\(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^6} \, dx\) [403]

Optimal result

Integrand size = 23, antiderivative size = 183 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b d^3 n}{25 x^5}-\frac {3 b d^2 e n x^{-5+r}}{(5-r)^2}-\frac {3 b d e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac {b e^3 n x^{-5+3 r}}{(5-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{-5+3 r} \left (a+b \log \left (c x^n\right )\right )}{5-3 r} \]

[Out]

-1/25*b*d^3*n/x^5-3*b*d^2*e*n*x^(-5+r)/(5-r)^2-3*b*d*e^2*n*x^(-5+2*r)/(5-2*r)^2-b*e^3*n*x^(-5+3*r)/(5-3*r)^2-1
/5*d^3*(a+b*ln(c*x^n))/x^5-3*d^2*e*x^(-5+r)*(a+b*ln(c*x^n))/(5-r)-3*d*e^2*x^(-5+2*r)*(a+b*ln(c*x^n))/(5-2*r)-e
^3*x^(-5+3*r)*(a+b*ln(c*x^n))/(5-3*r)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2372, 12, 14} \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{2 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{3 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-3 r}-\frac {b d^3 n}{25 x^5}-\frac {3 b d^2 e n x^{r-5}}{(5-r)^2}-\frac {3 b d e^2 n x^{2 r-5}}{(5-2 r)^2}-\frac {b e^3 n x^{3 r-5}}{(5-3 r)^2} \]

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/25*(b*d^3*n)/x^5 - (3*b*d^2*e*n*x^(-5 + r))/(5 - r)^2 - (3*b*d*e^2*n*x^(-5 + 2*r))/(5 - 2*r)^2 - (b*e^3*n*x
^(-5 + 3*r))/(5 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (3*d^2*e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r)
- (3*d*e^2*x^(-5 + 2*r)*(a + b*Log[c*x^n]))/(5 - 2*r) - (e^3*x^(-5 + 3*r)*(a + b*Log[c*x^n]))/(5 - 3*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 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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{-5+3 r} \left (a+b \log \left (c x^n\right )\right )}{5-3 r}-(b n) \int \frac {-d^3+\frac {15 d^2 e x^r}{-5+r}+\frac {15 d e^2 x^{2 r}}{-5+2 r}+\frac {5 e^3 x^{3 r}}{-5+3 r}}{5 x^6} \, dx \\ & = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{-5+3 r} \left (a+b \log \left (c x^n\right )\right )}{5-3 r}-\frac {1}{5} (b n) \int \frac {-d^3+\frac {15 d^2 e x^r}{-5+r}+\frac {15 d e^2 x^{2 r}}{-5+2 r}+\frac {5 e^3 x^{3 r}}{-5+3 r}}{x^6} \, dx \\ & = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{-5+3 r} \left (a+b \log \left (c x^n\right )\right )}{5-3 r}-\frac {1}{5} (b n) \int \left (-\frac {d^3}{x^6}+\frac {15 d^2 e x^{-6+r}}{-5+r}+\frac {15 d e^2 x^{2 (-3+r)}}{-5+2 r}+\frac {5 e^3 x^{3 (-2+r)}}{-5+3 r}\right ) \, dx \\ & = -\frac {b d^3 n}{25 x^5}-\frac {3 b d^2 e n x^{-5+r}}{(5-r)^2}-\frac {3 b d e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac {b e^3 n x^{-5+3 r}}{(5-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{-5+3 r} \left (a+b \log \left (c x^n\right )\right )}{5-3 r} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\frac {b n \left (-d^3-\frac {75 d^2 e x^r}{(-5+r)^2}-\frac {75 d e^2 x^{2 r}}{(5-2 r)^2}-\frac {25 e^3 x^{3 r}}{(5-3 r)^2}\right )+a \left (-5 d^3+\frac {75 d^2 e x^r}{-5+r}+\frac {75 d e^2 x^{2 r}}{-5+2 r}+\frac {25 e^3 x^{3 r}}{-5+3 r}\right )+5 b \left (-d^3+\frac {15 d^2 e x^r}{-5+r}+\frac {15 d e^2 x^{2 r}}{-5+2 r}+\frac {5 e^3 x^{3 r}}{-5+3 r}\right ) \log \left (c x^n\right )}{25 x^5} \]

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

(b*n*(-d^3 - (75*d^2*e*x^r)/(-5 + r)^2 - (75*d*e^2*x^(2*r))/(5 - 2*r)^2 - (25*e^3*x^(3*r))/(5 - 3*r)^2) + a*(-
5*d^3 + (75*d^2*e*x^r)/(-5 + r) + (75*d*e^2*x^(2*r))/(-5 + 2*r) + (25*e^3*x^(3*r))/(-5 + 3*r)) + 5*b*(-d^3 + (
15*d^2*e*x^r)/(-5 + r) + (15*d*e^2*x^(2*r))/(-5 + 2*r) + (5*e^3*x^(3*r))/(-5 + 3*r))*Log[c*x^n])/(25*x^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1040\) vs. \(2(179)=358\).

Time = 3.65 (sec) , antiderivative size = 1041, normalized size of antiderivative = 5.69

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1041\)
risch	\(\text {Expression too large to display}\)	\(4031\)

[In]

int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/25*(78125*b*ln(c*x^n)*d^3+234375*b*d*e^2*ln(c*x^n)*(x^r)^2+78125*e^3*(x^r)^3*a-18000*b*d^3*n*r^3+36250*b*d^
3*n*r^2-37500*b*d^3*n*r+356250*a*d*e^2*r^2*(x^r)^2-127500*a*d*e^2*r^3*(x^r)^2+234375*d*e^2*(x^r)^2*a+234375*d^
2*e*x^r*a+78125*a*d^3-31875*a*e^3*r^3*(x^r)^3+96875*a*e^3*r^2*(x^r)^3-140625*a*e^3*r*(x^r)^3-300*a*e^3*r^5*(x^
r)^3+5000*a*e^3*r^4*(x^r)^3-300*(x^r)^3*ln(c*x^n)*b*e^3*r^5+5000*(x^r)^3*ln(c*x^n)*b*e^3*r^4-31875*(x^r)^3*ln(
c*x^n)*b*e^3*r^3+96875*(x^r)^3*ln(c*x^n)*b*e^3*r^2-140625*(x^r)^3*ln(c*x^n)*b*e^3*r+234375*b*d^2*e*ln(c*x^n)*x
^r+36*b*d^3*n*r^6-660*b*d^3*n*r^5+4825*b*d^3*n*r^4-2700*x^r*ln(c*x^n)*b*d^2*e*r^5+36000*x^r*ln(c*x^n)*b*d^2*e*
r^4-181875*x^r*ln(c*x^n)*b*d^2*e*r^3+440625*x^r*ln(c*x^n)*b*d^2*e*r^2-515625*x^r*ln(c*x^n)*b*d^2*e*r-1350*(x^r
)^2*ln(c*x^n)*b*d*e^2*r^5+21375*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-127500*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3+356250*(x^r
)^2*ln(c*x^n)*b*d*e^2*r^2-468750*(x^r)^2*ln(c*x^n)*b*d*e^2*r+15625*b*d^3*n+180*ln(c*x^n)*b*d^3*r^6-3300*ln(c*x
^n)*b*d^3*r^5+24125*ln(c*x^n)*b*d^3*r^4-90000*ln(c*x^n)*b*d^3*r^3+181250*ln(c*x^n)*b*d^3*r^2-187500*ln(c*x^n)*
b*d^3*r+78125*e^3*b*ln(c*x^n)*(x^r)^3-90000*a*d^3*r^3+181250*a*d^3*r^2-187500*a*d^3*r+180*a*d^3*r^6-3300*a*d^3
*r^5+24125*a*d^3*r^4-181875*a*d^2*e*r^3*x^r+46875*b*d*e^2*n*(x^r)^2+46875*b*d^2*e*n*x^r+15625*b*e^3*n*(x^r)^3-
1500*b*e^3*n*r^3*(x^r)^3+8125*b*e^3*n*r^2*(x^r)^3-18750*b*e^3*n*r*(x^r)^3+440625*a*d^2*e*r^2*x^r+36000*a*d^2*e
*r^4*x^r-515625*a*d^2*e*r*x^r+100*b*e^3*n*r^4*(x^r)^3-468750*a*d*e^2*r*(x^r)^2-1350*a*d*e^2*r^5*(x^r)^2+21375*
a*d*e^2*r^4*(x^r)^2-2700*a*d^2*e*r^5*x^r+41250*b*d*e^2*n*r^2*(x^r)^2+69375*b*d^2*e*n*r^2*x^r-75000*b*d*e^2*n*r
*(x^r)^2-93750*b*d^2*e*n*r*x^r-22500*b*d^2*e*n*r^3*x^r+675*b*d*e^2*n*r^4*(x^r)^2-9000*b*d*e^2*n*r^3*(x^r)^2+27
00*b*d^2*e*n*r^4*x^r)/x^5/(-5+3*r)^2/(-5+2*r)^2/(-5+r)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (174) = 348\).

Time = 0.36 (sec) , antiderivative size = 981, normalized size of antiderivative = 5.36 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\text {Too large to display} \]

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^6,x, algorithm="fricas")

[Out]

-1/25*(36*(b*d^3*n + 5*a*d^3)*r^6 - 660*(b*d^3*n + 5*a*d^3)*r^5 + 15625*b*d^3*n + 4825*(b*d^3*n + 5*a*d^3)*r^4
+ 78125*a*d^3 - 18000*(b*d^3*n + 5*a*d^3)*r^3 + 36250*(b*d^3*n + 5*a*d^3)*r^2 - 37500*(b*d^3*n + 5*a*d^3)*r -
25*(12*a*e^3*r^5 - 625*b*e^3*n - 4*(b*e^3*n + 50*a*e^3)*r^4 - 3125*a*e^3 + 15*(4*b*e^3*n + 85*a*e^3)*r^3 - 25
*(13*b*e^3*n + 155*a*e^3)*r^2 + 375*(2*b*e^3*n + 15*a*e^3)*r + (12*b*e^3*r^5 - 200*b*e^3*r^4 + 1275*b*e^3*r^3
- 3875*b*e^3*r^2 + 5625*b*e^3*r - 3125*b*e^3)*log(c) + (12*b*e^3*n*r^5 - 200*b*e^3*n*r^4 + 1275*b*e^3*n*r^3 -
3875*b*e^3*n*r^2 + 5625*b*e^3*n*r - 3125*b*e^3*n)*log(x))*x^(3*r) - 75*(18*a*d*e^2*r^5 - 625*b*d*e^2*n - 3*(3*
b*d*e^2*n + 95*a*d*e^2)*r^4 - 3125*a*d*e^2 + 20*(6*b*d*e^2*n + 85*a*d*e^2)*r^3 - 50*(11*b*d*e^2*n + 95*a*d*e^2
)*r^2 + 250*(4*b*d*e^2*n + 25*a*d*e^2)*r + (18*b*d*e^2*r^5 - 285*b*d*e^2*r^4 + 1700*b*d*e^2*r^3 - 4750*b*d*e^2
*r^2 + 6250*b*d*e^2*r - 3125*b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 285*b*d*e^2*n*r^4 + 1700*b*d*e^2*n*r^3 - 47
50*b*d*e^2*n*r^2 + 6250*b*d*e^2*n*r - 3125*b*d*e^2*n)*log(x))*x^(2*r) - 75*(36*a*d^2*e*r^5 - 625*b*d^2*e*n - 1
2*(3*b*d^2*e*n + 40*a*d^2*e)*r^4 - 3125*a*d^2*e + 25*(12*b*d^2*e*n + 97*a*d^2*e)*r^3 - 25*(37*b*d^2*e*n + 235*
a*d^2*e)*r^2 + 625*(2*b*d^2*e*n + 11*a*d^2*e)*r + (36*b*d^2*e*r^5 - 480*b*d^2*e*r^4 + 2425*b*d^2*e*r^3 - 5875*
b*d^2*e*r^2 + 6875*b*d^2*e*r - 3125*b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 480*b*d^2*e*n*r^4 + 2425*b*d^2*e*n*r
^3 - 5875*b*d^2*e*n*r^2 + 6875*b*d^2*e*n*r - 3125*b*d^2*e*n)*log(x))*x^r + 5*(36*b*d^3*r^6 - 660*b*d^3*r^5 + 4
825*b*d^3*r^4 - 18000*b*d^3*r^3 + 36250*b*d^3*r^2 - 37500*b*d^3*r + 15625*b*d^3)*log(c) + 5*(36*b*d^3*n*r^6 -
660*b*d^3*n*r^5 + 4825*b*d^3*n*r^4 - 18000*b*d^3*n*r^3 + 36250*b*d^3*n*r^2 - 37500*b*d^3*n*r + 15625*b*d^3*n)*
log(x))/((36*r^6 - 660*r^5 + 4825*r^4 - 18000*r^3 + 36250*r^2 - 37500*r + 15625)*x^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\text {Timed out} \]

[In]

integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**6,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(r-6>0)', see `assume?` for mor
e details)Is

Giac [F]

\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{6}} \,d x } \]

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^6,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^6, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \]

[In]

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^6,x)

[Out]

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^6, x)