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

Optimal. Leaf size=238 \[ \frac{(f h-e i)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3}+\frac{2 i (e+f x) (f h-e i) (a+b \log (c (e+f x)))^2}{d f^3}+\frac{i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}-\frac{b i^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}-\frac{4 a b i x (f h-e i)}{d f^2}-\frac{4 b^2 i (e+f x) (f h-e i) \log (c (e+f x))}{d f^3}+\frac{4 b^2 i x (f h-e i)}{d f^2}+\frac{b^2 i^2 (e+f x)^2}{4 d f^3} \]

[Out]

(-4*a*b*i*(f*h - e*i)*x)/(d*f^2) + (4*b^2*i*(f*h - e*i)*x)/(d*f^2) + (b^2*i^2*(e + f*x)^2)/(4*d*f^3) - (4*b^2*
i*(f*h - e*i)*(e + f*x)*Log[c*(e + f*x)])/(d*f^3) - (b*i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^3) + (
2*i*(f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^3) + (i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(
2*d*f^3) + ((f*h - e*i)^2*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^3)

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Rubi [A]  time = 0.513442, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2411, 12, 2346, 2302, 30, 2296, 2295, 2330, 2305, 2304} \[ \frac{(f h-e i)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3}+\frac{2 i (e+f x) (f h-e i) (a+b \log (c (e+f x)))^2}{d f^3}+\frac{i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}-\frac{b i^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}-\frac{4 a b i x (f h-e i)}{d f^2}-\frac{4 b^2 i (e+f x) (f h-e i) \log (c (e+f x))}{d f^3}+\frac{4 b^2 i x (f h-e i)}{d f^2}+\frac{b^2 i^2 (e+f x)^2}{4 d f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*i*(f*h - e*i)*x)/(d*f^2) + (4*b^2*i*(f*h - e*i)*x)/(d*f^2) + (b^2*i^2*(e + f*x)^2)/(4*d*f^3) - (4*b^2*
i*(f*h - e*i)*(e + f*x)*Log[c*(e + f*x)])/(d*f^3) - (b*i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^3) + (
2*i*(f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^3) + (i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(
2*d*f^3) + ((f*h - e*i)^2*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^3)

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 12

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

Rule 2346

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

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 30

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

Rule 2296

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{(h+185 x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-185 e+f h}{f}+\frac{185 x}{f}\right )^2 (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-185 e+f h}{f}+\frac{185 x}{f}\right )^2 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{185 \operatorname{Subst}\left (\int \left (\frac{-185 e+f h}{f}+\frac{185 x}{f}\right ) (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}-\frac{(185 e-f h) \operatorname{Subst}\left (\int \frac{\left (\frac{-185 e+f h}{f}+\frac{185 x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac{185 \operatorname{Subst}\left (\int \left (\frac{(-185 e+f h) (a+b \log (c x))^2}{f}+\frac{185 x (a+b \log (c x))^2}{f}\right ) \, dx,x,e+f x\right )}{d f^2}-\frac{(185 (185 e-f h)) \operatorname{Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac{(185 e-f h)^2 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^3}\\ &=-\frac{185 (185 e-f h) (e+f x) (a+b \log (c (e+f x)))^2}{d f^3}+\frac{34225 \operatorname{Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}-\frac{(185 (185 e-f h)) \operatorname{Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac{(370 b (185 e-f h)) \operatorname{Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^3}+\frac{(185 e-f h)^2 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^3}\\ &=\frac{370 a b (185 e-f h) x}{d f^2}-\frac{370 (185 e-f h) (e+f x) (a+b \log (c (e+f x)))^2}{d f^3}+\frac{34225 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}+\frac{(185 e-f h)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3}-\frac{(34225 b) \operatorname{Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^3}+\frac{(370 b (185 e-f h)) \operatorname{Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^3}+\frac{\left (370 b^2 (185 e-f h)\right ) \operatorname{Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^3}\\ &=\frac{740 a b (185 e-f h) x}{d f^2}-\frac{370 b^2 (185 e-f h) x}{d f^2}+\frac{34225 b^2 (e+f x)^2}{4 d f^3}+\frac{370 b^2 (185 e-f h) (e+f x) \log (c (e+f x))}{d f^3}-\frac{34225 b (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}-\frac{370 (185 e-f h) (e+f x) (a+b \log (c (e+f x)))^2}{d f^3}+\frac{34225 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}+\frac{(185 e-f h)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3}+\frac{\left (370 b^2 (185 e-f h)\right ) \operatorname{Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^3}\\ &=\frac{740 a b (185 e-f h) x}{d f^2}-\frac{740 b^2 (185 e-f h) x}{d f^2}+\frac{34225 b^2 (e+f x)^2}{4 d f^3}+\frac{740 b^2 (185 e-f h) (e+f x) \log (c (e+f x))}{d f^3}-\frac{34225 b (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}-\frac{370 (185 e-f h) (e+f x) (a+b \log (c (e+f x)))^2}{d f^3}+\frac{34225 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}+\frac{(185 e-f h)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3}\\ \end{align*}

Mathematica [A]  time = 0.158805, size = 171, normalized size = 0.72 \[ \frac{\frac{4 (f h-e i)^2 (a+b \log (c (e+f x)))^3}{b}+24 i (e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-48 b i (f h-e i) (f x (a-b)+b (e+f x) \log (c (e+f x)))+6 i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2+3 b i^2 \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )}{12 d f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*i*(f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)])^2 + 6*i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2 + (4*(f*
h - e*i)^2*(a + b*Log[c*(e + f*x)])^3)/b - 48*b*i*(f*h - e*i)*((a - b)*f*x + b*(e + f*x)*Log[c*(e + f*x)]) + 3
*b*i^2*(b*f*x*(2*e + f*x) - 2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])))/(12*d*f^3)

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Maple [B]  time = 0.068, size = 825, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/f/d*b^2*i^2*x^2+1/2/f/d*a^2*i^2*x^2+1/3/f/d*b^2*h^2*ln(c*f*x+c*e)^3+1/f/d*a^2*h^2*ln(c*f*x+c*e)-1/2/f/d*b^
2*i^2*ln(c*f*x+c*e)*x^2+1/f/d*a*b*h^2*ln(c*f*x+c*e)^2+1/f^3/d*a^2*e^2*i^2*ln(c*f*x+c*e)+1/3/f^3/d*b^2*e^2*i^2*
ln(c*f*x+c*e)^3-3/2/f^3/d*b^2*e^2*i^2*ln(c*f*x+c*e)^2+7/2/f^3/d*b^2*e^2*i^2*ln(c*f*x+c*e)-1/f^2/d*a^2*e*i^2*x+
3/f^2/d*b^2*e*i^2*ln(c*f*x+c*e)*x+2/f/d*b^2*h*i*ln(c*f*x+c*e)^2*x-2/3/f^2/d*b^2*e*h*i*ln(c*f*x+c*e)^3+1/f^3/d*
a*b*e^2*i^2*ln(c*f*x+c*e)^2-2/f^2/d*a^2*e*h*i*ln(c*f*x+c*e)-3/f^3/d*a*b*e^2*i^2*ln(c*f*x+c*e)+1/f/d*a*b*i^2*ln
(c*f*x+c*e)*x^2-2/f^2/d*a*b*e*h*i*ln(c*f*x+c*e)^2-2/f^2/d*a*b*e*i^2*ln(c*f*x+c*e)*x-3/2/f^3/d*a^2*e^2*i^2-15/4
/f^3/d*b^2*e^2*i^2+2/f/d*a^2*h*i*x+1/2/f/d*b^2*i^2*ln(c*f*x+c*e)^2*x^2+4/f/d*b^2*h*i*x-1/2/f/d*a*b*i^2*x^2-7/2
/f^2/d*b^2*e*i^2*x+2/f^2/d*a^2*e*h*i-4/f^2/d*a*b*e*h*i+4/f/d*a*b*h*i*ln(c*f*x+c*e)*x+4/f^2/d*a*b*h*i*ln(c*f*x+
c*e)*e-4/f/d*a*b*h*i*x+3/f^2/d*a*b*e*i^2*x-4/f^2/d*b^2*h*i*ln(c*f*x+c*e)*e+2/f^2/d*b^2*h*i*ln(c*f*x+c*e)^2*e-4
/f/d*b^2*h*i*ln(c*f*x+c*e)*x-1/f^2/d*b^2*e*i^2*ln(c*f*x+c*e)^2*x+4/f^2/d*b^2*e*h*i+7/2/f^3/d*a*b*e^2*i^2

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Maxima [B]  time = 1.29231, size = 791, normalized size = 3.32 \begin{align*} 4 \, a b h i{\left (\frac{x}{d f} - \frac{e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) + a b i^{2}{\left (\frac{2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac{f x^{2} - 2 \, e x}{d f^{2}}\right )} \log \left (c f x + c e\right ) - a b h^{2}{\left (\frac{2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac{\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + 2 \, a^{2} h i{\left (\frac{x}{d f} - \frac{e \log \left (f x + e\right )}{d f^{2}}\right )} + \frac{1}{2} \, a^{2} i^{2}{\left (\frac{2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac{f x^{2} - 2 \, e x}{d f^{2}}\right )} + \frac{b^{2} h^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac{2 \, a b h^{2} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac{a^{2} h^{2} \log \left (d f x + d e\right )}{d f} + \frac{2 \,{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b h i}{d f^{2}} - \frac{{\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} a b i^{2}}{2 \, d f^{3}} - \frac{2 \,{\left (c^{2} e \log \left (c f x + c e\right )^{3} - 3 \,{\left (c f x + c e\right )}{\left (c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + 2 \, c\right )}\right )} b^{2} h i}{3 \, c^{2} d f^{2}} + \frac{{\left (4 \, c^{3} e^{2} \log \left (c f x + c e\right )^{3} + 3 \,{\left (c f x + c e\right )}^{2}{\left (2 \, c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + c\right )} - 24 \,{\left (c^{2} e \log \left (c f x + c e\right )^{2} - 2 \, c^{2} e \log \left (c f x + c e\right ) + 2 \, c^{2} e\right )}{\left (c f x + c e\right )}\right )} b^{2} i^{2}}{12 \, c^{3} d f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a*b*h*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + a*b*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 -
2*e*x)/(d*f^2))*log(c*f*x + c*e) - a*b*h^2*(2*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - (log(f*x + e)^2 + 2*l
og(f*x + e)*log(c))/(d*f)) + 2*a^2*h*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) + 1/2*a^2*i^2*(2*e^2*log(f*x + e)/(d
*f^3) + (f*x^2 - 2*e*x)/(d*f^2)) + 1/3*b^2*h^2*log(c*f*x + c*e)^3/(d*f) + 2*a*b*h^2*log(c*f*x + c*e)*log(d*f*x
+ d*e)/(d*f) + a^2*h^2*log(d*f*x + d*e)/(d*f) + 2*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*a*b*h*i/(d*f^
2) - 1/2*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*a*b*i^2/(d*f^3) - 2/3*(c^2*e*log(c*f*
x + c*e)^3 - 3*(c*f*x + c*e)*(c*log(c*f*x + c*e)^2 - 2*c*log(c*f*x + c*e) + 2*c))*b^2*h*i/(c^2*d*f^2) + 1/12*(
4*c^3*e^2*log(c*f*x + c*e)^3 + 3*(c*f*x + c*e)^2*(2*c*log(c*f*x + c*e)^2 - 2*c*log(c*f*x + c*e) + c) - 24*(c^2
*e*log(c*f*x + c*e)^2 - 2*c^2*e*log(c*f*x + c*e) + 2*c^2*e)*(c*f*x + c*e))*b^2*i^2/(c^3*d*f^3)

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Fricas [A]  time = 1.60253, size = 709, normalized size = 2.98 \begin{align*} \frac{3 \,{\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{2} i^{2} x^{2} + 4 \,{\left (b^{2} f^{2} h^{2} - 2 \, b^{2} e f h i + b^{2} e^{2} i^{2}\right )} \log \left (c f x + c e\right )^{3} + 6 \,{\left (b^{2} f^{2} i^{2} x^{2} + 2 \, a b f^{2} h^{2} - 4 \,{\left (a b - b^{2}\right )} e f h i +{\left (2 \, a b - 3 \, b^{2}\right )} e^{2} i^{2} + 2 \,{\left (2 \, b^{2} f^{2} h i - b^{2} e f i^{2}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 6 \,{\left (4 \,{\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{2} h i -{\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f i^{2}\right )} x + 6 \,{\left ({\left (2 \, a b - b^{2}\right )} f^{2} i^{2} x^{2} + 2 \, a^{2} f^{2} h^{2} - 4 \,{\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f h i +{\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} i^{2} + 2 \,{\left (4 \,{\left (a b - b^{2}\right )} f^{2} h i -{\left (2 \, a b - 3 \, b^{2}\right )} e f i^{2}\right )} x\right )} \log \left (c f x + c e\right )}{12 \, d f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*(2*a^2 - 2*a*b + b^2)*f^2*i^2*x^2 + 4*(b^2*f^2*h^2 - 2*b^2*e*f*h*i + b^2*e^2*i^2)*log(c*f*x + c*e)^3 +
6*(b^2*f^2*i^2*x^2 + 2*a*b*f^2*h^2 - 4*(a*b - b^2)*e*f*h*i + (2*a*b - 3*b^2)*e^2*i^2 + 2*(2*b^2*f^2*h*i - b^2
*e*f*i^2)*x)*log(c*f*x + c*e)^2 + 6*(4*(a^2 - 2*a*b + 2*b^2)*f^2*h*i - (2*a^2 - 6*a*b + 7*b^2)*e*f*i^2)*x + 6*
((2*a*b - b^2)*f^2*i^2*x^2 + 2*a^2*f^2*h^2 - 4*(a^2 - 2*a*b + 2*b^2)*e*f*h*i + (2*a^2 - 6*a*b + 7*b^2)*e^2*i^2
+ 2*(4*(a*b - b^2)*f^2*h*i - (2*a*b - 3*b^2)*e*f*i^2)*x)*log(c*f*x + c*e))/(d*f^3)

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Sympy [B]  time = 2.79055, size = 452, normalized size = 1.9 \begin{align*} \frac{x^{2} \left (2 a^{2} i^{2} - 2 a b i^{2} + b^{2} i^{2}\right )}{4 d f} - \frac{x \left (2 a^{2} e i^{2} - 4 a^{2} f h i - 6 a b e i^{2} + 8 a b f h i + 7 b^{2} e i^{2} - 8 b^{2} f h i\right )}{2 d f^{2}} + \frac{\left (- 4 a b e i^{2} x + 8 a b f h i x + 2 a b f i^{2} x^{2} + 6 b^{2} e i^{2} x - 8 b^{2} f h i x - b^{2} f i^{2} x^{2}\right ) \log{\left (c \left (e + f x\right ) \right )}}{2 d f^{2}} + \frac{\left (b^{2} e^{2} i^{2} - 2 b^{2} e f h i + b^{2} f^{2} h^{2}\right ) \log{\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{3}} + \frac{\left (2 a^{2} e^{2} i^{2} - 4 a^{2} e f h i + 2 a^{2} f^{2} h^{2} - 6 a b e^{2} i^{2} + 8 a b e f h i + 7 b^{2} e^{2} i^{2} - 8 b^{2} e f h i\right ) \log{\left (e + f x \right )}}{2 d f^{3}} + \frac{\left (2 a b e^{2} i^{2} - 4 a b e f h i + 2 a b f^{2} h^{2} - 3 b^{2} e^{2} i^{2} + 4 b^{2} e f h i - 2 b^{2} e f i^{2} x + 4 b^{2} f^{2} h i x + b^{2} f^{2} i^{2} x^{2}\right ) \log{\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*(2*a**2*i**2 - 2*a*b*i**2 + b**2*i**2)/(4*d*f) - x*(2*a**2*e*i**2 - 4*a**2*f*h*i - 6*a*b*e*i**2 + 8*a*b*f
*h*i + 7*b**2*e*i**2 - 8*b**2*f*h*i)/(2*d*f**2) + (-4*a*b*e*i**2*x + 8*a*b*f*h*i*x + 2*a*b*f*i**2*x**2 + 6*b**
2*e*i**2*x - 8*b**2*f*h*i*x - b**2*f*i**2*x**2)*log(c*(e + f*x))/(2*d*f**2) + (b**2*e**2*i**2 - 2*b**2*e*f*h*i
+ b**2*f**2*h**2)*log(c*(e + f*x))**3/(3*d*f**3) + (2*a**2*e**2*i**2 - 4*a**2*e*f*h*i + 2*a**2*f**2*h**2 - 6*
a*b*e**2*i**2 + 8*a*b*e*f*h*i + 7*b**2*e**2*i**2 - 8*b**2*e*f*h*i)*log(e + f*x)/(2*d*f**3) + (2*a*b*e**2*i**2
- 4*a*b*e*f*h*i + 2*a*b*f**2*h**2 - 3*b**2*e**2*i**2 + 4*b**2*e*f*h*i - 2*b**2*e*f*i**2*x + 4*b**2*f**2*h*i*x
+ b**2*f**2*i**2*x**2)*log(c*(e + f*x))**2/(2*d*f**3)

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Giac [B]  time = 1.17982, size = 756, normalized size = 3.18 \begin{align*} \frac{24 \, b^{2} f^{2} h i x \log \left (c f x + c e\right )^{2} + 4 \, b^{2} f^{2} h^{2} \log \left (c f x + c e\right )^{3} - 8 \, b^{2} f h i e \log \left (c f x + c e\right )^{3} + 48 \, a b f^{2} h i x \log \left (c f x + c e\right ) - 48 \, b^{2} f^{2} h i x \log \left (c f x + c e\right ) + 12 \, a b f^{2} h^{2} \log \left (c f x + c e\right )^{2} - 6 \, b^{2} f^{2} x^{2} \log \left (c f x + c e\right )^{2} - 24 \, a b f h i e \log \left (c f x + c e\right )^{2} + 24 \, b^{2} f h i e \log \left (c f x + c e\right )^{2} + 24 \, a^{2} f^{2} h i x - 48 \, a b f^{2} h i x + 48 \, b^{2} f^{2} h i x - 12 \, a b f^{2} x^{2} \log \left (c f x + c e\right ) + 6 \, b^{2} f^{2} x^{2} \log \left (c f x + c e\right ) + 12 \, b^{2} f x e \log \left (c f x + c e\right )^{2} + 12 \, a^{2} f^{2} h^{2} \log \left (f x + e\right ) - 24 \, a^{2} f h i e \log \left (f x + e\right ) + 48 \, a b f h i e \log \left (f x + e\right ) - 48 \, b^{2} f h i e \log \left (f x + e\right ) - 6 \, a^{2} f^{2} x^{2} + 6 \, a b f^{2} x^{2} - 3 \, b^{2} f^{2} x^{2} + 24 \, a b f x e \log \left (c f x + c e\right ) - 36 \, b^{2} f x e \log \left (c f x + c e\right ) - 4 \, b^{2} e^{2} \log \left (c f x + c e\right )^{3} + 12 \, a^{2} f x e - 36 \, a b f x e + 42 \, b^{2} f x e - 12 \, a b e^{2} \log \left (c f x + c e\right )^{2} + 18 \, b^{2} e^{2} \log \left (c f x + c e\right )^{2} - 12 \, a^{2} e^{2} \log \left (f x + e\right ) + 36 \, a b e^{2} \log \left (f x + e\right ) - 42 \, b^{2} e^{2} \log \left (f x + e\right )}{12 \, d f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(24*b^2*f^2*h*i*x*log(c*f*x + c*e)^2 + 4*b^2*f^2*h^2*log(c*f*x + c*e)^3 - 8*b^2*f*h*i*e*log(c*f*x + c*e)^
3 + 48*a*b*f^2*h*i*x*log(c*f*x + c*e) - 48*b^2*f^2*h*i*x*log(c*f*x + c*e) + 12*a*b*f^2*h^2*log(c*f*x + c*e)^2
- 6*b^2*f^2*x^2*log(c*f*x + c*e)^2 - 24*a*b*f*h*i*e*log(c*f*x + c*e)^2 + 24*b^2*f*h*i*e*log(c*f*x + c*e)^2 + 2
4*a^2*f^2*h*i*x - 48*a*b*f^2*h*i*x + 48*b^2*f^2*h*i*x - 12*a*b*f^2*x^2*log(c*f*x + c*e) + 6*b^2*f^2*x^2*log(c*
f*x + c*e) + 12*b^2*f*x*e*log(c*f*x + c*e)^2 + 12*a^2*f^2*h^2*log(f*x + e) - 24*a^2*f*h*i*e*log(f*x + e) + 48*
a*b*f*h*i*e*log(f*x + e) - 48*b^2*f*h*i*e*log(f*x + e) - 6*a^2*f^2*x^2 + 6*a*b*f^2*x^2 - 3*b^2*f^2*x^2 + 24*a*
b*f*x*e*log(c*f*x + c*e) - 36*b^2*f*x*e*log(c*f*x + c*e) - 4*b^2*e^2*log(c*f*x + c*e)^3 + 12*a^2*f*x*e - 36*a*
b*f*x*e + 42*b^2*f*x*e - 12*a*b*e^2*log(c*f*x + c*e)^2 + 18*b^2*e^2*log(c*f*x + c*e)^2 - 12*a^2*e^2*log(f*x +
e) + 36*a*b*e^2*log(f*x + e) - 42*b^2*e^2*log(f*x + e))/(d*f^3)