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

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

[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

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Rubi [A]
time = 0.28, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2458, 12, 45, 2372, 14, 2338} \begin {gather*} \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} \end {gather*}

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) + (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*} \int \frac {(h+176 x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx &=\frac {\text {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 {\text {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 \text {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 \text {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 \text {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) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.21, size = 375, normalized size = 1.54 \begin {gather*} \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} \end {gather*}

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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(620\) vs. \(2(234)=468\).
time = 0.56, size = 621, normalized size = 2.55 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)))/(d*f*x+d*e),x,method=_RETURNVERBOSE)

[Out]

1/c/f*(-c/f^3/d*a*e^3*i^3*ln(c*f*x+c*e)+3*c/f^2/d*a*e^2*h*i^2*ln(c*f*x+c*e)-3*c/f/d*a*e*h^2*i*ln(c*f*x+c*e)+c/
d*a*h^3*ln(c*f*x+c*e)+3/f^3/d*a*e^2*i^3*(c*f*x+c*e)-6/f^2/d*a*e*h*i^2*(c*f*x+c*e)+3/f/d*a*h^2*i*(c*f*x+c*e)-3/
2/c/f^3/d*a*e*i^3*(c*f*x+c*e)^2+3/2/c/f^2/d*a*h*i^2*(c*f*x+c*e)^2+1/3/c^2/f^3/d*a*i^3*(c*f*x+c*e)^3-1/2*c/f^3/
d*b*e^3*i^3*ln(c*f*x+c*e)^2+3/2*c/f^2/d*b*e^2*h*i^2*ln(c*f*x+c*e)^2-3/2*c/f/d*b*e*h^2*i*ln(c*f*x+c*e)^2+1/2*c/
d*b*h^3*ln(c*f*x+c*e)^2+3/f^3/d*b*e^2*i^3*((c*f*x+c*e)*ln(c*f*x+c*e)-c*f*x-c*e)-6/f^2/d*b*e*h*i^2*((c*f*x+c*e)
*ln(c*f*x+c*e)-c*f*x-c*e)+3/f/d*b*h^2*i*((c*f*x+c*e)*ln(c*f*x+c*e)-c*f*x-c*e)-3/c/f^3/d*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*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)+1/
c^2/f^3/d*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. 537 vs. \(2 (231) = 462\).
time = 0.31, size = 537, normalized size = 2.20 \begin {gather*} -\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 i \, 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 ) + 3 i \, a h^{2} {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - \frac {3}{2} \, 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 {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 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}{6} i \, 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 h^{3} \log \left (d f x + d e\right )}{d f} - \frac {1}{6} i \, a {\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 )} b h^{2}}{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 )} b h}{4 \, d f^{3}} + \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 )} b}{36 \, 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)))/(d*f*x+d*e),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.40, size = 269, normalized size = 1.10 \begin {gather*} -\frac {108 \, {\left (-i \, a + i \, b\right )} f^{3} h^{2} x + 27 \, {\left (2 \, a - b\right )} f^{3} h x^{2} + 4 \, {\left (3 i \, a - i \, b\right )} f^{3} x^{3} + 6 \, {\left (6 i \, a - 11 i \, b\right )} f x e^{2} - 18 \, {\left (b f^{3} h^{3} - 3 i \, b f^{2} h^{2} e - 3 \, b f h e^{2} + i \, b e^{3}\right )} \log \left (c f x + c e\right )^{2} - 3 \, {\left (18 \, {\left (2 \, a - 3 \, b\right )} f^{2} h x - {\left (-6 i \, a + 5 i \, b\right )} f^{2} x^{2}\right )} e - 6 \, {\left (6 \, a f^{3} h^{3} + 18 i \, b f^{3} h^{2} x - 9 \, b f^{3} h x^{2} - 2 i \, b f^{3} x^{3} - {\left (-6 i \, a + 11 i \, b\right )} e^{3} - 3 \, {\left (3 \, {\left (2 \, a - 3 \, b\right )} f h + 2 i \, b f x\right )} e^{2} - 3 \, {\left (6 \, {\left (i \, a - i \, b\right )} f^{2} h^{2} - 6 \, b f^{2} h x - i \, b f^{2} x^{2}\right )} e\right )} \log \left (c f x + c e\right )}{36 \, 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)))/(d*f*x+d*e),x, algorithm="fricas")

[Out]

-1/36*(108*(-I*a + I*b)*f^3*h^2*x + 27*(2*a - b)*f^3*h*x^2 + 4*(3*I*a - I*b)*f^3*x^3 + 6*(6*I*a - 11*I*b)*f*x*
e^2 - 18*(b*f^3*h^3 - 3*I*b*f^2*h^2*e - 3*b*f*h*e^2 + I*b*e^3)*log(c*f*x + c*e)^2 - 3*(18*(2*a - 3*b)*f^2*h*x
- (-6*I*a + 5*I*b)*f^2*x^2)*e - 6*(6*a*f^3*h^3 + 18*I*b*f^3*h^2*x - 9*b*f^3*h*x^2 - 2*I*b*f^3*x^3 - (-6*I*a +
11*I*b)*e^3 - 3*(3*(2*a - 3*b)*f*h + 2*I*b*f*x)*e^2 - 3*(6*(I*a - I*b)*f^2*h^2 - 6*b*f^2*h*x - I*b*f^2*x^2)*e)
*log(c*f*x + c*e))/(d*f^4)

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Sympy [A]
time = 0.72, size = 427, normalized size = 1.75 \begin {gather*} 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}} \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)))/(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)

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Giac [A]
time = 5.97, size = 424, normalized size = 1.74 \begin {gather*} \frac {18 \, b f^{3} h^{3} \log \left (c f x + c e\right )^{2} + 108 i \, b f^{3} h^{2} x \log \left (c f x + c e\right ) - 54 \, b f^{3} h x^{2} \log \left (c f x + c e\right ) - 12 i \, b f^{3} x^{3} \log \left (c f x + c e\right ) - 54 i \, b f^{2} h^{2} e \log \left (c f x + c e\right )^{2} + 36 \, a f^{3} h^{3} \log \left (f x + e\right ) + 108 i \, a f^{3} h^{2} x - 108 i \, b f^{3} h^{2} x - 54 \, a f^{3} h x^{2} + 27 \, b f^{3} h x^{2} - 12 i \, a f^{3} x^{3} + 4 i \, b f^{3} x^{3} + 108 \, b f^{2} h x e \log \left (c f x + c e\right ) + 18 i \, b f^{2} x^{2} e \log \left (c f x + c e\right ) - 108 i \, a f^{2} h^{2} e \log \left (f x + e\right ) + 108 i \, b f^{2} h^{2} e \log \left (f x + e\right ) + 108 \, a f^{2} h x e - 162 \, b f^{2} h x e + 18 i \, a f^{2} x^{2} e - 15 i \, b f^{2} x^{2} e - 54 \, b f h e^{2} \log \left (c f x + c e\right )^{2} - 36 i \, b f x e^{2} \log \left (c f x + c e\right ) - 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 i \, a f x e^{2} + 66 i \, b f x e^{2} + 18 i \, b e^{3} \log \left (c f x + c e\right )^{2} + 36 i \, a e^{3} \log \left (f x + e\right ) - 66 i \, b e^{3} \log \left (f x + e\right )}{36 \, 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)))/(d*f*x+d*e),x, algorithm="giac")

[Out]

1/36*(18*b*f^3*h^3*log(c*f*x + c*e)^2 + 108*I*b*f^3*h^2*x*log(c*f*x + c*e) - 54*b*f^3*h*x^2*log(c*f*x + c*e) -
12*I*b*f^3*x^3*log(c*f*x + c*e) - 54*I*b*f^2*h^2*e*log(c*f*x + c*e)^2 + 36*a*f^3*h^3*log(f*x + e) + 108*I*a*f
^3*h^2*x - 108*I*b*f^3*h^2*x - 54*a*f^3*h*x^2 + 27*b*f^3*h*x^2 - 12*I*a*f^3*x^3 + 4*I*b*f^3*x^3 + 108*b*f^2*h*
x*e*log(c*f*x + c*e) + 18*I*b*f^2*x^2*e*log(c*f*x + c*e) - 108*I*a*f^2*h^2*e*log(f*x + e) + 108*I*b*f^2*h^2*e*
log(f*x + e) + 108*a*f^2*h*x*e - 162*b*f^2*h*x*e + 18*I*a*f^2*x^2*e - 15*I*b*f^2*x^2*e - 54*b*f*h*e^2*log(c*f*
x + c*e)^2 - 36*I*b*f*x*e^2*log(c*f*x + c*e) - 108*a*f*h*e^2*log(f*x + e) + 162*b*f*h*e^2*log(f*x + e) - 36*I*
a*f*x*e^2 + 66*I*b*f*x*e^2 + 18*I*b*e^3*log(c*f*x + c*e)^2 + 36*I*a*e^3*log(f*x + e) - 66*I*b*e^3*log(f*x + e)
)/(d*f^4)

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Mupad [B]
time = 0.41, size = 393, normalized size = 1.61 \begin {gather*} 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} \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))))/(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)

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