3.176 \(\int \frac{(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx\)

Optimal. Leaf size=244 \[ \frac{3 i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))}{2 d f^4}+\frac{(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac{3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^4}+\frac{i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}-\frac{3 b i^2 (e+f x)^2 (f h-e i)}{4 d f^4}-\frac{3 b i x (f h-e i)^2}{d f^3}-\frac{b (f h-e i)^3 \log ^2(e+f x)}{2 d f^4}-\frac{b i^3 (e+f x)^3}{9 d f^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.383851, antiderivative size = 204, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2411, 12, 43, 2334, 14, 2301} \[ \frac{\left (\frac{9 i^2 (e+f x)^2 (f h-e i)}{f^3}+\frac{18 i (e+f x) (f h-e i)^2}{f^3}+\frac{6 (f h-e i)^3 \log (e+f x)}{f^3}+\frac{2 i^3 (e+f x)^3}{f^3}\right ) (a+b \log (c (e+f x)))}{6 d f}-\frac{3 b i^2 (e+f x)^2 (f h-e i)}{4 d f^4}-\frac{3 b i x (f h-e i)^2}{d f^3}-\frac{b (f h-e i)^3 \log ^2(e+f x)}{2 d f^4}-\frac{b i^3 (e+f x)^3}{9 d f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*i*(f*h - e*i)^2*x)/(d*f^3) - (3*b*i^2*(f*h - e*i)*(e + f*x)^2)/(4*d*f^4) - (b*i^3*(e + f*x)^3)/(9*d*f^4)
- (b*(f*h - e*i)^3*Log[e + f*x]^2)/(2*d*f^4) + (((18*i*(f*h - e*i)^2*(e + f*x))/f^3 + (9*i^2*(f*h - e*i)*(e +
f*x)^2)/f^3 + (2*i^3*(e + f*x)^3)/f^3 + (6*(f*h - e*i)^3*Log[e + f*x])/f^3)*(a + b*Log[c*(e + f*x)]))/(6*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 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 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+176 x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-176 e+f h}{f}+\frac{176 x}{f}\right )^3 (a+b \log (c x))}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-176 e+f h}{f}+\frac{176 x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\left (\frac{1584 (176 e-f h)^2 (e+f x)}{f^3}-\frac{139392 (176 e-f h) (e+f x)^2}{f^3}+\frac{5451776 (e+f x)^3}{f^3}-\frac{3 (176 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f}-\frac{b \operatorname{Subst}\left (\int \frac{176 x \left (278784 e^2+9 f^2 h^2+792 f h x+30976 x^2-3168 e (f h+44 x)\right )-3 (176 e-f h)^3 \log (x)}{3 f^3 x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\left (\frac{1584 (176 e-f h)^2 (e+f x)}{f^3}-\frac{139392 (176 e-f h) (e+f x)^2}{f^3}+\frac{5451776 (e+f x)^3}{f^3}-\frac{3 (176 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f}-\frac{b \operatorname{Subst}\left (\int \frac{176 x \left (278784 e^2+9 f^2 h^2+792 f h x+30976 x^2-3168 e (f h+44 x)\right )-3 (176 e-f h)^3 \log (x)}{x} \, dx,x,e+f x\right )}{3 d f^4}\\ &=\frac{\left (\frac{1584 (176 e-f h)^2 (e+f x)}{f^3}-\frac{139392 (176 e-f h) (e+f x)^2}{f^3}+\frac{5451776 (e+f x)^3}{f^3}-\frac{3 (176 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f}-\frac{b \operatorname{Subst}\left (\int \left (176 \left (9 (176 e-f h)^2-792 (176 e-f h) x+30976 x^2\right )-\frac{3 (176 e-f h)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{3 d f^4}\\ &=\frac{\left (\frac{1584 (176 e-f h)^2 (e+f x)}{f^3}-\frac{139392 (176 e-f h) (e+f x)^2}{f^3}+\frac{5451776 (e+f x)^3}{f^3}-\frac{3 (176 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f}-\frac{(176 b) \operatorname{Subst}\left (\int \left (9 (176 e-f h)^2-792 (176 e-f h) x+30976 x^2\right ) \, dx,x,e+f x\right )}{3 d f^4}+\frac{\left (b (176 e-f h)^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,e+f x\right )}{d f^4}\\ &=-\frac{528 b (176 e-f h)^2 x}{d f^3}+\frac{23232 b (176 e-f h) (e+f x)^2}{d f^4}-\frac{5451776 b (e+f x)^3}{9 d f^4}+\frac{b (176 e-f h)^3 \log ^2(e+f x)}{2 d f^4}+\frac{\left (\frac{1584 (176 e-f h)^2 (e+f x)}{f^3}-\frac{139392 (176 e-f h) (e+f x)^2}{f^3}+\frac{5451776 (e+f x)^3}{f^3}-\frac{3 (176 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f}\\ \end{align*}

Mathematica [A]  time = 0.303033, size = 375, normalized size = 1.54 \[ \frac{54 a^2 e^2 f h i^2-18 a^2 e^3 i^3-54 a^2 e f^2 h^2 i+18 a^2 f^3 h^3+6 b \log (c (e+f x)) \left (6 a (f h-e i)^3+b i \left (6 e^2 f i (i x-3 h)+6 e^3 i^2+3 e f^2 \left (6 h^2-6 h i x-i^2 x^2\right )+f^3 x \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right )+36 a b e^2 f i^3 x-108 a b e f^2 h i^2 x-18 a b e f^2 i^3 x^2+108 a b f^3 h^2 i x+54 a b f^3 h i^2 x^2+12 a b f^3 i^3 x^3+18 b^2 (f h-e i)^3 \log ^2(c (e+f x))+6 b^2 e^2 i^2 (5 e i-9 f h) \log (e+f x)-66 b^2 e^2 f i^3 x+162 b^2 e f^2 h i^2 x+15 b^2 e f^2 i^3 x^2-108 b^2 f^3 h^2 i x-27 b^2 f^3 h i^2 x^2-4 b^2 f^3 i^3 x^3}{36 b d f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*a^2*f^3*h^3 - 54*a^2*e*f^2*h^2*i + 54*a^2*e^2*f*h*i^2 - 18*a^2*e^3*i^3 + 108*a*b*f^3*h^2*i*x - 108*b^2*f^3
*h^2*i*x - 108*a*b*e*f^2*h*i^2*x + 162*b^2*e*f^2*h*i^2*x + 36*a*b*e^2*f*i^3*x - 66*b^2*e^2*f*i^3*x + 54*a*b*f^
3*h*i^2*x^2 - 27*b^2*f^3*h*i^2*x^2 - 18*a*b*e*f^2*i^3*x^2 + 15*b^2*e*f^2*i^3*x^2 + 12*a*b*f^3*i^3*x^3 - 4*b^2*
f^3*i^3*x^3 + 6*b^2*e^2*i^2*(-9*f*h + 5*e*i)*Log[e + f*x] + 6*b*(6*a*(f*h - e*i)^3 + b*i*(6*e^3*i^2 + 6*e^2*f*
i*(-3*h + i*x) + 3*e*f^2*(6*h^2 - 6*h*i*x - i^2*x^2) + f^3*x*(18*h^2 + 9*h*i*x + 2*i^2*x^2)))*Log[c*(e + f*x)]
+ 18*b^2*(f*h - e*i)^3*Log[c*(e + f*x)]^2)/(36*b*d*f^4)

________________________________________________________________________________________

Maple [B]  time = 0.065, size = 685, normalized size = 2.8 \begin{align*}{\frac{3\,ah{i}^{2}{x}^{2}}{2\,df}}-3\,{\frac{b{h}^{2}ix}{df}}+{\frac{5\,be{i}^{3}{x}^{2}}{12\,d{f}^{2}}}-{\frac{ae{i}^{3}{x}^{2}}{2\,d{f}^{2}}}+{\frac{b{i}^{3}\ln \left ( cfx+ce \right ){x}^{3}}{3\,df}}+{\frac{11\,b{e}^{3}{i}^{3}\ln \left ( cfx+ce \right ) }{6\,{f}^{4}d}}-{\frac{b{e}^{3}{i}^{3} \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{2\,{f}^{4}d}}-{\frac{a{e}^{3}{i}^{3}\ln \left ( cfx+ce \right ) }{{f}^{4}d}}+3\,{\frac{a{h}^{2}ix}{df}}+{\frac{a{e}^{2}{i}^{3}x}{d{f}^{3}}}-{\frac{3\,bh{i}^{2}{x}^{2}}{4\,df}}-{\frac{9\,a{e}^{2}h{i}^{2}}{2\,d{f}^{3}}}+{\frac{b{h}^{3} \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{2\,df}}-{\frac{b{i}^{3}{x}^{3}}{9\,df}}+{\frac{a{i}^{3}{x}^{3}}{3\,df}}+{\frac{a{h}^{3}\ln \left ( cfx+ce \right ) }{df}}-{\frac{11\,b{e}^{2}{i}^{3}x}{6\,d{f}^{3}}}+{\frac{11\,a{e}^{3}{i}^{3}}{6\,{f}^{4}d}}-{\frac{85\,b{e}^{3}{i}^{3}}{36\,{f}^{4}d}}-{\frac{3\,be{h}^{2}i \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{2\,d{f}^{2}}}+{\frac{b{e}^{2}{i}^{3}\ln \left ( cfx+ce \right ) x}{d{f}^{3}}}+{\frac{3\,b{e}^{2}h{i}^{2} \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{2\,d{f}^{3}}}+{\frac{9\,beh{i}^{2}x}{2\,d{f}^{2}}}+{\frac{3\,bh{i}^{2}\ln \left ( cfx+ce \right ){x}^{2}}{2\,df}}-{\frac{9\,b{e}^{2}h{i}^{2}\ln \left ( cfx+ce \right ) }{2\,d{f}^{3}}}-3\,{\frac{ae{h}^{2}i\ln \left ( cfx+ce \right ) }{d{f}^{2}}}-3\,{\frac{be{h}^{2}i}{d{f}^{2}}}+{\frac{21\,b{e}^{2}h{i}^{2}}{4\,d{f}^{3}}}+3\,{\frac{ae{h}^{2}i}{d{f}^{2}}}+3\,{\frac{b{h}^{2}i\ln \left ( cfx+ce \right ) x}{df}}+3\,{\frac{b{h}^{2}i\ln \left ( cfx+ce \right ) e}{d{f}^{2}}}+3\,{\frac{a{e}^{2}h{i}^{2}\ln \left ( cfx+ce \right ) }{d{f}^{3}}}-3\,{\frac{beh{i}^{2}\ln \left ( cfx+ce \right ) x}{d{f}^{2}}}-3\,{\frac{aeh{i}^{2}x}{d{f}^{2}}}-{\frac{be{i}^{3}\ln \left ( cfx+ce \right ){x}^{2}}{2\,d{f}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2/f/d*a*h*i^2*x^2-3/f/d*b*h^2*i*x+5/12/f^2/d*b*e*i^3*x^2-1/2/f^2/d*a*e*i^3*x^2+1/3/f/d*b*i^3*ln(c*f*x+c*e)*x
^3+11/6/f^4/d*b*e^3*i^3*ln(c*f*x+c*e)-1/2/f^4/d*b*e^3*i^3*ln(c*f*x+c*e)^2-1/f^4/d*a*e^3*i^3*ln(c*f*x+c*e)+3/f/
d*a*h^2*i*x+1/f^3/d*a*e^2*i^3*x-3/4/f/d*b*h*i^2*x^2-9/2/f^3/d*a*e^2*h*i^2+1/2/f/d*b*h^3*ln(c*f*x+c*e)^2-1/9/f/
d*b*i^3*x^3+1/3/f/d*a*i^3*x^3+1/f/d*a*h^3*ln(c*f*x+c*e)-11/6/f^3/d*b*e^2*i^3*x+11/6/f^4/d*a*e^3*i^3-85/36/f^4/
d*b*e^3*i^3-3/2/f^2/d*b*e*h^2*i*ln(c*f*x+c*e)^2+1/f^3/d*b*e^2*i^3*ln(c*f*x+c*e)*x+3/2/f^3/d*b*e^2*h*i^2*ln(c*f
*x+c*e)^2+9/2/f^2/d*b*e*h*i^2*x+3/2/f/d*b*h*i^2*ln(c*f*x+c*e)*x^2-9/2/f^3/d*b*e^2*h*i^2*ln(c*f*x+c*e)-3/f^2/d*
a*e*h^2*i*ln(c*f*x+c*e)-3/f^2/d*b*e*h^2*i+21/4/f^3/d*b*e^2*h*i^2+3/f^2/d*a*e*h^2*i+3/f/d*b*h^2*i*ln(c*f*x+c*e)
*x+3/f^2/d*b*h^2*i*ln(c*f*x+c*e)*e+3/f^3/d*a*e^2*h*i^2*ln(c*f*x+c*e)-3/f^2/d*b*e*h*i^2*ln(c*f*x+c*e)*x-3/f^2/d
*a*e*h*i^2*x-1/2/f^2/d*b*e*i^3*ln(c*f*x+c*e)*x^2

________________________________________________________________________________________

Maxima [B]  time = 1.17299, size = 728, normalized size = 2.98 \begin{align*} 3 \, b h^{2} 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 ) - \frac{1}{6} \, b i^{3}{\left (\frac{6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac{2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac{3}{2} \, b h 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 ) - \frac{1}{2} \, 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 )} + 3 \, a h^{2} i{\left (\frac{x}{d f} - \frac{e \log \left (f x + e\right )}{d f^{2}}\right )} - \frac{1}{6} \, a i^{3}{\left (\frac{6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac{2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} + \frac{3}{2} \, a h 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 h^{3} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac{a h^{3} \log \left (d f x + d e\right )}{d f} + \frac{3 \,{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} b h^{2} i}{2 \, d f^{2}} - \frac{3 \,{\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 )} b h i^{2}}{4 \, d f^{3}} - \frac{{\left (4 \, f^{3} x^{3} - 15 \, e f^{2} x^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} + 66 \, e^{2} f x - 66 \, e^{3} \log \left (f x + e\right )\right )} b i^{3}}{36 \, d f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*h^2*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) - 1/6*b*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) + 3/2*b*h*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^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)) + 3*a*h^2*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) - 1/6*a*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)) + 3/2*a*h*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2*e*x)/(d*
f^2)) + b*h^3*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) + a*h^3*log(d*f*x + d*e)/(d*f) + 3/2*(e*log(f*x + e)^2 -
2*f*x + 2*e*log(f*x + e))*b*h^2*i/(d*f^2) - 3/4*(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*i^2/(d*f^3) - 1/36*(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*i^3/(d*f^4)

________________________________________________________________________________________

Fricas [A]  time = 1.79446, size = 652, normalized size = 2.67 \begin{align*} \frac{4 \,{\left (3 \, a - b\right )} f^{3} i^{3} x^{3} + 3 \,{\left (9 \,{\left (2 \, a - b\right )} f^{3} h i^{2} -{\left (6 \, a - 5 \, b\right )} e f^{2} i^{3}\right )} x^{2} + 18 \,{\left (b f^{3} h^{3} - 3 \, b e f^{2} h^{2} i + 3 \, b e^{2} f h i^{2} - b e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{2} + 6 \,{\left (18 \,{\left (a - b\right )} f^{3} h^{2} i - 9 \,{\left (2 \, a - 3 \, b\right )} e f^{2} h i^{2} +{\left (6 \, a - 11 \, b\right )} e^{2} f i^{3}\right )} x + 6 \,{\left (2 \, b f^{3} i^{3} x^{3} + 6 \, a f^{3} h^{3} - 18 \,{\left (a - b\right )} e f^{2} h^{2} i + 9 \,{\left (2 \, a - 3 \, b\right )} e^{2} f h i^{2} -{\left (6 \, a - 11 \, b\right )} e^{3} i^{3} + 3 \,{\left (3 \, b f^{3} h i^{2} - b e f^{2} i^{3}\right )} x^{2} + 6 \,{\left (3 \, b f^{3} h^{2} i - 3 \, b e f^{2} h i^{2} + b e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )}{36 \, d f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(4*(3*a - b)*f^3*i^3*x^3 + 3*(9*(2*a - b)*f^3*h*i^2 - (6*a - 5*b)*e*f^2*i^3)*x^2 + 18*(b*f^3*h^3 - 3*b*e*
f^2*h^2*i + 3*b*e^2*f*h*i^2 - b*e^3*i^3)*log(c*f*x + c*e)^2 + 6*(18*(a - b)*f^3*h^2*i - 9*(2*a - 3*b)*e*f^2*h*
i^2 + (6*a - 11*b)*e^2*f*i^3)*x + 6*(2*b*f^3*i^3*x^3 + 6*a*f^3*h^3 - 18*(a - b)*e*f^2*h^2*i + 9*(2*a - 3*b)*e^
2*f*h*i^2 - (6*a - 11*b)*e^3*i^3 + 3*(3*b*f^3*h*i^2 - b*e*f^2*i^3)*x^2 + 6*(3*b*f^3*h^2*i - 3*b*e*f^2*h*i^2 +
b*e^2*f*i^3)*x)*log(c*f*x + c*e))/(d*f^4)

________________________________________________________________________________________

Sympy [A]  time = 2.1331, size = 400, normalized size = 1.64 \begin{align*} \frac{x^{3} \left (3 a i^{3} - b i^{3}\right )}{9 d f} - \frac{x^{2} \left (6 a e i^{3} - 18 a f h i^{2} - 5 b e i^{3} + 9 b f h i^{2}\right )}{12 d f^{2}} + \frac{x \left (6 a e^{2} i^{3} - 18 a e f h i^{2} + 18 a f^{2} h^{2} i - 11 b e^{2} i^{3} + 27 b e f h i^{2} - 18 b f^{2} h^{2} i\right )}{6 d f^{3}} + \frac{\left (6 b e^{2} i^{3} x - 18 b e f h i^{2} x - 3 b e f i^{3} x^{2} + 18 b f^{2} h^{2} i x + 9 b f^{2} h i^{2} x^{2} + 2 b f^{2} i^{3} x^{3}\right ) \log{\left (c \left (e + f x\right ) \right )}}{6 d f^{3}} + \frac{\left (- b e^{3} i^{3} + 3 b e^{2} f h i^{2} - 3 b e f^{2} h^{2} i + b f^{3} h^{3}\right ) \log{\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{4}} - \frac{\left (6 a e^{3} i^{3} - 18 a e^{2} f h i^{2} + 18 a e f^{2} h^{2} i - 6 a f^{3} h^{3} - 11 b e^{3} i^{3} + 27 b e^{2} f h i^{2} - 18 b e f^{2} h^{2} i\right ) \log{\left (e + f x \right )}}{6 d f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(3*a*i**3 - b*i**3)/(9*d*f) - x**2*(6*a*e*i**3 - 18*a*f*h*i**2 - 5*b*e*i**3 + 9*b*f*h*i**2)/(12*d*f**2) +
x*(6*a*e**2*i**3 - 18*a*e*f*h*i**2 + 18*a*f**2*h**2*i - 11*b*e**2*i**3 + 27*b*e*f*h*i**2 - 18*b*f**2*h**2*i)/
(6*d*f**3) + (6*b*e**2*i**3*x - 18*b*e*f*h*i**2*x - 3*b*e*f*i**3*x**2 + 18*b*f**2*h**2*i*x + 9*b*f**2*h*i**2*x
**2 + 2*b*f**2*i**3*x**3)*log(c*(e + f*x))/(6*d*f**3) + (-b*e**3*i**3 + 3*b*e**2*f*h*i**2 - 3*b*e*f**2*h**2*i
+ b*f**3*h**3)*log(c*(e + f*x))**2/(2*d*f**4) - (6*a*e**3*i**3 - 18*a*e**2*f*h*i**2 + 18*a*e*f**2*h**2*i - 6*a
*f**3*h**3 - 11*b*e**3*i**3 + 27*b*e**2*f*h*i**2 - 18*b*e*f**2*h**2*i)*log(e + f*x)/(6*d*f**4)

________________________________________________________________________________________

Giac [A]  time = 1.1887, size = 597, normalized size = 2.45 \begin{align*} \frac{108 \, b f^{3} h^{2} i x \log \left (c f x + c e\right ) - 12 \, b f^{3} i x^{3} \log \left (c f x + c e\right ) + 18 \, b f^{3} h^{3} \log \left (c f x + c e\right )^{2} - 54 \, b f^{2} h^{2} i e \log \left (c f x + c e\right )^{2} + 108 \, a f^{3} h^{2} i x - 108 \, b f^{3} h^{2} i x - 12 \, a f^{3} i x^{3} + 4 \, b f^{3} i x^{3} - 54 \, b f^{3} h x^{2} \log \left (c f x + c e\right ) + 18 \, b f^{2} i x^{2} e \log \left (c f x + c e\right ) + 36 \, a f^{3} h^{3} \log \left (f x + e\right ) - 108 \, a f^{2} h^{2} i e \log \left (f x + e\right ) + 108 \, b f^{2} h^{2} i e \log \left (f x + e\right ) - 54 \, a f^{3} h x^{2} + 27 \, b f^{3} h x^{2} + 18 \, a f^{2} i x^{2} e - 15 \, b f^{2} i x^{2} e + 108 \, b f^{2} h x e \log \left (c f x + c e\right ) + 108 \, a f^{2} h x e - 162 \, b f^{2} h x e - 36 \, b f i x e^{2} \log \left (c f x + c e\right ) - 54 \, b f h e^{2} \log \left (c f x + c e\right )^{2} - 36 \, a f i x e^{2} + 66 \, b f i x e^{2} + 18 \, b i e^{3} \log \left (c f x + c e\right )^{2} - 108 \, a f h e^{2} \log \left (f x + e\right ) + 162 \, b f h e^{2} \log \left (f x + e\right ) + 36 \, a i e^{3} \log \left (f x + e\right ) - 66 \, b i e^{3} \log \left (f x + e\right )}{36 \, d f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(108*b*f^3*h^2*i*x*log(c*f*x + c*e) - 12*b*f^3*i*x^3*log(c*f*x + c*e) + 18*b*f^3*h^3*log(c*f*x + c*e)^2 -
54*b*f^2*h^2*i*e*log(c*f*x + c*e)^2 + 108*a*f^3*h^2*i*x - 108*b*f^3*h^2*i*x - 12*a*f^3*i*x^3 + 4*b*f^3*i*x^3
- 54*b*f^3*h*x^2*log(c*f*x + c*e) + 18*b*f^2*i*x^2*e*log(c*f*x + c*e) + 36*a*f^3*h^3*log(f*x + e) - 108*a*f^2*
h^2*i*e*log(f*x + e) + 108*b*f^2*h^2*i*e*log(f*x + e) - 54*a*f^3*h*x^2 + 27*b*f^3*h*x^2 + 18*a*f^2*i*x^2*e - 1
5*b*f^2*i*x^2*e + 108*b*f^2*h*x*e*log(c*f*x + c*e) + 108*a*f^2*h*x*e - 162*b*f^2*h*x*e - 36*b*f*i*x*e^2*log(c*
f*x + c*e) - 54*b*f*h*e^2*log(c*f*x + c*e)^2 - 36*a*f*i*x*e^2 + 66*b*f*i*x*e^2 + 18*b*i*e^3*log(c*f*x + c*e)^2
- 108*a*f*h*e^2*log(f*x + e) + 162*b*f*h*e^2*log(f*x + e) + 36*a*i*e^3*log(f*x + e) - 66*b*i*e^3*log(f*x + e)
)/(d*f^4)