\(\int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx\) [176]

Optimal result

Integrand size = 30, antiderivative size = 244 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=-\frac {3 b i (f h-e i)^2 x}{d f^3}-\frac {3 b i^2 (f h-e i) (e+f x)^2}{4 d f^4}-\frac {b i^3 (e+f x)^3}{9 d f^4}-\frac {b (f h-e i)^3 \log ^2(e+f x)}{2 d f^4}+\frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2458, 12, 45, 2372, 14, 2338} \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {3 i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))}{2 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}-\frac {3 b i^2 (e+f x)^2 (f h-e i)}{4 d f^4}-\frac {b (f h-e i)^3 \log ^2(e+f x)}{2 d f^4}-\frac {b i^3 (e+f x)^3}{9 d f^4}-\frac {3 b i x (f h-e i)^2}{d f^3} \]

[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) + (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)

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 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 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 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 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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))}{d x} \, dx,x,e+f x\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{d f} \\ & = \frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {b \text {Subst}\left (\int \frac {i x \left (18 f^2 h^2+9 f h i (-4 e+x)+i^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f h-e i)^3 \log (x)}{6 f^3 x} \, dx,x,e+f x\right )}{d f} \\ & = \frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {b \text {Subst}\left (\int \frac {i x \left (18 f^2 h^2+9 f h i (-4 e+x)+i^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f h-e i)^3 \log (x)}{x} \, dx,x,e+f x\right )}{6 d f^4} \\ & = \frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {b \text {Subst}\left (\int \left (i \left (18 (f h-e i)^2+9 i (f h-e i) x+2 i^2 x^2\right )+\frac {6 (f h-e i)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{6 d f^4} \\ & = \frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {(b i) \text {Subst}\left (\int \left (18 (f h-e i)^2+9 i (f h-e i) x+2 i^2 x^2\right ) \, dx,x,e+f x\right )}{6 d f^4}-\frac {\left (b (f h-e i)^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{d f^4} \\ & = -\frac {3 b i (f h-e i)^2 x}{d f^3}-\frac {3 b i^2 (f h-e i) (e+f x)^2}{4 d f^4}-\frac {b i^3 (e+f x)^3}{9 d f^4}-\frac {b (f h-e i)^3 \log ^2(e+f x)}{2 d f^4}+\frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.54 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {18 a^2 f^3 h^3-54 a^2 e f^2 h^2 i+54 a^2 e^2 f h i^2-18 a^2 e^3 i^3+108 a b f^3 h^2 i x-108 b^2 f^3 h^2 i x-108 a b e f^2 h i^2 x+162 b^2 e f^2 h i^2 x+36 a b e^2 f i^3 x-66 b^2 e^2 f i^3 x+54 a b f^3 h i^2 x^2-27 b^2 f^3 h i^2 x^2-18 a b e f^2 i^3 x^2+15 b^2 e f^2 i^3 x^2+12 a b f^3 i^3 x^3-4 b^2 f^3 i^3 x^3+6 b^2 e^2 i^2 (-9 f h+5 e i) \log (e+f x)+6 b \left (6 a (f h-e i)^3+b i \left (6 e^3 i^2+6 e^2 f i (-3 h+i x)+3 e f^2 \left (6 h^2-6 h i x-i^2 x^2\right )+f^3 x \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right ) \log (c (e+f x))+18 b^2 (f h-e i)^3 \log ^2(c (e+f x))}{36 b d f^4} \]

[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 [A] (verified)

Time = 0.65 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.46

method	result	size
norman	\(\frac {b i \left (e^{2} i^{2}-3 e f h i +3 f^{2} h^{2}\right ) x \ln \left (c \left (f x +e \right )\right )}{d \,f^{3}}-\frac {\left (6 a \,e^{3} i^{3}-18 a \,e^{2} f h \,i^{2}+18 a e \,f^{2} h^{2} i -6 a \,f^{3} h^{3}-11 b \,e^{3} i^{3}+27 b \,e^{2} f h \,i^{2}-18 b e \,f^{2} h^{2} i \right ) \ln \left (c \left (f x +e \right )\right )}{6 d \,f^{4}}-\frac {b \left (e^{3} i^{3}-3 e^{2} f h \,i^{2}+3 e \,f^{2} h^{2} i -f^{3} h^{3}\right ) \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{4}}+\frac {i \left (6 a \,e^{2} i^{2}-18 a e f h i +18 a \,f^{2} h^{2}-11 b \,e^{2} i^{2}+27 b e f h i -18 b \,f^{2} h^{2}\right ) x}{6 d \,f^{3}}-\frac {i^{2} \left (6 a e i -18 a f h -5 b e i +9 b f h \right ) x^{2}}{12 d \,f^{2}}+\frac {i^{3} \left (3 a -b \right ) x^{3}}{9 d f}+\frac {b \,i^{3} x^{3} \ln \left (c \left (f x +e \right )\right )}{3 d f}-\frac {b \,i^{2} \left (e i -3 f h \right ) x^{2} \ln \left (c \left (f x +e \right )\right )}{2 d \,f^{2}}\)	\(356\)
parts	\(\frac {a \left (\frac {i \left (\frac {1}{3} f^{2} i^{2} x^{3}-\frac {1}{2} e f \,i^{2} x^{2}+\frac {3}{2} f^{2} h i \,x^{2}+x \,e^{2} i^{2}-3 x e f h i +3 x \,f^{2} h^{2}\right )}{f^{3}}+\frac {\left (-e^{3} i^{3}+3 e^{2} f h \,i^{2}-3 e \,f^{2} h^{2} i +f^{3} h^{3}\right ) \ln \left (f x +e \right )}{f^{4}}\right )}{d}+\frac {b \left (-\frac {c \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3}}+\frac {3 c \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2}}-\frac {3 c e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f}+\frac {c \,h^{3} \ln \left (c f x +c e \right )^{2}}{2}+\frac {3 e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3}}-\frac {6 e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2}}+\frac {3 h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f}-\frac {3 e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3}}+\frac {3 h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}+\frac {i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3}}\right )}{d c f}\)	\(463\)
risch	\(-\frac {b \ln \left (c \left (f x +e \right )\right )^{2} e^{3} i^{3}}{2 d \,f^{4}}+\frac {3 b \ln \left (c \left (f x +e \right )\right )^{2} e^{2} h \,i^{2}}{2 d \,f^{3}}-\frac {3 b \ln \left (c \left (f x +e \right )\right )^{2} e \,h^{2} i}{2 d \,f^{2}}+\frac {b \ln \left (c \left (f x +e \right )\right )^{2} h^{3}}{2 d f}+\frac {b i x \left (2 f^{2} i^{2} x^{2}-3 e f \,i^{2} x +9 f^{2} h i x +6 e^{2} i^{2}-18 e f h i +18 f^{2} h^{2}\right ) \ln \left (c \left (f x +e \right )\right )}{6 d \,f^{3}}+\frac {a \,i^{3} x^{3}}{3 d f}-\frac {b \,i^{3} x^{3}}{9 d f}-\frac {a e \,i^{3} x^{2}}{2 d \,f^{2}}+\frac {3 a h \,i^{2} x^{2}}{2 d f}+\frac {5 b e \,i^{3} x^{2}}{12 d \,f^{2}}-\frac {3 b h \,i^{2} x^{2}}{4 d f}-\frac {\ln \left (f x +e \right ) a \,e^{3} i^{3}}{d \,f^{4}}+\frac {3 \ln \left (f x +e \right ) a \,e^{2} h \,i^{2}}{d \,f^{3}}-\frac {3 \ln \left (f x +e \right ) a e \,h^{2} i}{d \,f^{2}}+\frac {\ln \left (f x +e \right ) a \,h^{3}}{d f}+\frac {11 \ln \left (f x +e \right ) b \,e^{3} i^{3}}{6 d \,f^{4}}-\frac {9 \ln \left (f x +e \right ) b \,e^{2} h \,i^{2}}{2 d \,f^{3}}+\frac {3 \ln \left (f x +e \right ) b e \,h^{2} i}{d \,f^{2}}+\frac {a \,e^{2} i^{3} x}{d \,f^{3}}-\frac {3 a e h \,i^{2} x}{d \,f^{2}}+\frac {3 a \,h^{2} i x}{d f}-\frac {11 b \,e^{2} i^{3} x}{6 d \,f^{3}}+\frac {9 b e h \,i^{2} x}{2 d \,f^{2}}-\frac {3 b \,h^{2} i x}{d f}\)	\(494\)
parallelrisch	\(\frac {-108 x \ln \left (c \left (f x +e \right )\right ) b e \,f^{2} h \,i^{2}+12 a \,f^{3} i^{3} x^{3}-4 b \,f^{3} i^{3} x^{3}+108 \ln \left (c \left (f x +e \right )\right ) a \,e^{2} f h \,i^{2}-108 \ln \left (c \left (f x +e \right )\right ) a e \,f^{2} h^{2} i -162 \ln \left (c \left (f x +e \right )\right ) b \,e^{2} f h \,i^{2}+108 \ln \left (c \left (f x +e \right )\right ) b e \,f^{2} h^{2} i -18 x^{2} \ln \left (c \left (f x +e \right )\right ) b e \,f^{2} i^{3}+54 x^{2} \ln \left (c \left (f x +e \right )\right ) b \,f^{3} h \,i^{2}+36 x \ln \left (c \left (f x +e \right )\right ) b \,e^{2} f \,i^{3}+108 x \ln \left (c \left (f x +e \right )\right ) b \,f^{3} h^{2} i +54 \ln \left (c \left (f x +e \right )\right )^{2} b \,e^{2} f h \,i^{2}-54 \ln \left (c \left (f x +e \right )\right )^{2} b e \,f^{2} h^{2} i -54 a \,e^{3} i^{3}+117 b \,e^{3} i^{3}-18 \ln \left (c \left (f x +e \right )\right )^{2} b \,e^{3} i^{3}+18 \ln \left (c \left (f x +e \right )\right )^{2} b \,f^{3} h^{3}-36 \ln \left (c \left (f x +e \right )\right ) a \,e^{3} i^{3}+36 \ln \left (c \left (f x +e \right )\right ) a \,f^{3} h^{3}+66 \ln \left (c \left (f x +e \right )\right ) b \,e^{3} i^{3}-108 a e \,f^{2} h \,i^{2} x +162 a \,e^{2} f h \,i^{2}-216 a e \,f^{2} h^{2} i -297 b \,e^{2} f h \,i^{2}+216 b e \,f^{2} h^{2} i +12 x^{3} \ln \left (c \left (f x +e \right )\right ) b \,f^{3} i^{3}-18 a e \,f^{2} i^{3} x^{2}+54 a \,f^{3} h \,i^{2} x^{2}+15 b e \,f^{2} i^{3} x^{2}-27 b \,f^{3} h \,i^{2} x^{2}+36 a \,e^{2} f \,i^{3} x +108 a \,f^{3} h^{2} i x -66 b \,e^{2} f \,i^{3} x -108 b \,f^{3} h^{2} i x +162 b e \,f^{2} h \,i^{2} x}{36 d \,f^{4}}\)	\(543\)
derivativedivides	\(\frac {-\frac {c a \,e^{3} i^{3} \ln \left (c f x +c e \right )}{f^{3} d}+\frac {3 c a \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )}{f^{2} d}-\frac {3 c a e \,h^{2} i \ln \left (c f x +c e \right )}{f d}+\frac {c a \,h^{3} \ln \left (c f x +c e \right )}{d}+\frac {3 a \,e^{2} i^{3} \left (c f x +c e \right )}{f^{3} d}-\frac {6 a e h \,i^{2} \left (c f x +c e \right )}{f^{2} d}+\frac {3 a \,h^{2} i \left (c f x +c e \right )}{f d}-\frac {3 a e \,i^{3} \left (c f x +c e \right )^{2}}{2 c \,f^{3} d}+\frac {3 a h \,i^{2} \left (c f x +c e \right )^{2}}{2 c \,f^{2} d}+\frac {a \,i^{3} \left (c f x +c e \right )^{3}}{3 c^{2} f^{3} d}-\frac {c b \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3} d}+\frac {3 c b \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2} d}-\frac {3 c b e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f d}+\frac {c b \,h^{3} \ln \left (c f x +c e \right )^{2}}{2 d}+\frac {3 b \,e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3} d}-\frac {6 b e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2} d}+\frac {3 b \,h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}-\frac {3 b e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3} d}+\frac {3 b h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}+\frac {b \,i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3} d}}{c f}\)	\(621\)
default	\(\frac {-\frac {c a \,e^{3} i^{3} \ln \left (c f x +c e \right )}{f^{3} d}+\frac {3 c a \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )}{f^{2} d}-\frac {3 c a e \,h^{2} i \ln \left (c f x +c e \right )}{f d}+\frac {c a \,h^{3} \ln \left (c f x +c e \right )}{d}+\frac {3 a \,e^{2} i^{3} \left (c f x +c e \right )}{f^{3} d}-\frac {6 a e h \,i^{2} \left (c f x +c e \right )}{f^{2} d}+\frac {3 a \,h^{2} i \left (c f x +c e \right )}{f d}-\frac {3 a e \,i^{3} \left (c f x +c e \right )^{2}}{2 c \,f^{3} d}+\frac {3 a h \,i^{2} \left (c f x +c e \right )^{2}}{2 c \,f^{2} d}+\frac {a \,i^{3} \left (c f x +c e \right )^{3}}{3 c^{2} f^{3} d}-\frac {c b \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3} d}+\frac {3 c b \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2} d}-\frac {3 c b e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f d}+\frac {c b \,h^{3} \ln \left (c f x +c e \right )^{2}}{2 d}+\frac {3 b \,e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3} d}-\frac {6 b e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2} d}+\frac {3 b \,h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}-\frac {3 b e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3} d}+\frac {3 b h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}+\frac {b \,i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3} d}}{c f}\)	\(621\)

[In]

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

[Out]

b*i*(e^2*i^2-3*e*f*h*i+3*f^2*h^2)/d/f^3*x*ln(c*(f*x+e))-1/6*(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)/d/f^4*ln(c*(f*x+e))-1/2*b*(e^3*i^3-3*e^2*f*h*i^2+3*e*
f^2*h^2*i-f^3*h^3)/d/f^4*ln(c*(f*x+e))^2+1/6*i*(6*a*e^2*i^2-18*a*e*f*h*i+18*a*f^2*h^2-11*b*e^2*i^2+27*b*e*f*h*
i-18*b*f^2*h^2)/d/f^3*x-1/12*i^2*(6*a*e*i-18*a*f*h-5*b*e*i+9*b*f*h)/d/f^2*x^2+1/9*i^3*(3*a-b)/d/f*x^3+1/3*b*i^
3/d/f*x^3*ln(c*(f*x+e))-1/2*b*i^2*(e*i-3*f*h)/d/f^2*x^2*ln(c*(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.26 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {4 \, {\left (3 \, a - b\right )} f^{3} i^{3} x^{3} + 3 \, {\left (9 \, {\left (2 \, a - b\right )} f^{3} h i^{2} - {\left (6 \, a - 5 \, b\right )} e f^{2} i^{3}\right )} x^{2} + 18 \, {\left (b f^{3} h^{3} - 3 \, b e f^{2} h^{2} i + 3 \, b e^{2} f h i^{2} - b e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{2} + 6 \, {\left (18 \, {\left (a - b\right )} f^{3} h^{2} i - 9 \, {\left (2 \, a - 3 \, b\right )} e f^{2} h i^{2} + {\left (6 \, a - 11 \, b\right )} e^{2} f i^{3}\right )} x + 6 \, {\left (2 \, b f^{3} i^{3} x^{3} + 6 \, a f^{3} h^{3} - 18 \, {\left (a - b\right )} e f^{2} h^{2} i + 9 \, {\left (2 \, a - 3 \, b\right )} e^{2} f h i^{2} - {\left (6 \, a - 11 \, b\right )} e^{3} i^{3} + 3 \, {\left (3 \, b f^{3} h i^{2} - b e f^{2} i^{3}\right )} x^{2} + 6 \, {\left (3 \, b f^{3} h^{2} i - 3 \, b e f^{2} h i^{2} + b e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )}{36 \, d f^{4}} \]

[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] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.75 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=x^{3} \left (\frac {a i^{3}}{3 d f} - \frac {b i^{3}}{9 d f}\right ) + x^{2} \left (- \frac {a e i^{3}}{2 d f^{2}} + \frac {3 a h i^{2}}{2 d f} + \frac {5 b e i^{3}}{12 d f^{2}} - \frac {3 b h i^{2}}{4 d f}\right ) + x \left (\frac {a e^{2} i^{3}}{d f^{3}} - \frac {3 a e h i^{2}}{d f^{2}} + \frac {3 a h^{2} i}{d f} - \frac {11 b e^{2} i^{3}}{6 d f^{3}} + \frac {9 b e h i^{2}}{2 d f^{2}} - \frac {3 b h^{2} i}{d f}\right ) + \frac {\left (6 b e^{2} i^{3} x - 18 b e f h i^{2} x - 3 b e f i^{3} x^{2} + 18 b f^{2} h^{2} i x + 9 b f^{2} h i^{2} x^{2} + 2 b f^{2} i^{3} x^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}}{6 d f^{3}} + \frac {\left (- b e^{3} i^{3} + 3 b e^{2} f h i^{2} - 3 b e f^{2} h^{2} i + b f^{3} h^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{4}} - \frac {\left (6 a e^{3} i^{3} - 18 a e^{2} f h i^{2} + 18 a e f^{2} h^{2} i - 6 a f^{3} h^{3} - 11 b e^{3} i^{3} + 27 b e^{2} f h i^{2} - 18 b e f^{2} h^{2} i\right ) \log {\left (e + f x \right )}}{6 d f^{4}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (234) = 468\).

Time = 0.23 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.21 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=3 \, b h^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) - \frac {1}{6} \, b i^{3} {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac {3}{2} \, b h i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} \log \left (c f x + c e\right ) - \frac {1}{2} \, b h^{3} {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + 3 \, a h^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - \frac {1}{6} \, a i^{3} {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} + \frac {3}{2} \, a h i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} + \frac {b h^{3} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a h^{3} \log \left (d f x + d e\right )}{d f} + \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} b h^{2} i}{2 \, d f^{2}} - \frac {3 \, {\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} b h i^{2}}{4 \, d f^{3}} - \frac {{\left (4 \, f^{3} x^{3} - 15 \, e f^{2} x^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} + 66 \, e^{2} f x - 66 \, e^{3} \log \left (f x + e\right )\right )} b i^{3}}{36 \, d f^{4}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.51 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {{\left (3 \, a i^{3} - b i^{3}\right )} x^{3}}{9 \, d f} + \frac {1}{6} \, {\left (\frac {2 \, b i^{3} x^{3}}{d f} + \frac {3 \, {\left (3 \, b f h i^{2} - b e i^{3}\right )} x^{2}}{d f^{2}} + \frac {6 \, {\left (3 \, b f^{2} h^{2} i - 3 \, b e f h i^{2} + b e^{2} i^{3}\right )} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac {{\left (18 \, a f h i^{2} - 9 \, b f h i^{2} - 6 \, a e i^{3} + 5 \, b e i^{3}\right )} x^{2}}{12 \, d f^{2}} + \frac {{\left (18 \, a f^{2} h^{2} i - 18 \, b f^{2} h^{2} i - 18 \, a e f h i^{2} + 27 \, b e f h i^{2} + 6 \, a e^{2} i^{3} - 11 \, b e^{2} i^{3}\right )} x}{6 \, d f^{3}} + \frac {{\left (b f^{3} h^{3} - 3 \, b e f^{2} h^{2} i + 3 \, b e^{2} f h i^{2} - b e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{2}}{2 \, d f^{4}} + \frac {{\left (6 \, a f^{3} h^{3} - 18 \, a e f^{2} h^{2} i + 18 \, b e f^{2} h^{2} i + 18 \, a e^{2} f h i^{2} - 27 \, b e^{2} f h i^{2} - 6 \, a e^{3} i^{3} + 11 \, b e^{3} i^{3}\right )} \log \left (f x + e\right )}{6 \, d f^{4}} \]

[In]

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

[Out]

1/9*(3*a*i^3 - b*i^3)*x^3/(d*f) + 1/6*(2*b*i^3*x^3/(d*f) + 3*(3*b*f*h*i^2 - b*e*i^3)*x^2/(d*f^2) + 6*(3*b*f^2*
h^2*i - 3*b*e*f*h*i^2 + b*e^2*i^3)*x/(d*f^3))*log(c*f*x + c*e) + 1/12*(18*a*f*h*i^2 - 9*b*f*h*i^2 - 6*a*e*i^3
+ 5*b*e*i^3)*x^2/(d*f^2) + 1/6*(18*a*f^2*h^2*i - 18*b*f^2*h^2*i - 18*a*e*f*h*i^2 + 27*b*e*f*h*i^2 + 6*a*e^2*i^
3 - 11*b*e^2*i^3)*x/(d*f^3) + 1/2*(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/(d*f^4) + 1/6*(6*a*f^3*h^3 - 18*a*e*f^2*h^2*i + 18*b*e*f^2*h^2*i + 18*a*e^2*f*h*i^2 - 27*b*e^2*f*h*i^2 - 6*
a*e^3*i^3 + 11*b*e^3*i^3)*log(f*x + e)/(d*f^4)

Mupad [B] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.61 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=x^2\,\left (\frac {i^2\,\left (6\,a\,f\,h+b\,e\,i-3\,b\,f\,h\right )}{4\,d\,f^2}-\frac {e\,i^3\,\left (3\,a-b\right )}{6\,d\,f^2}\right )-x\,\left (\frac {e\,\left (\frac {i^2\,\left (6\,a\,f\,h+b\,e\,i-3\,b\,f\,h\right )}{2\,d\,f^2}-\frac {e\,i^3\,\left (3\,a-b\right )}{3\,d\,f^2}\right )}{f}-\frac {i\,\left (3\,a\,f^2\,h^2-b\,e^2\,i^2-3\,b\,f^2\,h^2+3\,b\,e\,f\,h\,i\right )}{d\,f^3}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {b\,i^3\,x^3}{3\,d\,f^2}+\frac {b\,i\,x\,\left (e^2\,i^2-3\,e\,f\,h\,i+3\,f^2\,h^2\right )}{d\,f^4}-\frac {b\,i^2\,x^2\,\left (e\,i-3\,f\,h\right )}{2\,d\,f^3}\right )+\frac {\ln \left (e+f\,x\right )\,\left (6\,a\,f^3\,h^3-6\,a\,e^3\,i^3+11\,b\,e^3\,i^3-18\,a\,e\,f^2\,h^2\,i+18\,a\,e^2\,f\,h\,i^2+18\,b\,e\,f^2\,h^2\,i-27\,b\,e^2\,f\,h\,i^2\right )}{6\,d\,f^4}+\frac {i^3\,x^3\,\left (3\,a-b\right )}{9\,d\,f}-\frac {b\,{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (e^3\,i^3-3\,e^2\,f\,h\,i^2+3\,e\,f^2\,h^2\,i-f^3\,h^3\right )}{2\,d\,f^4} \]

[In]

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

[Out]

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