3.2.91 \(\int \frac {(h+i x)^4}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx\) [191]

Optimal. Leaf size=230 \[ \frac {4 e^{-\frac {a}{b}} i (f h-e i)^3 \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^5}+\frac {6 e^{-\frac {2 a}{b}} i^2 (f h-e i)^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^5}+\frac {4 e^{-\frac {3 a}{b}} i^3 (f h-e i) \text {Ei}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^5}+\frac {e^{-\frac {4 a}{b}} i^4 \text {Ei}\left (\frac {4 (a+b \log (c (e+f x)))}{b}\right )}{b c^4 d f^5}+\frac {(f h-e i)^4 \log (a+b \log (c (e+f x)))}{b d f^5} \]

[Out]

4*i*(-e*i+f*h)^3*Ei((a+b*ln(c*(f*x+e)))/b)/b/c/d/exp(a/b)/f^5+6*i^2*(-e*i+f*h)^2*Ei(2*(a+b*ln(c*(f*x+e)))/b)/b
/c^2/d/exp(2*a/b)/f^5+4*i^3*(-e*i+f*h)*Ei(3*(a+b*ln(c*(f*x+e)))/b)/b/c^3/d/exp(3*a/b)/f^5+i^4*Ei(4*(a+b*ln(c*(
f*x+e)))/b)/b/c^4/d/exp(4*a/b)/f^5+(-e*i+f*h)^4*ln(a+b*ln(c*(f*x+e)))/b/d/f^5

________________________________________________________________________________________

Rubi [A]
time = 0.47, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2458, 12, 2395, 2336, 2209, 2339, 29, 2346} \begin {gather*} \frac {i^4 e^{-\frac {4 a}{b}} \text {Ei}\left (\frac {4 (a+b \log (c (e+f x)))}{b}\right )}{b c^4 d f^5}+\frac {4 i^3 e^{-\frac {3 a}{b}} (f h-e i) \text {Ei}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^5}+\frac {6 i^2 e^{-\frac {2 a}{b}} (f h-e i)^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^5}+\frac {4 i e^{-\frac {a}{b}} (f h-e i)^3 \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^5}+\frac {(f h-e i)^4 \log (a+b \log (c (e+f x)))}{b d f^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*i*(f*h - e*i)^3*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(b*c*d*E^(a/b)*f^5) + (6*i^2*(f*h - e*i)^2*ExpIn
tegralEi[(2*(a + b*Log[c*(e + f*x)]))/b])/(b*c^2*d*E^((2*a)/b)*f^5) + (4*i^3*(f*h - e*i)*ExpIntegralEi[(3*(a +
b*Log[c*(e + f*x)]))/b])/(b*c^3*d*E^((3*a)/b)*f^5) + (i^4*ExpIntegralEi[(4*(a + b*Log[c*(e + f*x)]))/b])/(b*c
^4*d*E^((4*a)/b)*f^5) + ((f*h - e*i)^4*Log[a + b*Log[c*(e + f*x)]])/(b*d*f^5)

Rule 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

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 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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+191 x)^4}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-191 e+f h}{f}+\frac {191 x}{f}\right )^4}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-191 e+f h}{f}+\frac {191 x}{f}\right )^4}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {764 (191 e-f h)^3}{f^4 (a+b \log (c x))}+\frac {(191 e-f h)^4}{f^4 x (a+b \log (c x))}+\frac {218886 (191 e-f h)^2 x}{f^4 (a+b \log (c x))}-\frac {27871484 (191 e-f h) x^2}{f^4 (a+b \log (c x))}+\frac {1330863361 x^3}{f^4 (a+b \log (c x))}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac {1330863361 \text {Subst}\left (\int \frac {x^3}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^5}-\frac {(27871484 (191 e-f h)) \text {Subst}\left (\int \frac {x^2}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^5}+\frac {\left (218886 (191 e-f h)^2\right ) \text {Subst}\left (\int \frac {x}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^5}-\frac {\left (764 (191 e-f h)^3\right ) \text {Subst}\left (\int \frac {1}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^5}+\frac {(191 e-f h)^4 \text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^5}\\ &=\frac {1330863361 \text {Subst}\left (\int \frac {e^{4 x}}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c^4 d f^5}-\frac {(27871484 (191 e-f h)) \text {Subst}\left (\int \frac {e^{3 x}}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c^3 d f^5}+\frac {\left (218886 (191 e-f h)^2\right ) \text {Subst}\left (\int \frac {e^{2 x}}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c^2 d f^5}-\frac {\left (764 (191 e-f h)^3\right ) \text {Subst}\left (\int \frac {e^x}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c d f^5}+\frac {(191 e-f h)^4 \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f^5}\\ &=-\frac {764 e^{-\frac {a}{b}} (191 e-f h)^3 \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^5}+\frac {218886 e^{-\frac {2 a}{b}} (191 e-f h)^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^5}-\frac {27871484 e^{-\frac {3 a}{b}} (191 e-f h) \text {Ei}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^5}+\frac {1330863361 e^{-\frac {4 a}{b}} \text {Ei}\left (\frac {4 (a+b \log (c (e+f x)))}{b}\right )}{b c^4 d f^5}+\frac {(191 e-f h)^4 \log (a+b \log (c (e+f x)))}{b d f^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.80, size = 397, normalized size = 1.73 \begin {gather*} \frac {e^{-\frac {4 a}{b}} \left (4 c^3 e^{\frac {3 a}{b}} i (f h-e i)^3 \text {Ei}\left (\frac {a}{b}+\log (c (e+f x))\right )+6 c^2 e e^{\frac {2 a}{b}} i^3 (-2 f h+e i) \text {Ei}\left (2 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )+4 c e^{a/b} f h i^3 \text {Ei}\left (3 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )-4 c e e^{a/b} i^4 \text {Ei}\left (3 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )+i^4 \text {Ei}\left (4 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )+6 c^2 e^{\frac {2 a}{b}} f^2 h^2 i^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )-4 c^4 e e^{\frac {4 a}{b}} f^3 h^3 i \log (a+b \log (c (e+f x)))+6 c^4 e^2 e^{\frac {4 a}{b}} f^2 h^2 i^2 \log (a+b \log (c (e+f x)))-4 c^4 e^3 e^{\frac {4 a}{b}} f h i^3 \log (a+b \log (c (e+f x)))+c^4 e^4 e^{\frac {4 a}{b}} i^4 \log (a+b \log (c (e+f x)))+c^4 e^{\frac {4 a}{b}} f^4 h^4 \log (f (a+b \log (c (e+f x))))\right )}{b c^4 d f^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c^3*E^((3*a)/b)*i*(f*h - e*i)^3*ExpIntegralEi[a/b + Log[c*(e + f*x)]] + 6*c^2*e*E^((2*a)/b)*i^3*(-2*f*h + e
*i)*ExpIntegralEi[2*(a/b + Log[c*(e + f*x)])] + 4*c*E^(a/b)*f*h*i^3*ExpIntegralEi[3*(a/b + Log[c*(e + f*x)])]
- 4*c*e*E^(a/b)*i^4*ExpIntegralEi[3*(a/b + Log[c*(e + f*x)])] + i^4*ExpIntegralEi[4*(a/b + Log[c*(e + f*x)])]
+ 6*c^2*E^((2*a)/b)*f^2*h^2*i^2*ExpIntegralEi[(2*(a + b*Log[c*(e + f*x)]))/b] - 4*c^4*e*E^((4*a)/b)*f^3*h^3*i*
Log[a + b*Log[c*(e + f*x)]] + 6*c^4*e^2*E^((4*a)/b)*f^2*h^2*i^2*Log[a + b*Log[c*(e + f*x)]] - 4*c^4*e^3*E^((4*
a)/b)*f*h*i^3*Log[a + b*Log[c*(e + f*x)]] + c^4*e^4*E^((4*a)/b)*i^4*Log[a + b*Log[c*(e + f*x)]] + c^4*E^((4*a)
/b)*f^4*h^4*Log[f*(a + b*Log[c*(e + f*x)])])/(b*c^4*d*E^((4*a)/b)*f^5)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(566\) vs. \(2(233)=466\).
time = 3.64, size = 567, normalized size = 2.47 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x+h)^4/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x,method=_RETURNVERBOSE)

[Out]

1/c^4/f^5/d*(-i^4/b*exp(-4*a/b)*Ei(1,-4*ln(c*f*x+c*e)-4*a/b)+c^4*e^4*i^4*ln(a+b*ln(c*f*x+c*e))/b+c^4*f^4*h^4*l
n(a+b*ln(c*f*x+c*e))/b+4*c*e*i^4/b*exp(-3*a/b)*Ei(1,-3*ln(c*f*x+c*e)-3*a/b)-6*c^2*e^2*i^4/b*exp(-2*a/b)*Ei(1,-
2*ln(c*f*x+c*e)-2*a/b)+4*c^3*e^3*i^4/b*exp(-a/b)*Ei(1,-ln(c*f*x+c*e)-a/b)-4*c*f*h*i^3/b*exp(-3*a/b)*Ei(1,-3*ln
(c*f*x+c*e)-3*a/b)-6*c^2*f^2*h^2*i^2/b*exp(-2*a/b)*Ei(1,-2*ln(c*f*x+c*e)-2*a/b)-4*c^3*f^3*h^3*i/b*exp(-a/b)*Ei
(1,-ln(c*f*x+c*e)-a/b)-4*c^4*e*f^3*h^3*i*ln(a+b*ln(c*f*x+c*e))/b+6*c^4*e^2*f^2*h^2*i^2*ln(a+b*ln(c*f*x+c*e))/b
-4*c^4*e^3*f*h*i^3*ln(a+b*ln(c*f*x+c*e))/b+12*c^2*e*f*h*i^3/b*exp(-2*a/b)*Ei(1,-2*ln(c*f*x+c*e)-2*a/b)+12*c^3*
e*f^2*h^2*i^2/b*exp(-a/b)*Ei(1,-ln(c*f*x+c*e)-a/b)-12*c^3*e^2*f*h*i^3/b*exp(-a/b)*Ei(1,-ln(c*f*x+c*e)-a/b))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 364, normalized size = 1.58 \begin {gather*} \frac {{\left ({\left (c^{4} f^{4} h^{4} - 4 i \, c^{4} f^{3} h^{3} e - 6 \, c^{4} f^{2} h^{2} e^{2} + 4 i \, c^{4} f h e^{3} + c^{4} e^{4}\right )} e^{\left (\frac {4 \, a}{b}\right )} \log \left (\frac {b \log \left (c f x + c e\right ) + a}{b}\right ) - 4 \, {\left (i \, c f h + c e\right )} e^{\frac {a}{b}} \operatorname {log\_integral}\left ({\left (c^{3} f^{3} x^{3} + 3 \, c^{3} f^{2} x^{2} e + 3 \, c^{3} f x e^{2} + c^{3} e^{3}\right )} e^{\left (\frac {3 \, a}{b}\right )}\right ) - 6 \, {\left (c^{2} f^{2} h^{2} - 2 i \, c^{2} f h e - c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )} \operatorname {log\_integral}\left ({\left (c^{2} f^{2} x^{2} + 2 \, c^{2} f x e + c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )}\right ) - 4 \, {\left (-i \, c^{3} f^{3} h^{3} - 3 \, c^{3} f^{2} h^{2} e + 3 i \, c^{3} f h e^{2} + c^{3} e^{3}\right )} e^{\left (\frac {3 \, a}{b}\right )} \operatorname {log\_integral}\left ({\left (c f x + c e\right )} e^{\frac {a}{b}}\right ) + \operatorname {log\_integral}\left ({\left (c^{4} f^{4} x^{4} + 4 \, c^{4} f^{3} x^{3} e + 6 \, c^{4} f^{2} x^{2} e^{2} + 4 \, c^{4} f x e^{3} + c^{4} e^{4}\right )} e^{\left (\frac {4 \, a}{b}\right )}\right )\right )} e^{\left (-\frac {4 \, a}{b}\right )}}{b c^{4} d f^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c^4*f^4*h^4 - 4*I*c^4*f^3*h^3*e - 6*c^4*f^2*h^2*e^2 + 4*I*c^4*f*h*e^3 + c^4*e^4)*e^(4*a/b)*log((b*log(c*f*x
+ c*e) + a)/b) - 4*(I*c*f*h + c*e)*e^(a/b)*log_integral((c^3*f^3*x^3 + 3*c^3*f^2*x^2*e + 3*c^3*f*x*e^2 + c^3*e
^3)*e^(3*a/b)) - 6*(c^2*f^2*h^2 - 2*I*c^2*f*h*e - c^2*e^2)*e^(2*a/b)*log_integral((c^2*f^2*x^2 + 2*c^2*f*x*e +
c^2*e^2)*e^(2*a/b)) - 4*(-I*c^3*f^3*h^3 - 3*c^3*f^2*h^2*e + 3*I*c^3*f*h*e^2 + c^3*e^3)*e^(3*a/b)*log_integral
((c*f*x + c*e)*e^(a/b)) + log_integral((c^4*f^4*x^4 + 4*c^4*f^3*x^3*e + 6*c^4*f^2*x^2*e^2 + 4*c^4*f*x*e^3 + c^
4*e^4)*e^(4*a/b)))*e^(-4*a/b)/(b*c^4*d*f^5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {h^{4}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {i^{4} x^{4}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {4 h i^{3} x^{3}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {6 h^{2} i^{2} x^{2}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {4 h^{3} i x}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(h**4/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x)), x) + Integral(i**4*x**4/(a*e + a
*f*x + b*e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x)), x) + Integral(4*h*i**3*x**3/(a*e + a*f*x + b*e*log(c*e
+ c*f*x) + b*f*x*log(c*e + c*f*x)), x) + Integral(6*h**2*i**2*x**2/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x
*log(c*e + c*f*x)), x) + Integral(4*h**3*i*x/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x)), x)
)/d

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((h + I*x)^4/((d*f*x + d*e)*(b*log((f*x + e)*c) + a)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (h+i\,x\right )}^4}{\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h + i*x)^4/((d*e + d*f*x)*(a + b*log(c*(e + f*x)))),x)

[Out]

int((h + i*x)^4/((d*e + d*f*x)*(a + b*log(c*(e + f*x)))), x)

________________________________________________________________________________________