^{2}\right )} \log \left (i x + h\right )}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(f*log(f*x + e)/(d*f^2*h^2 - 2*d*e*f*h*i + d*e^2*i^2) - f*log(i*x + h)/(d*f^2*h^2 - 2*d*e*f*h*i + d*e^2*i^
2) + 1/(d*f*h^2 - d*e*h*i + (d*f*h*i - d*e*i^2)*x)) - (log(f*x + e)^2*log((f*i*x + e*i)/(f*h - e*i) + 1) + 2*d
ilog(-(f*i*x + e*i)/(f*h - e*i))*log(f*x + e) - 2*polylog(3, -(f*i*x + e*i)/(f*h - e*i)))*b^2*f/((f^2*h^2 - 2*
e*f*h*i + e^2*i^2)*d) + 1/3*(3*(f*h - e*i)*b^2*log(c)^2 + (b^2*f*i*x + b^2*f*h)*log(f*x + e)^3 + 6*(f*h - e*i)
*a*b*log(c) + 3*(a*b*f*h + (f*h*log(c) - e*i)*b^2 + (a*b*f*i + (f*i*log(c) - f*i)*b^2)*x)*log(f*x + e)^2 + 3*(
2*(f*h*log(c) - e*i)*a*b + (f*h*log(c)^2 - 2*e*i*log(c))*b^2 + (2*(f*i*log(c) - f*i)*a*b + (f*i*log(c)^2 - 2*f
*i*log(c))*b^2)*x)*log(f*x + e))/((f^2*h^2*i - 2*e*f*h*i^2 + e^2*i^3)*d*x + (f^2*h^3 - 2*e*f*h^2*i + e^2*h*i^2
)*d) - 2*((f*log(c) - f)*b^2 + a*b*f)*(log(f*x + e)*log((f*i*x + e*i)/(f*h - e*i) + 1) + dilog(-(f*i*x + e*i)/
(f*h - e*i)))/((f^2*h^2 - 2*e*f*h*i + e^2*i^2)*d) - (2*(f*log(c) - f)*a*b + (f*log(c)^2 - 2*f*log(c))*b^2)*log
(i*x + h)/((f^2*h^2 - 2*e*f*h*i + e^2*i^2)*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^2}{{\left (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a^{2}}{e h^{2} + 2 e h i x + e i^{2} x^{2} + f h^{2} x + 2 f h i x^{2} + f i^{2} x^{3}}\, dx + \int \frac {b^{2} \log {\left (c e + c f x \right )}^{2}}{e h^{2} + 2 e h i x + e i^{2} x^{2} + f h^{2} x + 2 f h i x^{2} + f i^{2} x^{3}}\, dx + \int \frac {2 a b \log {\left (c e + c f x \right )}}{e h^{2} + 2 e h i x + e i^{2} x^{2} + f h^{2} x + 2 f h i x^{2} + f i^{2} x^{3}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2/(e*h**2 + 2*e*h*i*x + e*i**2*x**2 + f*h**2*x + 2*f*h*i*x**2 + f*i**2*x**3), x) + Integral(b**2*
log(c*e + c*f*x)**2/(e*h**2 + 2*e*h*i*x + e*i**2*x**2 + f*h**2*x + 2*f*h*i*x**2 + f*i**2*x**3), x) + Integral(
2*a*b*log(c*e + c*f*x)/(e*h**2 + 2*e*h*i*x + e*i**2*x**2 + f*h**2*x + 2*f*h*i*x**2 + f*i**2*x**3), x))/d

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