3.175 \(\int \frac{(h+i x)^4 (a+b \log (c (e+f x)))}{d e+d f x} \, dx\)

Optimal. Leaf size=315 \[ \frac{3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^5}+\frac{4 i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))}{3 d f^5}+\frac{(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{d f^5}+\frac{4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))}{d f^5}+\frac{i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{4 d f^5}-\frac{3 b i^2 (e+f x)^2 (f h-e i)^2}{2 d f^5}-\frac{4 b i^3 (e+f x)^3 (f h-e i)}{9 d f^5}-\frac{4 b i x (f h-e i)^3}{d f^4}-\frac{b (f h-e i)^4 \log ^2(e+f x)}{2 d f^5}-\frac{b i^4 (e+f x)^4}{16 d f^5} \]

[Out]

(-4*b*i*(f*h - e*i)^3*x)/(d*f^4) - (3*b*i^2*(f*h - e*i)^2*(e + f*x)^2)/(2*d*f^5) - (4*b*i^3*(f*h - e*i)*(e + f
*x)^3)/(9*d*f^5) - (b*i^4*(e + f*x)^4)/(16*d*f^5) - (b*(f*h - e*i)^4*Log[e + f*x]^2)/(2*d*f^5) + (4*i*(f*h - e
*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^5) + (3*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))
/(d*f^5) + (4*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/(3*d*f^5) + (i^4*(e + f*x)^4*(a + b*Log[c*
(e + f*x)]))/(4*d*f^5) + ((f*h - e*i)^4*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(d*f^5)

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Rubi [A]  time = 0.506015, antiderivative size = 260, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2411, 12, 43, 2334, 2301} \[ \frac{\left (\frac{36 i^2 (e+f x)^2 (f h-e i)^2}{f^4}+\frac{16 i^3 (e+f x)^3 (f h-e i)}{f^4}+\frac{48 i (e+f x) (f h-e i)^3}{f^4}+\frac{12 (f h-e i)^4 \log (e+f x)}{f^4}+\frac{3 i^4 (e+f x)^4}{f^4}\right ) (a+b \log (c (e+f x)))}{12 d f}-\frac{3 b i^2 (e+f x)^2 (f h-e i)^2}{2 d f^5}-\frac{4 b i^3 (e+f x)^3 (f h-e i)}{9 d f^5}-\frac{4 b i x (f h-e i)^3}{d f^4}-\frac{b (f h-e i)^4 \log ^2(e+f x)}{2 d f^5}-\frac{b i^4 (e+f x)^4}{16 d f^5} \]

Antiderivative was successfully verified.

[In]

Int[((h + i*x)^4*(a + b*Log[c*(e + f*x)]))/(d*e + d*f*x),x]

[Out]

(-4*b*i*(f*h - e*i)^3*x)/(d*f^4) - (3*b*i^2*(f*h - e*i)^2*(e + f*x)^2)/(2*d*f^5) - (4*b*i^3*(f*h - e*i)*(e + f
*x)^3)/(9*d*f^5) - (b*i^4*(e + f*x)^4)/(16*d*f^5) - (b*(f*h - e*i)^4*Log[e + f*x]^2)/(2*d*f^5) + (((48*i*(f*h
- e*i)^3*(e + f*x))/f^4 + (36*i^2*(f*h - e*i)^2*(e + f*x)^2)/f^4 + (16*i^3*(f*h - e*i)*(e + f*x)^3)/f^4 + (3*i
^4*(e + f*x)^4)/f^4 + (12*(f*h - e*i)^4*Log[e + f*x])/f^4)*(a + b*Log[c*(e + f*x)]))/(12*d*f)

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 43

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 2301

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]

Rubi steps

\begin{align*} \int \frac{(h+175 x)^4 (a+b \log (c (e+f x)))}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-175 e+f h}{f}+\frac{175 x}{f}\right )^4 (a+b \log (c x))}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-175 e+f h}{f}+\frac{175 x}{f}\right )^4 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{d f}\\ &=-\frac{\left (\frac{8400 (175 e-f h)^3 (e+f x)}{f^4}-\frac{1102500 (175 e-f h)^2 (e+f x)^2}{f^4}+\frac{85750000 (175 e-f h) (e+f x)^3}{f^4}-\frac{2813671875 (e+f x)^4}{f^4}-\frac{12 (175 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{12 d f}-\frac{b \operatorname{Subst}\left (\int \frac{-8400 (175 e-f h)^3+1102500 (-175 e+f h)^2 x-85750000 (175 e-f h) x^2+2813671875 x^3+\frac{12 (-175 e+f h)^4 \log (x)}{x}}{12 f^4} \, dx,x,e+f x\right )}{d f}\\ &=-\frac{\left (\frac{8400 (175 e-f h)^3 (e+f x)}{f^4}-\frac{1102500 (175 e-f h)^2 (e+f x)^2}{f^4}+\frac{85750000 (175 e-f h) (e+f x)^3}{f^4}-\frac{2813671875 (e+f x)^4}{f^4}-\frac{12 (175 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{12 d f}-\frac{b \operatorname{Subst}\left (\int \left (-8400 (175 e-f h)^3+1102500 (-175 e+f h)^2 x-85750000 (175 e-f h) x^2+2813671875 x^3+\frac{12 (-175 e+f h)^4 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{12 d f^5}\\ &=\frac{700 b (175 e-f h)^3 x}{d f^4}-\frac{91875 b (175 e-f h)^2 (e+f x)^2}{2 d f^5}+\frac{21437500 b (175 e-f h) (e+f x)^3}{9 d f^5}-\frac{937890625 b (e+f x)^4}{16 d f^5}-\frac{\left (\frac{8400 (175 e-f h)^3 (e+f x)}{f^4}-\frac{1102500 (175 e-f h)^2 (e+f x)^2}{f^4}+\frac{85750000 (175 e-f h) (e+f x)^3}{f^4}-\frac{2813671875 (e+f x)^4}{f^4}-\frac{12 (175 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{12 d f}-\frac{\left (b (175 e-f h)^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,e+f x\right )}{d f^5}\\ &=\frac{700 b (175 e-f h)^3 x}{d f^4}-\frac{91875 b (175 e-f h)^2 (e+f x)^2}{2 d f^5}+\frac{21437500 b (175 e-f h) (e+f x)^3}{9 d f^5}-\frac{937890625 b (e+f x)^4}{16 d f^5}-\frac{b (175 e-f h)^4 \log ^2(e+f x)}{2 d f^5}-\frac{\left (\frac{8400 (175 e-f h)^3 (e+f x)}{f^4}-\frac{1102500 (175 e-f h)^2 (e+f x)^2}{f^4}+\frac{85750000 (175 e-f h) (e+f x)^3}{f^4}-\frac{2813671875 (e+f x)^4}{f^4}-\frac{12 (175 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{12 d f}\\ \end{align*}

Mathematica [A]  time = 0.54832, size = 589, normalized size = 1.87 \[ \frac{432 a^2 e^2 f^2 h^2 i^2-288 a^2 e^3 f h i^3+72 a^2 e^4 i^4-288 a^2 e f^3 h^3 i+72 a^2 f^4 h^4+12 b \log (c (e+f x)) \left (12 a (f h-e i)^4+b i \left (6 e^2 f^2 i \left (-12 h^2+8 h i x+i^2 x^2\right )-12 e^3 f i^2 (i x-4 h)-12 e^4 i^3+4 e f^3 \left (-18 h^2 i x+12 h^3-6 h i^2 x^2-i^3 x^3\right )+f^4 x \left (36 h^2 i x+48 h^3+16 h i^2 x^2+3 i^3 x^3\right )\right )\right )+576 a b e^2 f^2 h i^3 x+72 a b e^2 f^2 i^4 x^2-144 a b e^3 f i^4 x-864 a b e f^3 h^2 i^2 x-288 a b e f^3 h i^3 x^2-48 a b e f^3 i^4 x^3+432 a b f^4 h^2 i^2 x^2+576 a b f^4 h^3 i x+192 a b f^4 h i^3 x^3+36 a b f^4 i^4 x^4+72 b^2 (f h-e i)^4 \log ^2(c (e+f x))-12 b^2 e^2 i^2 \left (13 e^2 i^2-40 e f h i+36 f^2 h^2\right ) \log (e+f x)-1056 b^2 e^2 f^2 h i^3 x-78 b^2 e^2 f^2 i^4 x^2+300 b^2 e^3 f i^4 x+1296 b^2 e f^3 h^2 i^2 x+240 b^2 e f^3 h i^3 x^2+28 b^2 e f^3 i^4 x^3-216 b^2 f^4 h^2 i^2 x^2-576 b^2 f^4 h^3 i x-64 b^2 f^4 h i^3 x^3-9 b^2 f^4 i^4 x^4}{144 b d f^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((h + i*x)^4*(a + b*Log[c*(e + f*x)]))/(d*e + d*f*x),x]

[Out]

(72*a^2*f^4*h^4 - 288*a^2*e*f^3*h^3*i + 432*a^2*e^2*f^2*h^2*i^2 - 288*a^2*e^3*f*h*i^3 + 72*a^2*e^4*i^4 + 576*a
*b*f^4*h^3*i*x - 576*b^2*f^4*h^3*i*x - 864*a*b*e*f^3*h^2*i^2*x + 1296*b^2*e*f^3*h^2*i^2*x + 576*a*b*e^2*f^2*h*
i^3*x - 1056*b^2*e^2*f^2*h*i^3*x - 144*a*b*e^3*f*i^4*x + 300*b^2*e^3*f*i^4*x + 432*a*b*f^4*h^2*i^2*x^2 - 216*b
^2*f^4*h^2*i^2*x^2 - 288*a*b*e*f^3*h*i^3*x^2 + 240*b^2*e*f^3*h*i^3*x^2 + 72*a*b*e^2*f^2*i^4*x^2 - 78*b^2*e^2*f
^2*i^4*x^2 + 192*a*b*f^4*h*i^3*x^3 - 64*b^2*f^4*h*i^3*x^3 - 48*a*b*e*f^3*i^4*x^3 + 28*b^2*e*f^3*i^4*x^3 + 36*a
*b*f^4*i^4*x^4 - 9*b^2*f^4*i^4*x^4 - 12*b^2*e^2*i^2*(36*f^2*h^2 - 40*e*f*h*i + 13*e^2*i^2)*Log[e + f*x] + 12*b
*(12*a*(f*h - e*i)^4 + b*i*(-12*e^4*i^3 - 12*e^3*f*i^2*(-4*h + i*x) + 6*e^2*f^2*i*(-12*h^2 + 8*h*i*x + i^2*x^2
) + 4*e*f^3*(12*h^3 - 18*h^2*i*x - 6*h*i^2*x^2 - i^3*x^3) + f^4*x*(48*h^3 + 36*h^2*i*x + 16*h*i^2*x^2 + 3*i^3*
x^3)))*Log[c*(e + f*x)] + 72*b^2*(f*h - e*i)^4*Log[c*(e + f*x)]^2)/(144*b*d*f^5)

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Maple [B]  time = 0.064, size = 1057, normalized size = 3.4 \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)^4*(a+b*ln(c*(f*x+e)))/(d*f*x+d*e),x)

[Out]

7/36/f^2/d*b*i^4*x^3*e-13/24/f^3/d*b*i^4*x^2*e^2+25/12/f^4/d*b*i^4*x*e^3-1/3/f^2/d*a*i^4*x^3*e+1/2/f^3/d*a*i^4
*x^2*e^2-4/9/f/d*b*h*i^3*x^3+3/f/d*a*h^2*i^2*x^2+4/3/f/d*a*h*i^3*x^3-3/2/f/d*b*h^2*i^2*x^2-1/16/f/d*b*i^4*x^4+
1/4/f/d*a*i^4*x^4+1/2/f/d*b*h^4*ln(c*f*x+c*e)^2+1/f/d*a*h^4*ln(c*f*x+c*e)+1/2/f^5/d*b*e^4*i^4*ln(c*f*x+c*e)^2+
4/f/d*a*h^3*i*x-1/f^4/d*a*e^3*i^4*x-4/f/d*b*h^3*i*x+1/f^5/d*a*e^4*i^4*ln(c*f*x+c*e)+1/4/f/d*b*i^4*ln(c*f*x+c*e
)*x^4-25/12/f^5/d*b*i^4*ln(c*f*x+c*e)*e^4+415/144/f^5/d*b*e^4*i^4-25/12/f^5/d*a*e^4*i^4+5/3/f^2/d*b*e*h*i^3*x^
2-22/3/f^3/d*b*e^2*h*i^3*x+4/f^2/d*b*h^3*i*ln(c*f*x+c*e)*e+22/3/f^4/d*b*e^3*h*i^3*ln(c*f*x+c*e)-4/f^4/d*a*e^3*
h*i^3*ln(c*f*x+c*e)-1/3/f^2/d*b*i^4*ln(c*f*x+c*e)*x^3*e+3/f^3/d*b*e^2*h^2*i^2*ln(c*f*x+c*e)^2-2/f^2/d*b*e*h^3*
i*ln(c*f*x+c*e)^2-9/f^3/d*b*e^2*h^2*i^2*ln(c*f*x+c*e)+6/f^3/d*a*e^2*h^2*i^2*ln(c*f*x+c*e)-4/f^2/d*a*e*h^3*i*ln
(c*f*x+c*e)-2/f^4/d*b*e^3*h*i^3*ln(c*f*x+c*e)^2+3/f/d*b*h^2*i^2*ln(c*f*x+c*e)*x^2+4/3/f/d*b*h*i^3*ln(c*f*x+c*e
)*x^3+4/f^2/d*a*e*h^3*i-9/f^3/d*a*e^2*h^2*i^2+22/3/f^4/d*a*e^3*h*i^3+9/f^2/d*b*e*h^2*i^2*x-2/f^2/d*a*e*h*i^3*x
^2+4/f^3/d*a*e^2*h*i^3*x-6/f^2/d*a*e*h^2*i^2*x+1/2/f^3/d*b*i^4*ln(c*f*x+c*e)*x^2*e^2-1/f^4/d*b*i^4*ln(c*f*x+c*
e)*x*e^3+4/f/d*b*h^3*i*ln(c*f*x+c*e)*x-4/f^2/d*b*e*h^3*i+21/2/f^3/d*b*e^2*h^2*i^2-85/9/f^4/d*b*e^3*h*i^3-2/f^2
/d*b*e*h*i^3*ln(c*f*x+c*e)*x^2+4/f^3/d*b*e^2*h*i^3*ln(c*f*x+c*e)*x-6/f^2/d*b*e*h^2*i^2*ln(c*f*x+c*e)*x

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Maxima [B]  time = 1.2568, size = 1022, normalized size = 3.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^4*(a+b*log(c*(f*x+e)))/(d*f*x+d*e),x, algorithm="maxima")

[Out]

4*b*h^3*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + 1/12*b*i^4*(12*e^4*log(f*x + e)/(d*f^5) + (3*f
^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/(d*f^4))*log(c*f*x + c*e) - 2/3*b*h*i^3*(6*e^3*log(f*x + e)/(d*
f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3))*log(c*f*x + c*e) + 3*b*h^2*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) - 1/2*b*h^4*(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*a*h^3*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) + 1/12*a*i^4*(12*e^4*log(
f*x + e)/(d*f^5) + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/(d*f^4)) - 2/3*a*h*i^3*(6*e^3*log(f*x +
e)/(d*f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3)) + 3*a*h^2*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2
*e*x)/(d*f^2)) + b*h^4*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) + a*h^4*log(d*f*x + d*e)/(d*f) + 2*(e*log(f*x +
e)^2 - 2*f*x + 2*e*log(f*x + e))*b*h^3*i/(d*f^2) - 3/2*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(
f*x + e))*b*h^2*i^2/(d*f^3) - 1/9*(4*f^3*x^3 - 15*e*f^2*x^2 - 18*e^3*log(f*x + e)^2 + 66*e^2*f*x - 66*e^3*log(
f*x + e))*b*h*i^3/(d*f^4) - 1/144*(9*f^4*x^4 - 28*e*f^3*x^3 + 78*e^2*f^2*x^2 + 72*e^4*log(f*x + e)^2 - 300*e^3
*f*x + 300*e^4*log(f*x + e))*b*i^4/(d*f^5)

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Fricas [A]  time = 1.76316, size = 1019, normalized size = 3.23 \begin{align*} \frac{9 \,{\left (4 \, a - b\right )} f^{4} i^{4} x^{4} + 4 \,{\left (16 \,{\left (3 \, a - b\right )} f^{4} h i^{3} -{\left (12 \, a - 7 \, b\right )} e f^{3} i^{4}\right )} x^{3} + 6 \,{\left (36 \,{\left (2 \, a - b\right )} f^{4} h^{2} i^{2} - 8 \,{\left (6 \, a - 5 \, b\right )} e f^{3} h i^{3} +{\left (12 \, a - 13 \, b\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 72 \,{\left (b f^{4} h^{4} - 4 \, b e f^{3} h^{3} i + 6 \, b e^{2} f^{2} h^{2} i^{2} - 4 \, b e^{3} f h i^{3} + b e^{4} i^{4}\right )} \log \left (c f x + c e\right )^{2} + 12 \,{\left (48 \,{\left (a - b\right )} f^{4} h^{3} i - 36 \,{\left (2 \, a - 3 \, b\right )} e f^{3} h^{2} i^{2} + 8 \,{\left (6 \, a - 11 \, b\right )} e^{2} f^{2} h i^{3} -{\left (12 \, a - 25 \, b\right )} e^{3} f i^{4}\right )} x + 12 \,{\left (3 \, b f^{4} i^{4} x^{4} + 12 \, a f^{4} h^{4} - 48 \,{\left (a - b\right )} e f^{3} h^{3} i + 36 \,{\left (2 \, a - 3 \, b\right )} e^{2} f^{2} h^{2} i^{2} - 8 \,{\left (6 \, a - 11 \, b\right )} e^{3} f h i^{3} +{\left (12 \, a - 25 \, b\right )} e^{4} i^{4} + 4 \,{\left (4 \, b f^{4} h i^{3} - b e f^{3} i^{4}\right )} x^{3} + 6 \,{\left (6 \, b f^{4} h^{2} i^{2} - 4 \, b e f^{3} h i^{3} + b e^{2} f^{2} i^{4}\right )} x^{2} + 12 \,{\left (4 \, b f^{4} h^{3} i - 6 \, b e f^{3} h^{2} i^{2} + 4 \, b e^{2} f^{2} h i^{3} - b e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )}{144 \, d f^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^4*(a+b*log(c*(f*x+e)))/(d*f*x+d*e),x, algorithm="fricas")

[Out]

1/144*(9*(4*a - b)*f^4*i^4*x^4 + 4*(16*(3*a - b)*f^4*h*i^3 - (12*a - 7*b)*e*f^3*i^4)*x^3 + 6*(36*(2*a - b)*f^4
*h^2*i^2 - 8*(6*a - 5*b)*e*f^3*h*i^3 + (12*a - 13*b)*e^2*f^2*i^4)*x^2 + 72*(b*f^4*h^4 - 4*b*e*f^3*h^3*i + 6*b*
e^2*f^2*h^2*i^2 - 4*b*e^3*f*h*i^3 + b*e^4*i^4)*log(c*f*x + c*e)^2 + 12*(48*(a - b)*f^4*h^3*i - 36*(2*a - 3*b)*
e*f^3*h^2*i^2 + 8*(6*a - 11*b)*e^2*f^2*h*i^3 - (12*a - 25*b)*e^3*f*i^4)*x + 12*(3*b*f^4*i^4*x^4 + 12*a*f^4*h^4
- 48*(a - b)*e*f^3*h^3*i + 36*(2*a - 3*b)*e^2*f^2*h^2*i^2 - 8*(6*a - 11*b)*e^3*f*h*i^3 + (12*a - 25*b)*e^4*i^
4 + 4*(4*b*f^4*h*i^3 - b*e*f^3*i^4)*x^3 + 6*(6*b*f^4*h^2*i^2 - 4*b*e*f^3*h*i^3 + b*e^2*f^2*i^4)*x^2 + 12*(4*b*
f^4*h^3*i - 6*b*e*f^3*h^2*i^2 + 4*b*e^2*f^2*h*i^3 - b*e^3*f*i^4)*x)*log(c*f*x + c*e))/(d*f^5)

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Sympy [B]  time = 2.79228, size = 636, normalized size = 2.02 \begin{align*} \frac{x^{4} \left (4 a i^{4} - b i^{4}\right )}{16 d f} - \frac{x^{3} \left (12 a e i^{4} - 48 a f h i^{3} - 7 b e i^{4} + 16 b f h i^{3}\right )}{36 d f^{2}} + \frac{x^{2} \left (12 a e^{2} i^{4} - 48 a e f h i^{3} + 72 a f^{2} h^{2} i^{2} - 13 b e^{2} i^{4} + 40 b e f h i^{3} - 36 b f^{2} h^{2} i^{2}\right )}{24 d f^{3}} - \frac{x \left (12 a e^{3} i^{4} - 48 a e^{2} f h i^{3} + 72 a e f^{2} h^{2} i^{2} - 48 a f^{3} h^{3} i - 25 b e^{3} i^{4} + 88 b e^{2} f h i^{3} - 108 b e f^{2} h^{2} i^{2} + 48 b f^{3} h^{3} i\right )}{12 d f^{4}} + \frac{\left (- 12 b e^{3} i^{4} x + 48 b e^{2} f h i^{3} x + 6 b e^{2} f i^{4} x^{2} - 72 b e f^{2} h^{2} i^{2} x - 24 b e f^{2} h i^{3} x^{2} - 4 b e f^{2} i^{4} x^{3} + 48 b f^{3} h^{3} i x + 36 b f^{3} h^{2} i^{2} x^{2} + 16 b f^{3} h i^{3} x^{3} + 3 b f^{3} i^{4} x^{4}\right ) \log{\left (c \left (e + f x\right ) \right )}}{12 d f^{4}} + \frac{\left (b e^{4} i^{4} - 4 b e^{3} f h i^{3} + 6 b e^{2} f^{2} h^{2} i^{2} - 4 b e f^{3} h^{3} i + b f^{4} h^{4}\right ) \log{\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{5}} + \frac{\left (12 a e^{4} i^{4} - 48 a e^{3} f h i^{3} + 72 a e^{2} f^{2} h^{2} i^{2} - 48 a e f^{3} h^{3} i + 12 a f^{4} h^{4} - 25 b e^{4} i^{4} + 88 b e^{3} f h i^{3} - 108 b e^{2} f^{2} h^{2} i^{2} + 48 b e f^{3} h^{3} i\right ) \log{\left (e + f x \right )}}{12 d f^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)**4*(a+b*ln(c*(f*x+e)))/(d*f*x+d*e),x)

[Out]

x**4*(4*a*i**4 - b*i**4)/(16*d*f) - x**3*(12*a*e*i**4 - 48*a*f*h*i**3 - 7*b*e*i**4 + 16*b*f*h*i**3)/(36*d*f**2
) + x**2*(12*a*e**2*i**4 - 48*a*e*f*h*i**3 + 72*a*f**2*h**2*i**2 - 13*b*e**2*i**4 + 40*b*e*f*h*i**3 - 36*b*f**
2*h**2*i**2)/(24*d*f**3) - x*(12*a*e**3*i**4 - 48*a*e**2*f*h*i**3 + 72*a*e*f**2*h**2*i**2 - 48*a*f**3*h**3*i -
25*b*e**3*i**4 + 88*b*e**2*f*h*i**3 - 108*b*e*f**2*h**2*i**2 + 48*b*f**3*h**3*i)/(12*d*f**4) + (-12*b*e**3*i*
*4*x + 48*b*e**2*f*h*i**3*x + 6*b*e**2*f*i**4*x**2 - 72*b*e*f**2*h**2*i**2*x - 24*b*e*f**2*h*i**3*x**2 - 4*b*e
*f**2*i**4*x**3 + 48*b*f**3*h**3*i*x + 36*b*f**3*h**2*i**2*x**2 + 16*b*f**3*h*i**3*x**3 + 3*b*f**3*i**4*x**4)*
log(c*(e + f*x))/(12*d*f**4) + (b*e**4*i**4 - 4*b*e**3*f*h*i**3 + 6*b*e**2*f**2*h**2*i**2 - 4*b*e*f**3*h**3*i
+ b*f**4*h**4)*log(c*(e + f*x))**2/(2*d*f**5) + (12*a*e**4*i**4 - 48*a*e**3*f*h*i**3 + 72*a*e**2*f**2*h**2*i**
2 - 48*a*e*f**3*h**3*i + 12*a*f**4*h**4 - 25*b*e**4*i**4 + 88*b*e**3*f*h*i**3 - 108*b*e**2*f**2*h**2*i**2 + 48
*b*e*f**3*h**3*i)*log(e + f*x)/(12*d*f**5)

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Giac [B]  time = 1.20372, size = 921, normalized size = 2.92 \begin{align*} \frac{576 \, b f^{4} h^{3} i x \log \left (c f x + c e\right ) - 192 \, b f^{4} h i x^{3} \log \left (c f x + c e\right ) + 72 \, b f^{4} h^{4} \log \left (c f x + c e\right )^{2} - 288 \, b f^{3} h^{3} i e \log \left (c f x + c e\right )^{2} + 576 \, a f^{4} h^{3} i x - 576 \, b f^{4} h^{3} i x - 192 \, a f^{4} h i x^{3} + 64 \, b f^{4} h i x^{3} - 432 \, b f^{4} h^{2} x^{2} \log \left (c f x + c e\right ) + 36 \, b f^{4} x^{4} \log \left (c f x + c e\right ) + 288 \, b f^{3} h i x^{2} e \log \left (c f x + c e\right ) + 144 \, a f^{4} h^{4} \log \left (f x + e\right ) - 576 \, a f^{3} h^{3} i e \log \left (f x + e\right ) + 576 \, b f^{3} h^{3} i e \log \left (f x + e\right ) - 432 \, a f^{4} h^{2} x^{2} + 216 \, b f^{4} h^{2} x^{2} + 36 \, a f^{4} x^{4} - 9 \, b f^{4} x^{4} + 288 \, a f^{3} h i x^{2} e - 240 \, b f^{3} h i x^{2} e + 864 \, b f^{3} h^{2} x e \log \left (c f x + c e\right ) - 48 \, b f^{3} x^{3} e \log \left (c f x + c e\right ) + 864 \, a f^{3} h^{2} x e - 1296 \, b f^{3} h^{2} x e - 48 \, a f^{3} x^{3} e + 28 \, b f^{3} x^{3} e - 576 \, b f^{2} h i x e^{2} \log \left (c f x + c e\right ) - 432 \, b f^{2} h^{2} e^{2} \log \left (c f x + c e\right )^{2} - 576 \, a f^{2} h i x e^{2} + 1056 \, b f^{2} h i x e^{2} + 72 \, b f^{2} x^{2} e^{2} \log \left (c f x + c e\right ) + 288 \, b f h i e^{3} \log \left (c f x + c e\right )^{2} - 864 \, a f^{2} h^{2} e^{2} \log \left (f x + e\right ) + 1296 \, b f^{2} h^{2} e^{2} \log \left (f x + e\right ) + 72 \, a f^{2} x^{2} e^{2} - 78 \, b f^{2} x^{2} e^{2} + 576 \, a f h i e^{3} \log \left (f x + e\right ) - 1056 \, b f h i e^{3} \log \left (f x + e\right ) - 144 \, b f x e^{3} \log \left (c f x + c e\right ) - 144 \, a f x e^{3} + 300 \, b f x e^{3} + 72 \, b e^{4} \log \left (c f x + c e\right )^{2} + 144 \, a e^{4} \log \left (f x + e\right ) - 300 \, b e^{4} \log \left (f x + e\right )}{144 \, d f^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^4*(a+b*log(c*(f*x+e)))/(d*f*x+d*e),x, algorithm="giac")

[Out]

1/144*(576*b*f^4*h^3*i*x*log(c*f*x + c*e) - 192*b*f^4*h*i*x^3*log(c*f*x + c*e) + 72*b*f^4*h^4*log(c*f*x + c*e)
^2 - 288*b*f^3*h^3*i*e*log(c*f*x + c*e)^2 + 576*a*f^4*h^3*i*x - 576*b*f^4*h^3*i*x - 192*a*f^4*h*i*x^3 + 64*b*f
^4*h*i*x^3 - 432*b*f^4*h^2*x^2*log(c*f*x + c*e) + 36*b*f^4*x^4*log(c*f*x + c*e) + 288*b*f^3*h*i*x^2*e*log(c*f*
x + c*e) + 144*a*f^4*h^4*log(f*x + e) - 576*a*f^3*h^3*i*e*log(f*x + e) + 576*b*f^3*h^3*i*e*log(f*x + e) - 432*
a*f^4*h^2*x^2 + 216*b*f^4*h^2*x^2 + 36*a*f^4*x^4 - 9*b*f^4*x^4 + 288*a*f^3*h*i*x^2*e - 240*b*f^3*h*i*x^2*e + 8
64*b*f^3*h^2*x*e*log(c*f*x + c*e) - 48*b*f^3*x^3*e*log(c*f*x + c*e) + 864*a*f^3*h^2*x*e - 1296*b*f^3*h^2*x*e -
48*a*f^3*x^3*e + 28*b*f^3*x^3*e - 576*b*f^2*h*i*x*e^2*log(c*f*x + c*e) - 432*b*f^2*h^2*e^2*log(c*f*x + c*e)^2
- 576*a*f^2*h*i*x*e^2 + 1056*b*f^2*h*i*x*e^2 + 72*b*f^2*x^2*e^2*log(c*f*x + c*e) + 288*b*f*h*i*e^3*log(c*f*x
+ c*e)^2 - 864*a*f^2*h^2*e^2*log(f*x + e) + 1296*b*f^2*h^2*e^2*log(f*x + e) + 72*a*f^2*x^2*e^2 - 78*b*f^2*x^2*
e^2 + 576*a*f*h*i*e^3*log(f*x + e) - 1056*b*f*h*i*e^3*log(f*x + e) - 144*b*f*x*e^3*log(c*f*x + c*e) - 144*a*f*
x*e^3 + 300*b*f*x*e^3 + 72*b*e^4*log(c*f*x + c*e)^2 + 144*a*e^4*log(f*x + e) - 300*b*e^4*log(f*x + e))/(d*f^5)