3.2.84 \(\int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) [184]

Optimal. Leaf size=464 \[ -\frac {4 a b i (f h-e i)^2 x}{d f^3}+\frac {6 b^2 i (f h-e i)^2 x}{d f^3}+\frac {3 b^2 i^2 (f h-e i) (e+f x)^2}{4 d f^4}+\frac {2 b^2 i^3 (e+f x)^3}{27 d f^4}+\frac {b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}-\frac {4 b^2 i (f h-e i)^2 (e+f x) \log (c (e+f x))}{d f^4}-\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {3 b i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}-\frac {2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^4}-\frac {2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac {2 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac {i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4} \]

[Out]

-4*a*b*i*(-e*i+f*h)^2*x/d/f^3+6*b^2*i*(-e*i+f*h)^2*x/d/f^3+3/4*b^2*i^2*(-e*i+f*h)*(f*x+e)^2/d/f^4+2/27*b^2*i^3
*(f*x+e)^3/d/f^4+1/3*b^2*(-e*i+f*h)^3*ln(f*x+e)^2/d/f^4-4*b^2*i*(-e*i+f*h)^2*(f*x+e)*ln(c*(f*x+e))/d/f^4-2*b*i
*(-e*i+f*h)^2*(f*x+e)*(a+b*ln(c*(f*x+e)))/d/f^4-3/2*b*i^2*(-e*i+f*h)*(f*x+e)^2*(a+b*ln(c*(f*x+e)))/d/f^4-2/9*b
*i^3*(f*x+e)^3*(a+b*ln(c*(f*x+e)))/d/f^4-2/3*b*(-e*i+f*h)^3*ln(f*x+e)*(a+b*ln(c*(f*x+e)))/d/f^4+2*i*(-e*i+f*h)
^2*(f*x+e)*(a+b*ln(c*(f*x+e)))^2/d/f^4+1/2*i^2*(-e*i+f*h)*(f*x+e)^2*(a+b*ln(c*(f*x+e)))^2/d/f^4+1/3*(i*x+h)^3*
(a+b*ln(c*(f*x+e)))^2/d/f+1/3*(-e*i+f*h)^3*(a+b*ln(c*(f*x+e)))^3/b/d/f^4

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Rubi [A]
time = 0.69, antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 15, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2458, 12, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341, 2356, 45, 2372, 14, 2338} \begin {gather*} \frac {i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))^2}{2 d f^4}-\frac {3 b i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))}{2 d f^4}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}-\frac {2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac {2 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac {2 b i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac {2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac {4 a b i x (f h-e i)^2}{d f^3}-\frac {4 b^2 i (e+f x) (f h-e i)^2 \log (c (e+f x))}{d f^4}+\frac {3 b^2 i^2 (e+f x)^2 (f h-e i)}{4 d f^4}+\frac {b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}+\frac {2 b^2 i^3 (e+f x)^3}{27 d f^4}+\frac {6 b^2 i x (f h-e i)^2}{d f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*i*(f*h - e*i)^2*x)/(d*f^3) + (6*b^2*i*(f*h - e*i)^2*x)/(d*f^3) + (3*b^2*i^2*(f*h - e*i)*(e + f*x)^2)/(
4*d*f^4) + (2*b^2*i^3*(e + f*x)^3)/(27*d*f^4) + (b^2*(f*h - e*i)^3*Log[e + f*x]^2)/(3*d*f^4) - (4*b^2*i*(f*h -
e*i)^2*(e + f*x)*Log[c*(e + f*x)])/(d*f^4) - (2*b*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^4)
- (3*b*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^4) - (2*b*i^3*(e + f*x)^3*(a + b*Log[c*(e
+ f*x)]))/(9*d*f^4) - (2*b*(f*h - e*i)^3*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(3*d*f^4) + (2*i*(f*h - e*i)^
2*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^4) + (i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(2*
d*f^4) + ((h + i*x)^3*(a + b*Log[c*(e + f*x)])^2)/(3*d*f) + ((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*
f^4)

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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 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 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 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

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+184 x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right )^3 (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right )^3 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {184 \text {Subst}\left (\int \left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right )^2 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}-\frac {(184 e-f h) \text {Subst}\left (\int \frac {\left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right )^2 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac {(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac {(2 b) \text {Subst}\left (\int \frac {\left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{3 d f}-\frac {(184 (184 e-f h)) \text {Subst}\left (\int \left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right ) (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac {(184 e-f h)^2 \text {Subst}\left (\int \frac {\left (\frac {-184 e+f h}{f}+\frac {184 x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^3}\\ &=-\frac {2 b \left (\frac {1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac {152352 (184 e-f h) (e+f x)^2}{f^3}+\frac {6229504 (e+f x)^3}{f^3}-\frac {3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac {(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {184 x \left (304704 e^2+9 f^2 h^2+828 f h x+33856 x^2-3312 e (f h+46 x)\right )-3 (184 e-f h)^3 \log (x)}{3 f^3 x} \, dx,x,e+f x\right )}{3 d f}-\frac {(184 (184 e-f h)) \text {Subst}\left (\int \left (\frac {(-184 e+f h) (a+b \log (c x))^2}{f}+\frac {184 x (a+b \log (c x))^2}{f}\right ) \, dx,x,e+f x\right )}{d f^3}+\frac {\left (184 (184 e-f h)^2\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}-\frac {(184 e-f h)^3 \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^4}\\ &=-\frac {2 b \left (\frac {1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac {152352 (184 e-f h) (e+f x)^2}{f^3}+\frac {6229504 (e+f x)^3}{f^3}-\frac {3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac {(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {184 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {184 x \left (304704 e^2+9 f^2 h^2+828 f h x+33856 x^2-3312 e (f h+46 x)\right )-3 (184 e-f h)^3 \log (x)}{x} \, dx,x,e+f x\right )}{9 d f^4}-\frac {(33856 (184 e-f h)) \text {Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}+\frac {\left (184 (184 e-f h)^2\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}-\frac {\left (368 b (184 e-f h)^2\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac {(184 e-f h)^3 \text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^4}\\ &=-\frac {368 a b (184 e-f h)^2 x}{d f^3}-\frac {2 b \left (\frac {1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac {152352 (184 e-f h) (e+f x)^2}{f^3}+\frac {6229504 (e+f x)^3}{f^3}-\frac {3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac {(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {368 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}-\frac {16928 (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac {(184 e-f h)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \left (184 \left (9 (184 e-f h)^2-828 (184 e-f h) x+33856 x^2\right )-\frac {3 (184 e-f h)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{9 d f^4}+\frac {(33856 b (184 e-f h)) \text {Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac {\left (368 b (184 e-f h)^2\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac {\left (368 b^2 (184 e-f h)^2\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^4}\\ &=-\frac {736 a b (184 e-f h)^2 x}{d f^3}+\frac {368 b^2 (184 e-f h)^2 x}{d f^3}-\frac {8464 b^2 (184 e-f h) (e+f x)^2}{d f^4}-\frac {368 b^2 (184 e-f h)^2 (e+f x) \log (c (e+f x))}{d f^4}+\frac {16928 b (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac {2 b \left (\frac {1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac {152352 (184 e-f h) (e+f x)^2}{f^3}+\frac {6229504 (e+f x)^3}{f^3}-\frac {3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac {(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {368 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}-\frac {16928 (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac {(184 e-f h)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac {\left (368 b^2\right ) \text {Subst}\left (\int \left (9 (184 e-f h)^2-828 (184 e-f h) x+33856 x^2\right ) \, dx,x,e+f x\right )}{9 d f^4}-\frac {\left (368 b^2 (184 e-f h)^2\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^4}-\frac {\left (2 b^2 (184 e-f h)^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{3 d f^4}\\ &=-\frac {736 a b (184 e-f h)^2 x}{d f^3}+\frac {1104 b^2 (184 e-f h)^2 x}{d f^3}-\frac {25392 b^2 (184 e-f h) (e+f x)^2}{d f^4}+\frac {12459008 b^2 (e+f x)^3}{27 d f^4}-\frac {b^2 (184 e-f h)^3 \log ^2(e+f x)}{3 d f^4}-\frac {736 b^2 (184 e-f h)^2 (e+f x) \log (c (e+f x))}{d f^4}+\frac {16928 b (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac {2 b \left (\frac {1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac {152352 (184 e-f h) (e+f x)^2}{f^3}+\frac {6229504 (e+f x)^3}{f^3}-\frac {3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac {(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {368 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}-\frac {16928 (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac {(184 e-f h)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 460, normalized size = 0.99 \begin {gather*} \frac {f i x \left (18 a^2 \left (6 e^2 i^2-3 e f i (6 h+i x)+f^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )-6 a b \left (66 e^2 i^2-3 e f i (54 h+5 i x)+f^2 \left (108 h^2+27 h i x+4 i^2 x^2\right )\right )+b^2 \left (510 e^2 i^2-3 e f i (378 h+19 i x)+f^2 \left (648 h^2+81 h i x+8 i^2 x^2\right )\right )\right )+6 \left (18 a^2 (f h-e i)^3+b^2 e i \left (-108 f^2 h^2+189 e f h i-85 e^2 i^2\right )+6 a b e i \left (18 f^2 h^2-27 e f h i+11 e^2 i^2\right )\right ) \log (e+f x)+6 b f i x \left (6 a \left (6 e^2 i^2-3 e f i (6 h+i x)+f^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )-b \left (66 e^2 i^2-3 e f i (54 h+5 i x)+f^2 \left (108 h^2+27 h i x+4 i^2 x^2\right )\right )\right ) \log (c (e+f x))+18 b \left (6 a (f h-e i)^3+b i (e+f x) \left (11 e^2 i^2-e f i (27 h+5 i x)+f^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right ) \log ^2(c (e+f x))+36 b^2 (f h-e i)^3 \log ^3(c (e+f x))}{108 d f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f*i*x*(18*a^2*(6*e^2*i^2 - 3*e*f*i*(6*h + i*x) + f^2*(18*h^2 + 9*h*i*x + 2*i^2*x^2)) - 6*a*b*(66*e^2*i^2 - 3*
e*f*i*(54*h + 5*i*x) + f^2*(108*h^2 + 27*h*i*x + 4*i^2*x^2)) + b^2*(510*e^2*i^2 - 3*e*f*i*(378*h + 19*i*x) + f
^2*(648*h^2 + 81*h*i*x + 8*i^2*x^2))) + 6*(18*a^2*(f*h - e*i)^3 + b^2*e*i*(-108*f^2*h^2 + 189*e*f*h*i - 85*e^2
*i^2) + 6*a*b*e*i*(18*f^2*h^2 - 27*e*f*h*i + 11*e^2*i^2))*Log[e + f*x] + 6*b*f*i*x*(6*a*(6*e^2*i^2 - 3*e*f*i*(
6*h + i*x) + f^2*(18*h^2 + 9*h*i*x + 2*i^2*x^2)) - b*(66*e^2*i^2 - 3*e*f*i*(54*h + 5*i*x) + f^2*(108*h^2 + 27*
h*i*x + 4*i^2*x^2)))*Log[c*(e + f*x)] + 18*b*(6*a*(f*h - e*i)^3 + b*i*(e + f*x)*(11*e^2*i^2 - e*f*i*(27*h + 5*
i*x) + f^2*(18*h^2 + 9*h*i*x + 2*i^2*x^2)))*Log[c*(e + f*x)]^2 + 36*b^2*(f*h - e*i)^3*Log[c*(e + f*x)]^3)/(108
*d*f^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1177\) vs. \(2(446)=892\).
time = 0.54, size = 1178, normalized size = 2.54 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x+h)^3*(a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e),x,method=_RETURNVERBOSE)

[Out]

1/c/f*(6/c/f^2/d*a*b*h*i^2*(1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)-1/4*(c*f*x+c*e)^2)-c/f/d*b^2*e*h^2*i*ln(c*f*x+c*e)
^3-12/f^2/d*a*b*e*h*i^2*((c*f*x+c*e)*ln(c*f*x+c*e)-c*f*x-c*e)+c/f^2/d*b^2*e^2*h*i^2*ln(c*f*x+c*e)^3+1/c^2/f^3/
d*b^2*i^3*(1/3*(c*f*x+c*e)^3*ln(c*f*x+c*e)^2-2/9*(c*f*x+c*e)^3*ln(c*f*x+c*e)+2/27*(c*f*x+c*e)^3)+3/f^3/d*b^2*e
^2*i^3*((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)+6/f/d*a*b*h^2*i*((c*f*x+c*e)*ln
(c*f*x+c*e)-c*f*x-c*e)+c/d*a*b*h^3*ln(c*f*x+c*e)^2+3/f/d*a^2*h^2*i*(c*f*x+c*e)+1/3/c^2/f^3/d*a^2*i^3*(c*f*x+c*
e)^3+3/f^3/d*a^2*e^2*i^3*(c*f*x+c*e)+3/f/d*b^2*h^2*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)+6/f^3/d*a*b*e^2*i^3*((c*f*x+c*e)*ln(c*f*x+c*e)-c*f*x-c*e)-3/c/f^3/d*b^2*e*i^3*(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)-6/f^2/d*b^2*e*h*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)-3/2/c/f^3/d*a^2*e*i^3*(c*f*x+c*e)^2-c/f^3/d*a^2*e^3*i^3*ln(
c*f*x+c*e)+3*c/f^2/d*a*b*e^2*h*i^2*ln(c*f*x+c*e)^2-3*c/f/d*a*b*e*h^2*i*ln(c*f*x+c*e)^2-6/c/f^3/d*a*b*e*i^3*(1/
2*(c*f*x+c*e)^2*ln(c*f*x+c*e)-1/4*(c*f*x+c*e)^2)+3*c/f^2/d*a^2*e^2*h*i^2*ln(c*f*x+c*e)-3*c/f/d*a^2*e*h^2*i*ln(
c*f*x+c*e)+1/3*c/d*b^2*h^3*ln(c*f*x+c*e)^3+c/d*a^2*h^3*ln(c*f*x+c*e)-c/f^3/d*a*b*e^3*i^3*ln(c*f*x+c*e)^2-6/f^2
/d*a^2*e*h*i^2*(c*f*x+c*e)+3/2/c/f^2/d*a^2*h*i^2*(c*f*x+c*e)^2+3/c/f^2/d*b^2*h*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)-1/3*c/f^3/d*b^2*e^3*i^3*ln(c*f*x+c*e)^3+2/c^2/f^3/d
*a*b*i^3*(1/3*(c*f*x+c*e)^3*ln(c*f*x+c*e)-1/9*(c*f*x+c*e)^3))

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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. 979 vs. \(2 (445) = 890\).
time = 0.32, size = 979, normalized size = 2.11 \begin {gather*} -a b h^{3} {\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 )} + 6 i \, a b h^{2} {\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^{3} \log \left (c f x + c e\right )^{3}}{3 \, d f} + 3 i \, a^{2} h^{2} {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - 3 \, a b h {\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^{3} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {3}{2} \, a^{2} h {\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 {1}{3} i \, a b {\left (\frac {2 \, f^{2} x^{3} - 3 \, f x^{2} e + 6 \, x e^{2}}{d f^{3}} - \frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}}\right )} \log \left (c f x + c e\right ) + \frac {a^{2} h^{3} \log \left (d f x + d e\right )}{d f} - \frac {1}{6} i \, a^{2} {\left (\frac {2 \, f^{2} x^{3} - 3 \, f x^{2} e + 6 \, x e^{2}}{d f^{3}} - \frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}}\right )} + \frac {3 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^{2}}{d f^{2}} + \frac {3 \, {\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 h}{2 \, d f^{3}} - \frac {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^{2}}{c^{2} d f^{2}} + \frac {i \, {\left (4 \, f^{3} x^{3} - 15 \, f^{2} x^{2} e + 66 \, f x e^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} - 66 \, e^{3} \log \left (f x + e\right )\right )} a b}{18 \, d f^{4}} - \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} h}{4 \, c^{3} d f^{3}} + \frac {i \, {\left (36 \, c^{4} e^{3} \log \left (c f x + c e\right )^{3} - 4 \, {\left (c f x + c e\right )}^{3} {\left (9 \, c \log \left (c f x + c e\right )^{2} - 6 \, c \log \left (c f x + c e\right ) + 2 \, c\right )} + 81 \, {\left (2 \, c^{2} e \log \left (c f x + c e\right )^{2} - 2 \, c^{2} e \log \left (c f x + c e\right ) + c^{2} e\right )} {\left (c f x + c e\right )}^{2} - 324 \, {\left (c^{3} e^{2} \log \left (c f x + c e\right )^{2} - 2 \, c^{3} e^{2} \log \left (c f x + c e\right ) + 2 \, c^{3} e^{2}\right )} {\left (c f x + c e\right )}\right )} b^{2}}{108 \, c^{4} d f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^3*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="maxima")

[Out]

-a*b*h^3*(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)) + 6*I*a*
b*h^2*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + 1/3*b^2*h^3*log(c*f*x + c*e)^3/(d*f) + 3*I*a^2*h^2
*(x/(d*f) - e*log(f*x + e)/(d*f^2)) - 3*a*b*h*((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^3*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - 3/2*a^2*h*((f*x^2 - 2*x*e)/(d*f^2) + 2*e^2*log(f
*x + e)/(d*f^3)) - 1/3*I*a*b*((2*f^2*x^3 - 3*f*x^2*e + 6*x*e^2)/(d*f^3) - 6*e^3*log(f*x + e)/(d*f^4))*log(c*f*
x + c*e) + a^2*h^3*log(d*f*x + d*e)/(d*f) - 1/6*I*a^2*((2*f^2*x^3 - 3*f*x^2*e + 6*x*e^2)/(d*f^3) - 6*e^3*log(f
*x + e)/(d*f^4)) + 3*I*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*a*b*h^2/(d*f^2) + 3/2*(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*h/(d*f^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^2/(c^2*d*f^2) + 1/18*I*(4*f^3*x^3 - 15*f^2*x^2*e + 66*
f*x*e^2 - 18*e^3*log(f*x + e)^2 - 66*e^3*log(f*x + e))*a*b/(d*f^4) - 1/4*(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*h/(c^3*d*f^3) + 1/108*I*(36*c^4*e^3*log(c*f*x + c*e)^3 - 4*(c*f*x +
c*e)^3*(9*c*log(c*f*x + c*e)^2 - 6*c*log(c*f*x + c*e) + 2*c) + 81*(2*c^2*e*log(c*f*x + c*e)^2 - 2*c^2*e*log(c
*f*x + c*e) + c^2*e)*(c*f*x + c*e)^2 - 324*(c^3*e^2*log(c*f*x + c*e)^2 - 2*c^3*e^2*log(c*f*x + c*e) + 2*c^3*e^
2)*(c*f*x + c*e))*b^2/(c^4*d*f^4)

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Fricas [A]
time = 0.38, size = 546, normalized size = 1.18 \begin {gather*} -\frac {324 \, {\left (-i \, a^{2} + 2 i \, a b - 2 i \, b^{2}\right )} f^{3} h^{2} x + 81 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{3} h x^{2} + 4 \, {\left (9 i \, a^{2} - 6 i \, a b + 2 i \, b^{2}\right )} f^{3} x^{3} + 6 \, {\left (18 i \, a^{2} - 66 i \, a b + 85 i \, b^{2}\right )} f x e^{2} - 36 \, {\left (b^{2} f^{3} h^{3} - 3 i \, b^{2} f^{2} h^{2} e - 3 \, b^{2} f h e^{2} + i \, b^{2} e^{3}\right )} \log \left (c f x + c e\right )^{3} - 18 \, {\left (6 \, a b f^{3} h^{3} + 18 i \, b^{2} f^{3} h^{2} x - 9 \, b^{2} f^{3} h x^{2} - 2 i \, b^{2} f^{3} x^{3} - {\left (-6 i \, a b + 11 i \, b^{2}\right )} e^{3} - 3 \, {\left (2 i \, b^{2} f x + 3 \, {\left (2 \, a b - 3 \, b^{2}\right )} f h\right )} e^{2} + 3 \, {\left (6 \, b^{2} f^{2} h x + i \, b^{2} f^{2} x^{2} - 6 \, {\left (i \, a b - i \, b^{2}\right )} f^{2} h^{2}\right )} e\right )} \log \left (c f x + c e\right )^{2} - 3 \, {\left (54 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} f^{2} h x - {\left (-18 i \, a^{2} + 30 i \, a b - 19 i \, b^{2}\right )} f^{2} x^{2}\right )} e - 6 \, {\left (18 \, a^{2} f^{3} h^{3} - 108 \, {\left (-i \, a b + i \, b^{2}\right )} f^{3} h^{2} x - 27 \, {\left (2 \, a b - b^{2}\right )} f^{3} h x^{2} - 4 \, {\left (3 i \, a b - i \, b^{2}\right )} f^{3} x^{3} - {\left (-18 i \, a^{2} + 66 i \, a b - 85 i \, b^{2}\right )} e^{3} - 3 \, {\left (9 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} f h + 2 \, {\left (6 i \, a b - 11 i \, b^{2}\right )} f x\right )} e^{2} - 3 \, {\left (18 \, {\left (i \, a^{2} - 2 i \, a b + 2 i \, b^{2}\right )} f^{2} h^{2} - 18 \, {\left (2 \, a b - 3 \, b^{2}\right )} f^{2} h x + {\left (-6 i \, a b + 5 i \, b^{2}\right )} f^{2} x^{2}\right )} e\right )} \log \left (c f x + c e\right )}{108 \, d f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^3*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="fricas")

[Out]

-1/108*(324*(-I*a^2 + 2*I*a*b - 2*I*b^2)*f^3*h^2*x + 81*(2*a^2 - 2*a*b + b^2)*f^3*h*x^2 + 4*(9*I*a^2 - 6*I*a*b
+ 2*I*b^2)*f^3*x^3 + 6*(18*I*a^2 - 66*I*a*b + 85*I*b^2)*f*x*e^2 - 36*(b^2*f^3*h^3 - 3*I*b^2*f^2*h^2*e - 3*b^2
*f*h*e^2 + I*b^2*e^3)*log(c*f*x + c*e)^3 - 18*(6*a*b*f^3*h^3 + 18*I*b^2*f^3*h^2*x - 9*b^2*f^3*h*x^2 - 2*I*b^2*
f^3*x^3 - (-6*I*a*b + 11*I*b^2)*e^3 - 3*(2*I*b^2*f*x + 3*(2*a*b - 3*b^2)*f*h)*e^2 + 3*(6*b^2*f^2*h*x + I*b^2*f
^2*x^2 - 6*(I*a*b - I*b^2)*f^2*h^2)*e)*log(c*f*x + c*e)^2 - 3*(54*(2*a^2 - 6*a*b + 7*b^2)*f^2*h*x - (-18*I*a^2
+ 30*I*a*b - 19*I*b^2)*f^2*x^2)*e - 6*(18*a^2*f^3*h^3 - 108*(-I*a*b + I*b^2)*f^3*h^2*x - 27*(2*a*b - b^2)*f^3
*h*x^2 - 4*(3*I*a*b - I*b^2)*f^3*x^3 - (-18*I*a^2 + 66*I*a*b - 85*I*b^2)*e^3 - 3*(9*(2*a^2 - 6*a*b + 7*b^2)*f*
h + 2*(6*I*a*b - 11*I*b^2)*f*x)*e^2 - 3*(18*(I*a^2 - 2*I*a*b + 2*I*b^2)*f^2*h^2 - 18*(2*a*b - 3*b^2)*f^2*h*x +
(-6*I*a*b + 5*I*b^2)*f^2*x^2)*e)*log(c*f*x + c*e))/(d*f^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (428) = 856\).
time = 1.35, size = 918, normalized size = 1.98 \begin {gather*} x^{3} \left (\frac {a^{2} i^{3}}{3 d f} - \frac {2 a b i^{3}}{9 d f} + \frac {2 b^{2} i^{3}}{27 d f}\right ) + x^{2} \left (- \frac {a^{2} e i^{3}}{2 d f^{2}} + \frac {3 a^{2} h i^{2}}{2 d f} + \frac {5 a b e i^{3}}{6 d f^{2}} - \frac {3 a b h i^{2}}{2 d f} - \frac {19 b^{2} e i^{3}}{36 d f^{2}} + \frac {3 b^{2} h i^{2}}{4 d f}\right ) + x \left (\frac {a^{2} e^{2} i^{3}}{d f^{3}} - \frac {3 a^{2} e h i^{2}}{d f^{2}} + \frac {3 a^{2} h^{2} i}{d f} - \frac {11 a b e^{2} i^{3}}{3 d f^{3}} + \frac {9 a b e h i^{2}}{d f^{2}} - \frac {6 a b h^{2} i}{d f} + \frac {85 b^{2} e^{2} i^{3}}{18 d f^{3}} - \frac {21 b^{2} e h i^{2}}{2 d f^{2}} + \frac {6 b^{2} h^{2} i}{d f}\right ) + \frac {\left (36 a b e^{2} i^{3} x - 108 a b e f h i^{2} x - 18 a b e f i^{3} x^{2} + 108 a b f^{2} h^{2} i x + 54 a b f^{2} h i^{2} x^{2} + 12 a b f^{2} i^{3} x^{3} - 66 b^{2} e^{2} i^{3} x + 162 b^{2} e f h i^{2} x + 15 b^{2} e f i^{3} x^{2} - 108 b^{2} f^{2} h^{2} i x - 27 b^{2} f^{2} h i^{2} x^{2} - 4 b^{2} f^{2} i^{3} x^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}}{18 d f^{3}} + \frac {\left (- b^{2} e^{3} i^{3} + 3 b^{2} e^{2} f h i^{2} - 3 b^{2} e f^{2} h^{2} i + b^{2} f^{3} h^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{4}} - \frac {\left (18 a^{2} e^{3} i^{3} - 54 a^{2} e^{2} f h i^{2} + 54 a^{2} e f^{2} h^{2} i - 18 a^{2} f^{3} h^{3} - 66 a b e^{3} i^{3} + 162 a b e^{2} f h i^{2} - 108 a b e f^{2} h^{2} i + 85 b^{2} e^{3} i^{3} - 189 b^{2} e^{2} f h i^{2} + 108 b^{2} e f^{2} h^{2} i\right ) \log {\left (e + f x \right )}}{18 d f^{4}} + \frac {\left (- 6 a b e^{3} i^{3} + 18 a b e^{2} f h i^{2} - 18 a b e f^{2} h^{2} i + 6 a b f^{3} h^{3} + 11 b^{2} e^{3} i^{3} - 27 b^{2} e^{2} f h i^{2} + 6 b^{2} e^{2} f i^{3} x + 18 b^{2} e f^{2} h^{2} i - 18 b^{2} e f^{2} h i^{2} x - 3 b^{2} e f^{2} i^{3} x^{2} + 18 b^{2} f^{3} h^{2} i x + 9 b^{2} f^{3} h i^{2} x^{2} + 2 b^{2} f^{3} i^{3} x^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{6 d f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(a**2*i**3/(3*d*f) - 2*a*b*i**3/(9*d*f) + 2*b**2*i**3/(27*d*f)) + x**2*(-a**2*e*i**3/(2*d*f**2) + 3*a**2*
h*i**2/(2*d*f) + 5*a*b*e*i**3/(6*d*f**2) - 3*a*b*h*i**2/(2*d*f) - 19*b**2*e*i**3/(36*d*f**2) + 3*b**2*h*i**2/(
4*d*f)) + x*(a**2*e**2*i**3/(d*f**3) - 3*a**2*e*h*i**2/(d*f**2) + 3*a**2*h**2*i/(d*f) - 11*a*b*e**2*i**3/(3*d*
f**3) + 9*a*b*e*h*i**2/(d*f**2) - 6*a*b*h**2*i/(d*f) + 85*b**2*e**2*i**3/(18*d*f**3) - 21*b**2*e*h*i**2/(2*d*f
**2) + 6*b**2*h**2*i/(d*f)) + (36*a*b*e**2*i**3*x - 108*a*b*e*f*h*i**2*x - 18*a*b*e*f*i**3*x**2 + 108*a*b*f**2
*h**2*i*x + 54*a*b*f**2*h*i**2*x**2 + 12*a*b*f**2*i**3*x**3 - 66*b**2*e**2*i**3*x + 162*b**2*e*f*h*i**2*x + 15
*b**2*e*f*i**3*x**2 - 108*b**2*f**2*h**2*i*x - 27*b**2*f**2*h*i**2*x**2 - 4*b**2*f**2*i**3*x**3)*log(c*(e + f*
x))/(18*d*f**3) + (-b**2*e**3*i**3 + 3*b**2*e**2*f*h*i**2 - 3*b**2*e*f**2*h**2*i + b**2*f**3*h**3)*log(c*(e +
f*x))**3/(3*d*f**4) - (18*a**2*e**3*i**3 - 54*a**2*e**2*f*h*i**2 + 54*a**2*e*f**2*h**2*i - 18*a**2*f**3*h**3 -
66*a*b*e**3*i**3 + 162*a*b*e**2*f*h*i**2 - 108*a*b*e*f**2*h**2*i + 85*b**2*e**3*i**3 - 189*b**2*e**2*f*h*i**2
+ 108*b**2*e*f**2*h**2*i)*log(e + f*x)/(18*d*f**4) + (-6*a*b*e**3*i**3 + 18*a*b*e**2*f*h*i**2 - 18*a*b*e*f**2
*h**2*i + 6*a*b*f**3*h**3 + 11*b**2*e**3*i**3 - 27*b**2*e**2*f*h*i**2 + 6*b**2*e**2*f*i**3*x + 18*b**2*e*f**2*
h**2*i - 18*b**2*e*f**2*h*i**2*x - 3*b**2*e*f**2*i**3*x**2 + 18*b**2*f**3*h**2*i*x + 9*b**2*f**3*h*i**2*x**2 +
2*b**2*f**3*i**3*x**3)*log(c*(e + f*x))**2/(6*d*f**4)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (445) = 890\).
time = 7.02, size = 1005, normalized size = 2.17 \begin {gather*} \frac {36 \, b^{2} f^{3} h^{3} \log \left (c f x + c e\right )^{3} + 108 \, a b f^{3} h^{3} \log \left (c f x + c e\right )^{2} + 324 i \, b^{2} f^{3} h^{2} x \log \left (c f x + c e\right )^{2} - 162 \, b^{2} f^{3} h x^{2} \log \left (c f x + c e\right )^{2} - 36 i \, b^{2} f^{3} x^{3} \log \left (c f x + c e\right )^{2} - 108 i \, b^{2} f^{2} h^{2} e \log \left (c f x + c e\right )^{3} + 648 i \, a b f^{3} h^{2} x \log \left (c f x + c e\right ) - 648 i \, b^{2} f^{3} h^{2} x \log \left (c f x + c e\right ) - 324 \, a b f^{3} h x^{2} \log \left (c f x + c e\right ) + 162 \, b^{2} f^{3} h x^{2} \log \left (c f x + c e\right ) - 72 i \, a b f^{3} x^{3} \log \left (c f x + c e\right ) + 24 i \, b^{2} f^{3} x^{3} \log \left (c f x + c e\right ) - 324 i \, a b f^{2} h^{2} e \log \left (c f x + c e\right )^{2} + 324 i \, b^{2} f^{2} h^{2} e \log \left (c f x + c e\right )^{2} + 324 \, b^{2} f^{2} h x e \log \left (c f x + c e\right )^{2} + 54 i \, b^{2} f^{2} x^{2} e \log \left (c f x + c e\right )^{2} + 108 \, a^{2} f^{3} h^{3} \log \left (f x + e\right ) + 324 i \, a^{2} f^{3} h^{2} x - 648 i \, a b f^{3} h^{2} x + 648 i \, b^{2} f^{3} h^{2} x - 162 \, a^{2} f^{3} h x^{2} + 162 \, a b f^{3} h x^{2} - 81 \, b^{2} f^{3} h x^{2} - 36 i \, a^{2} f^{3} x^{3} + 24 i \, a b f^{3} x^{3} - 8 i \, b^{2} f^{3} x^{3} + 648 \, a b f^{2} h x e \log \left (c f x + c e\right ) - 972 \, b^{2} f^{2} h x e \log \left (c f x + c e\right ) + 108 i \, a b f^{2} x^{2} e \log \left (c f x + c e\right ) - 90 i \, b^{2} f^{2} x^{2} e \log \left (c f x + c e\right ) - 108 \, b^{2} f h e^{2} \log \left (c f x + c e\right )^{3} - 324 i \, a^{2} f^{2} h^{2} e \log \left (f x + e\right ) + 648 i \, a b f^{2} h^{2} e \log \left (f x + e\right ) - 648 i \, b^{2} f^{2} h^{2} e \log \left (f x + e\right ) + 324 \, a^{2} f^{2} h x e - 972 \, a b f^{2} h x e + 1134 \, b^{2} f^{2} h x e + 54 i \, a^{2} f^{2} x^{2} e - 90 i \, a b f^{2} x^{2} e + 57 i \, b^{2} f^{2} x^{2} e - 324 \, a b f h e^{2} \log \left (c f x + c e\right )^{2} + 486 \, b^{2} f h e^{2} \log \left (c f x + c e\right )^{2} - 108 i \, b^{2} f x e^{2} \log \left (c f x + c e\right )^{2} - 216 i \, a b f x e^{2} \log \left (c f x + c e\right ) + 396 i \, b^{2} f x e^{2} \log \left (c f x + c e\right ) + 36 i \, b^{2} e^{3} \log \left (c f x + c e\right )^{3} - 324 \, a^{2} f h e^{2} \log \left (f x + e\right ) + 972 \, a b f h e^{2} \log \left (f x + e\right ) - 1134 \, b^{2} f h e^{2} \log \left (f x + e\right ) - 108 i \, a^{2} f x e^{2} + 396 i \, a b f x e^{2} - 510 i \, b^{2} f x e^{2} + 108 i \, a b e^{3} \log \left (c f x + c e\right )^{2} - 198 i \, b^{2} e^{3} \log \left (c f x + c e\right )^{2} + 108 i \, a^{2} e^{3} \log \left (f x + e\right ) - 396 i \, a b e^{3} \log \left (f x + e\right ) + 510 i \, b^{2} e^{3} \log \left (f x + e\right )}{108 \, d f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^3*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="giac")

[Out]

1/108*(36*b^2*f^3*h^3*log(c*f*x + c*e)^3 + 108*a*b*f^3*h^3*log(c*f*x + c*e)^2 + 324*I*b^2*f^3*h^2*x*log(c*f*x
+ c*e)^2 - 162*b^2*f^3*h*x^2*log(c*f*x + c*e)^2 - 36*I*b^2*f^3*x^3*log(c*f*x + c*e)^2 - 108*I*b^2*f^2*h^2*e*lo
g(c*f*x + c*e)^3 + 648*I*a*b*f^3*h^2*x*log(c*f*x + c*e) - 648*I*b^2*f^3*h^2*x*log(c*f*x + c*e) - 324*a*b*f^3*h
*x^2*log(c*f*x + c*e) + 162*b^2*f^3*h*x^2*log(c*f*x + c*e) - 72*I*a*b*f^3*x^3*log(c*f*x + c*e) + 24*I*b^2*f^3*
x^3*log(c*f*x + c*e) - 324*I*a*b*f^2*h^2*e*log(c*f*x + c*e)^2 + 324*I*b^2*f^2*h^2*e*log(c*f*x + c*e)^2 + 324*b
^2*f^2*h*x*e*log(c*f*x + c*e)^2 + 54*I*b^2*f^2*x^2*e*log(c*f*x + c*e)^2 + 108*a^2*f^3*h^3*log(f*x + e) + 324*I
*a^2*f^3*h^2*x - 648*I*a*b*f^3*h^2*x + 648*I*b^2*f^3*h^2*x - 162*a^2*f^3*h*x^2 + 162*a*b*f^3*h*x^2 - 81*b^2*f^
3*h*x^2 - 36*I*a^2*f^3*x^3 + 24*I*a*b*f^3*x^3 - 8*I*b^2*f^3*x^3 + 648*a*b*f^2*h*x*e*log(c*f*x + c*e) - 972*b^2
*f^2*h*x*e*log(c*f*x + c*e) + 108*I*a*b*f^2*x^2*e*log(c*f*x + c*e) - 90*I*b^2*f^2*x^2*e*log(c*f*x + c*e) - 108
*b^2*f*h*e^2*log(c*f*x + c*e)^3 - 324*I*a^2*f^2*h^2*e*log(f*x + e) + 648*I*a*b*f^2*h^2*e*log(f*x + e) - 648*I*
b^2*f^2*h^2*e*log(f*x + e) + 324*a^2*f^2*h*x*e - 972*a*b*f^2*h*x*e + 1134*b^2*f^2*h*x*e + 54*I*a^2*f^2*x^2*e -
90*I*a*b*f^2*x^2*e + 57*I*b^2*f^2*x^2*e - 324*a*b*f*h*e^2*log(c*f*x + c*e)^2 + 486*b^2*f*h*e^2*log(c*f*x + c*
e)^2 - 108*I*b^2*f*x*e^2*log(c*f*x + c*e)^2 - 216*I*a*b*f*x*e^2*log(c*f*x + c*e) + 396*I*b^2*f*x*e^2*log(c*f*x
+ c*e) + 36*I*b^2*e^3*log(c*f*x + c*e)^3 - 324*a^2*f*h*e^2*log(f*x + e) + 972*a*b*f*h*e^2*log(f*x + e) - 1134
*b^2*f*h*e^2*log(f*x + e) - 108*I*a^2*f*x*e^2 + 396*I*a*b*f*x*e^2 - 510*I*b^2*f*x*e^2 + 108*I*a*b*e^3*log(c*f*
x + c*e)^2 - 198*I*b^2*e^3*log(c*f*x + c*e)^2 + 108*I*a^2*e^3*log(f*x + e) - 396*I*a*b*e^3*log(f*x + e) + 510*
I*b^2*e^3*log(f*x + e))/(d*f^4)

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Mupad [B]
time = 0.69, size = 803, normalized size = 1.73 \begin {gather*} x^2\,\left (\frac {i^2\,\left (18\,a^2\,f\,h-5\,b^2\,e\,i+9\,b^2\,f\,h+6\,a\,b\,e\,i-18\,a\,b\,f\,h\right )}{12\,d\,f^2}-\frac {e\,i^3\,\left (9\,a^2-6\,a\,b+2\,b^2\right )}{18\,d\,f^2}\right )+{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^3\,x^3}{3\,d\,f^2}-\frac {b^2\,i^2\,x^2\,\left (e\,i-3\,f\,h\right )}{2\,d\,f^3}+\frac {b^2\,i\,x\,\left (e^2\,i^2-3\,e\,f\,h\,i+3\,f^2\,h^2\right )}{d\,f^4}\right )+\frac {11\,b^2\,e^3\,i^3-27\,b^2\,e^2\,f\,h\,i^2+18\,b^2\,e\,f^2\,h^2\,i-6\,a\,b\,e^3\,i^3+18\,a\,b\,e^2\,f\,h\,i^2-18\,a\,b\,e\,f^2\,h^2\,i+6\,a\,b\,f^3\,h^3}{6\,d\,f^4}\right )+x\,\left (\frac {54\,a^2\,f^2\,h^2\,i-36\,a\,b\,e^2\,i^3+108\,a\,b\,e\,f\,h\,i^2-108\,a\,b\,f^2\,h^2\,i+66\,b^2\,e^2\,i^3-162\,b^2\,e\,f\,h\,i^2+108\,b^2\,f^2\,h^2\,i}{18\,d\,f^3}-\frac {e\,\left (\frac {i^2\,\left (18\,a^2\,f\,h-5\,b^2\,e\,i+9\,b^2\,f\,h+6\,a\,b\,e\,i-18\,a\,b\,f\,h\right )}{6\,d\,f^2}-\frac {e\,i^3\,\left (9\,a^2-6\,a\,b+2\,b^2\right )}{9\,d\,f^2}\right )}{f}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x^2\,\left (5\,e\,b^2\,i^3-9\,f\,h\,b^2\,i^2-6\,a\,e\,b\,i^3+18\,a\,f\,h\,b\,i^2\right )}{6\,d\,f^3}-\frac {x\,\left (11\,b^2\,e^2\,i^3-27\,b^2\,e\,f\,h\,i^2+18\,b^2\,f^2\,h^2\,i-6\,a\,b\,e^2\,i^3+18\,a\,b\,e\,f\,h\,i^2-18\,a\,b\,f^2\,h^2\,i\right )}{3\,d\,f^4}+\frac {2\,b\,i^3\,x^3\,\left (3\,a-b\right )}{9\,d\,f^2}\right )-\frac {\ln \left (e+f\,x\right )\,\left (18\,a^2\,e^3\,i^3-54\,a^2\,e^2\,f\,h\,i^2+54\,a^2\,e\,f^2\,h^2\,i-18\,a^2\,f^3\,h^3-66\,a\,b\,e^3\,i^3+162\,a\,b\,e^2\,f\,h\,i^2-108\,a\,b\,e\,f^2\,h^2\,i+85\,b^2\,e^3\,i^3-189\,b^2\,e^2\,f\,h\,i^2+108\,b^2\,e\,f^2\,h^2\,i\right )}{18\,d\,f^4}+\frac {i^3\,x^3\,\left (9\,a^2-6\,a\,b+2\,b^2\right )}{27\,d\,f}-\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^3\,i^3-3\,e^2\,f\,h\,i^2+3\,e\,f^2\,h^2\,i-f^3\,h^3\right )}{3\,d\,f^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*((i^2*(18*a^2*f*h - 5*b^2*e*i + 9*b^2*f*h + 6*a*b*e*i - 18*a*b*f*h))/(12*d*f^2) - (e*i^3*(9*a^2 - 6*a*b +
2*b^2))/(18*d*f^2)) + log(c*(e + f*x))^2*(f*((b^2*i^3*x^3)/(3*d*f^2) - (b^2*i^2*x^2*(e*i - 3*f*h))/(2*d*f^3) +
(b^2*i*x*(e^2*i^2 + 3*f^2*h^2 - 3*e*f*h*i))/(d*f^4)) + (11*b^2*e^3*i^3 - 6*a*b*e^3*i^3 + 6*a*b*f^3*h^3 + 18*b
^2*e*f^2*h^2*i - 27*b^2*e^2*f*h*i^2 - 18*a*b*e*f^2*h^2*i + 18*a*b*e^2*f*h*i^2)/(6*d*f^4)) + x*((66*b^2*e^2*i^3
+ 54*a^2*f^2*h^2*i + 108*b^2*f^2*h^2*i - 36*a*b*e^2*i^3 - 108*a*b*f^2*h^2*i - 162*b^2*e*f*h*i^2 + 108*a*b*e*f
*h*i^2)/(18*d*f^3) - (e*((i^2*(18*a^2*f*h - 5*b^2*e*i + 9*b^2*f*h + 6*a*b*e*i - 18*a*b*f*h))/(6*d*f^2) - (e*i^
3*(9*a^2 - 6*a*b + 2*b^2))/(9*d*f^2)))/f) + f*log(c*(e + f*x))*((x^2*(5*b^2*e*i^3 - 6*a*b*e*i^3 - 9*b^2*f*h*i^
2 + 18*a*b*f*h*i^2))/(6*d*f^3) - (x*(11*b^2*e^2*i^3 + 18*b^2*f^2*h^2*i - 6*a*b*e^2*i^3 - 18*a*b*f^2*h^2*i - 27
*b^2*e*f*h*i^2 + 18*a*b*e*f*h*i^2))/(3*d*f^4) + (2*b*i^3*x^3*(3*a - b))/(9*d*f^2)) - (log(e + f*x)*(18*a^2*e^3
*i^3 - 18*a^2*f^3*h^3 + 85*b^2*e^3*i^3 - 66*a*b*e^3*i^3 + 54*a^2*e*f^2*h^2*i - 54*a^2*e^2*f*h*i^2 + 108*b^2*e*
f^2*h^2*i - 189*b^2*e^2*f*h*i^2 - 108*a*b*e*f^2*h^2*i + 162*a*b*e^2*f*h*i^2))/(18*d*f^4) + (i^3*x^3*(9*a^2 - 6
*a*b + 2*b^2))/(27*d*f) - (b^2*log(c*(e + f*x))^3*(e^3*i^3 - f^3*h^3 + 3*e*f^2*h^2*i - 3*e^2*f*h*i^2))/(3*d*f^
4)

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