3.2.85 \(\int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) [185]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 238, normalized size of antiderivative = 1.00, 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 = {2458, 12, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341} \begin {gather*} \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 {b^2 i^2 (e+f x)^2}{4 d f^3}+\frac {4 b^2 i x (f h-e i)}{d f^2} \end {gather*}

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 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 2332

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

Rule 2333

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 2339

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 2341

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
]

Rule 2342

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 2367

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 2388

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 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 {(h+185 x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac {\text {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 {\text {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 \text {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) \text {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 \text {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)) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac {(185 e-f h)^2 \text {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 \text {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)) \text {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)) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^3}+\frac {(185 e-f h)^2 \text {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) \text {Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^3}+\frac {(370 b (185 e-f h)) \text {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 ) \text {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 ) \text {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 = 0.12, size = 171, normalized size = 0.72 \begin {gather*} \frac {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+\frac {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 \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} \end {gather*}

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)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(631\) vs. \(2(230)=460\).
time = 0.52, size = 632, normalized size = 2.66 Too large to display

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (228) = 456\).
time = 0.31, size = 606, normalized size = 2.55 \begin {gather*} -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 )} + 4 i \, a b h {\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 ) + \frac {b^{2} h^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + 2 i \, a^{2} h {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - a b {\left (\frac {f x^{2} - 2 \, x e}{d f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \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 {1}{2} \, a^{2} {\left (\frac {f x^{2} - 2 \, x e}{d f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}}\right )} + \frac {a^{2} h^{2} \log \left (d f x + d e\right )}{d f} + \frac {2 i \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b h}{d f^{2}} + \frac {{\left (f^{2} x^{2} - 6 \, f x e + 2 \, e^{2} \log \left (f x + e\right )^{2} + 6 \, e^{2} \log \left (f x + e\right )\right )} a b}{2 \, d f^{3}} - \frac {2 i \, {\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}{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}}{12 \, c^{3} d f^{3}} \end {gather*}

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]

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

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 315, normalized size = 1.32 \begin {gather*} -\frac {24 \, {\left (-i \, a^{2} + 2 i \, a b - 2 i \, b^{2}\right )} f^{2} h x + 3 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{2} x^{2} - 6 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} f x e - 4 \, {\left (b^{2} f^{2} h^{2} - 2 i \, b^{2} f h e - b^{2} e^{2}\right )} \log \left (c f x + c e\right )^{3} - 6 \, {\left (2 \, a b f^{2} h^{2} + 4 i \, b^{2} f^{2} h x - b^{2} f^{2} x^{2} - {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} + 2 \, {\left (b^{2} f x - 2 \, {\left (i \, a b - i \, b^{2}\right )} f h\right )} e\right )} \log \left (c f x + c e\right )^{2} - 6 \, {\left (2 \, a^{2} f^{2} h^{2} - 8 \, {\left (-i \, a b + i \, b^{2}\right )} f^{2} h x - {\left (2 \, a b - b^{2}\right )} f^{2} x^{2} - {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} - 2 \, {\left (2 \, {\left (i \, a^{2} - 2 i \, a b + 2 i \, b^{2}\right )} f h - {\left (2 \, a b - 3 \, b^{2}\right )} f x\right )} e\right )} \log \left (c f x + c e\right )}{12 \, d f^{3}} \end {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (218) = 436\).
time = 0.68, size = 473, normalized size = 1.99 \begin {gather*} x^{2} \left (\frac {a^{2} i^{2}}{2 d f} - \frac {a b i^{2}}{2 d f} + \frac {b^{2} i^{2}}{4 d f}\right ) + x \left (- \frac {a^{2} e i^{2}}{d f^{2}} + \frac {2 a^{2} h i}{d f} + \frac {3 a b e i^{2}}{d f^{2}} - \frac {4 a b h i}{d f} - \frac {7 b^{2} e i^{2}}{2 d f^{2}} + \frac {4 b^{2} h i}{d f}\right ) + \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 {gather*}

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*(a**2*i**2/(2*d*f) - a*b*i**2/(2*d*f) + b**2*i**2/(4*d*f)) + x*(-a**2*e*i**2/(d*f**2) + 2*a**2*h*i/(d*f)
+ 3*a*b*e*i**2/(d*f**2) - 4*a*b*h*i/(d*f) - 7*b**2*e*i**2/(2*d*f**2) + 4*b**2*h*i/(d*f)) + (-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)*lo
g(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)

________________________________________________________________________________________

Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (228) = 456\).
time = 5.27, size = 548, normalized size = 2.30 \begin {gather*} \frac {4 \, b^{2} f^{2} h^{2} \log \left (c f x + c e\right )^{3} + 12 \, a b f^{2} h^{2} \log \left (c f x + c e\right )^{2} + 24 i \, b^{2} f^{2} h x \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} - 8 i \, b^{2} f h e \log \left (c f x + c e\right )^{3} + 48 i \, a b f^{2} h x \log \left (c f x + c e\right ) - 48 i \, b^{2} f^{2} h x \log \left (c f x + c e\right ) - 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 ) - 24 i \, a b f h e \log \left (c f x + c e\right )^{2} + 24 i \, b^{2} f h e \log \left (c f x + c e\right )^{2} + 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 i \, a^{2} f^{2} h x - 48 i \, a b f^{2} h x + 48 i \, b^{2} f^{2} h x - 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} - 24 i \, a^{2} f h e \log \left (f x + e\right ) + 48 i \, a b f h e \log \left (f x + e\right ) - 48 i \, b^{2} f h e \log \left (f x + e\right ) + 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 {gather*}

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*(4*b^2*f^2*h^2*log(c*f*x + c*e)^3 + 12*a*b*f^2*h^2*log(c*f*x + c*e)^2 + 24*I*b^2*f^2*h*x*log(c*f*x + c*e)
^2 - 6*b^2*f^2*x^2*log(c*f*x + c*e)^2 - 8*I*b^2*f*h*e*log(c*f*x + c*e)^3 + 48*I*a*b*f^2*h*x*log(c*f*x + c*e) -
48*I*b^2*f^2*h*x*log(c*f*x + c*e) - 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) - 24*I*a
*b*f*h*e*log(c*f*x + c*e)^2 + 24*I*b^2*f*h*e*log(c*f*x + c*e)^2 + 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*I*a^2*f^2*h*x - 48*I*a*b*f^2*h*x + 48*I*b^2*f^2*h*x - 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 -
24*I*a^2*f*h*e*log(f*x + e) + 48*I*a*b*f*h*e*log(f*x + e) - 48*I*b^2*f*h*e*log(f*x + e) + 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)

________________________________________________________________________________________

Mupad [B]
time = 0.50, size = 408, normalized size = 1.71 \begin {gather*} x\,\left (\frac {i\,\left (2\,a^2\,f\,h-3\,b^2\,e\,i+4\,b^2\,f\,h+2\,a\,b\,e\,i-4\,a\,b\,f\,h\right )}{d\,f^2}-\frac {e\,i^2\,\left (2\,a^2-2\,a\,b+b^2\right )}{2\,d\,f^2}\right )+{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^2\,x^2}{2\,d\,f^2}-\frac {b^2\,i\,x\,\left (e\,i-2\,f\,h\right )}{d\,f^3}\right )+\frac {-3\,b^2\,e^2\,i^2+4\,b^2\,e\,f\,h\,i+2\,a\,b\,e^2\,i^2-4\,a\,b\,e\,f\,h\,i+2\,a\,b\,f^2\,h^2}{2\,d\,f^3}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x\,\left (3\,e\,b^2\,i^2-4\,f\,h\,b^2\,i-2\,a\,e\,b\,i^2+4\,a\,f\,h\,b\,i\right )}{d\,f^3}+\frac {b\,i^2\,x^2\,\left (2\,a-b\right )}{2\,d\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\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 )}{2\,d\,f^3}+\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^2\,i^2-2\,e\,f\,h\,i+f^2\,h^2\right )}{3\,d\,f^3}+\frac {i^2\,x^2\,\left (2\,a^2-2\,a\,b+b^2\right )}{4\,d\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________