3.4.89 \(\int \frac {(d+e x^r)^2 (a+b \log (c x^n))}{x^4} \, dx\) [389]

Optimal. Leaf size=127 \[ -\frac {b d^2 n}{9 x^3}-\frac {2 b d e n x^{-3+r}}{(3-r)^2}-\frac {b e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \]

[Out]

-1/9*b*d^2*n/x^3-2*b*d*e*n*x^(-3+r)/(3-r)^2-b*e^2*n*x^(-3+2*r)/(3-2*r)^2-1/3*d^2*(a+b*ln(c*x^n))/x^3-2*d*e*x^(
-3+r)*(a+b*ln(c*x^n))/(3-r)-e^2*x^(-3+2*r)*(a+b*ln(c*x^n))/(3-2*r)

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Rubi [A]
time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2372, 12, 14} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n x^{r-3}}{(3-r)^2}-\frac {b e^2 n x^{2 r-3}}{(3-2 r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-1/9*(b*d^2*n)/x^3 - (2*b*d*e*n*x^(-3 + r))/(3 - r)^2 - (b*e^2*n*x^(-3 + 2*r))/(3 - 2*r)^2 - (d^2*(a + b*Log[c
*x^n]))/(3*x^3) - (2*d*e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r) - (e^2*x^(-3 + 2*r)*(a + b*Log[c*x^n]))/(3 - 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 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*} \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2+\frac {6 d e x^r}{-3+r}+\frac {3 e^2 x^{2 r}}{-3+2 r}}{3 x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \frac {-d^2+\frac {6 d e x^r}{-3+r}+\frac {3 e^2 x^{2 r}}{-3+2 r}}{x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (-\frac {d^2}{x^4}+\frac {6 d e x^{-4+r}}{-3+r}+\frac {3 e^2 x^{2 (-2+r)}}{-3+2 r}\right ) \, dx\\ &=-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n x^{-3+r}}{(3-r)^2}-\frac {b e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 119, normalized size = 0.94 \begin {gather*} \frac {-3 b d^2 n \log (x)-d^2 \left (3 a+b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+\frac {18 d e x^r \left (-b n+a (-3+r)+b (-3+r) \log \left (c x^n\right )\right )}{(-3+r)^2}+\frac {9 e^2 x^{2 r} \left (-b n+a (-3+2 r)+b (-3+2 r) \log \left (c x^n\right )\right )}{(3-2 r)^2}}{9 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^4,x]

[Out]

(-3*b*d^2*n*Log[x] - d^2*(3*a + b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + (18*d*e*x^r*(-(b*n) + a*(-3 + r) + b*(-
3 + r)*Log[c*x^n]))/(-3 + r)^2 + (9*e^2*x^(2*r)*(-(b*n) + a*(-3 + 2*r) + b*(-3 + 2*r)*Log[c*x^n]))/(3 - 2*r)^2
)/(9*x^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.25, size = 1930, normalized size = 15.20

method	result	size
risch	\(\text {Expression too large to display}\)	\(1930\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)^2*(a+b*ln(c*x^n))/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*b*(-3*e^2*(x^r)^2*r+2*d^2*r^2-12*d*e*x^r*r+9*e^2*(x^r)^2-9*d^2*r+18*d*e*x^r+9*d^2)/x^3/(-3+2*r)/(-3+r)*ln
(x^n)-1/18*(486*e^2*(x^r)^2*a-243*I*Pi*b*d^2*csgn(I*c*x^n)^3+972*d*e*x^r*a+18*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)*(x^r)^2-72*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r+234*b*d^2*n*r^2-324*b*d^2*n*r+702*l
n(c)*b*d^2*r^2-972*ln(c)*b*d^2*r+24*ln(c)*b*d^2*r^4-216*ln(c)*b*d^2*r^3-486*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*c*x^
n)^2+486*d^2*b*ln(c)+72*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+810*I*Pi*b*d*e*r*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)*x^r+162*b*d^2*n+486*a*d^2+8*b*d^2*n*r^4-72*b*d^2*n*r^3+24*a*d^2*r^4-216*a*d^2*r^3+702*a*
d^2*r^2-972*a*d^2*r+486*ln(c)*b*e^2*(x^r)^2+432*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-36*a*e^2*r^3*(x
^r)^2+270*a*e^2*r^2*(x^r)^2-648*a*e^2*r*(x^r)^2+162*b*e^2*n*(x^r)^2+864*a*d*e*r^2*x^r-1620*a*d*e*r*x^r-108*b*e
^2*n*r*(x^r)^2+324*b*d*e*n*x^r+18*b*e^2*n*r^2*(x^r)^2-144*a*d*e*r^3*x^r+270*ln(c)*b*e^2*r^2*(x^r)^2-648*ln(c)*
b*e^2*r*(x^r)^2-36*ln(c)*b*e^2*r^3*(x^r)^2+972*ln(c)*b*d*e*x^r-243*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2-351*I*Pi
*b*d^2*r^2*csgn(I*c*x^n)^3-135*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+18*I*Pi*b*e^2*r^3*cs
gn(I*c*x^n)^3*(x^r)^2+324*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-108*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2+
486*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-486*I*Pi*b*d*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+432*I*Pi
*b*d*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r-810*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-810*I*Pi*b*d*e*r*csgn(
I*x^n)*csgn(I*c*x^n)^2*x^r+108*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+12*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*c*x^n)^2+351*
I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-243*I*Pi*b*d^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+135*I*Pi*b*e^2*r
^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-432*b*d*e*n*r*x^r+864*ln(c)*b*d*e*r^2*x^r-243*I*Pi*b*e^2*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)*(x^r)^2+243*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-486*I*Pi*b*d*e*csgn(I*c*x^n)^
3*x^r-135*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-486*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-108*I*Pi*b*d^2*r
^3*csgn(I*c)*csgn(I*c*x^n)^2-18*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+810*I*Pi*b*d*e*r*csgn(I*c*x
^n)^3*x^r+108*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+351*I*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2+
243*I*Pi*b*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+135*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-324*I*Pi
*b*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-324*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-12*I*Pi*b*d^2*
r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+72*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-351*I*Pi*b*d^2*r^2*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)+486*I*Pi*b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r-1620*ln(c)*b*d*e*r*x^r+12*I*Pi*b*d^2*r^4*c
sgn(I*x^n)*csgn(I*c*x^n)^2+486*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-18*I*Pi*b*e^2*r^3*csgn(I*c)*cs
gn(I*c*x^n)^2*(x^r)^2-432*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r-72*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r
+324*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-144*ln(c)*b*d*e*r^3*x^r+144*b*d*e*n*r^2*x^r+486*
I*Pi*b*d^2*r*csgn(I*c*x^n)^3+243*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2-432*I*Pi*b*d*e*r^2*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)*x^r-12*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3+243*I*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2)/(-3+2*r)^2/x^
3/(-3+r)^2

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^4,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-4>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (118) = 236\).
time = 0.38, size = 422, normalized size = 3.32 \begin {gather*} -\frac {4 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r^{4} + 81 \, b d^{2} n - 36 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r^{3} + 243 \, a d^{2} + 117 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r^{2} - 162 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r - 9 \, {\left ({\left (2 \, b r^{3} - 15 \, b r^{2} + 36 \, b r - 27 \, b\right )} e^{2} \log \left (c\right ) + {\left (2 \, b n r^{3} - 15 \, b n r^{2} + 36 \, b n r - 27 \, b n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a r^{3} - {\left (b n + 15 \, a\right )} r^{2} - 9 \, b n + 6 \, {\left (b n + 6 \, a\right )} r - 27 \, a\right )} e^{2}\right )} x^{2 \, r} - 18 \, {\left ({\left (4 \, b d r^{3} - 24 \, b d r^{2} + 45 \, b d r - 27 \, b d\right )} e \log \left (c\right ) + {\left (4 \, b d n r^{3} - 24 \, b d n r^{2} + 45 \, b d n r - 27 \, b d n\right )} e \log \left (x\right ) + {\left (4 \, a d r^{3} - 9 \, b d n - 4 \, {\left (b d n + 6 \, a d\right )} r^{2} - 27 \, a d + 3 \, {\left (4 \, b d n + 15 \, a d\right )} r\right )} e\right )} x^{r} + 3 \, {\left (4 \, b d^{2} r^{4} - 36 \, b d^{2} r^{3} + 117 \, b d^{2} r^{2} - 162 \, b d^{2} r + 81 \, b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (4 \, b d^{2} n r^{4} - 36 \, b d^{2} n r^{3} + 117 \, b d^{2} n r^{2} - 162 \, b d^{2} n r + 81 \, b d^{2} n\right )} \log \left (x\right )}{9 \, {\left (4 \, r^{4} - 36 \, r^{3} + 117 \, r^{2} - 162 \, r + 81\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")

[Out]

-1/9*(4*(b*d^2*n + 3*a*d^2)*r^4 + 81*b*d^2*n - 36*(b*d^2*n + 3*a*d^2)*r^3 + 243*a*d^2 + 117*(b*d^2*n + 3*a*d^2
)*r^2 - 162*(b*d^2*n + 3*a*d^2)*r - 9*((2*b*r^3 - 15*b*r^2 + 36*b*r - 27*b)*e^2*log(c) + (2*b*n*r^3 - 15*b*n*r
^2 + 36*b*n*r - 27*b*n)*e^2*log(x) + (2*a*r^3 - (b*n + 15*a)*r^2 - 9*b*n + 6*(b*n + 6*a)*r - 27*a)*e^2)*x^(2*r
) - 18*((4*b*d*r^3 - 24*b*d*r^2 + 45*b*d*r - 27*b*d)*e*log(c) + (4*b*d*n*r^3 - 24*b*d*n*r^2 + 45*b*d*n*r - 27*
b*d*n)*e*log(x) + (4*a*d*r^3 - 9*b*d*n - 4*(b*d*n + 6*a*d)*r^2 - 27*a*d + 3*(4*b*d*n + 15*a*d)*r)*e)*x^r + 3*(
4*b*d^2*r^4 - 36*b*d^2*r^3 + 117*b*d^2*r^2 - 162*b*d^2*r + 81*b*d^2)*log(c) + 3*(4*b*d^2*n*r^4 - 36*b*d^2*n*r^
3 + 117*b*d^2*n*r^2 - 162*b*d^2*n*r + 81*b*d^2*n)*log(x))/((4*r^4 - 36*r^3 + 117*r^2 - 162*r + 81)*x^3)

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Sympy [A]
time = 36.09, size = 228, normalized size = 1.80 \begin {gather*} - \frac {a d^{2}}{3 x^{3}} + 2 a d e \left (\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + a e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{2} n}{9 x^{3}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} - 2 b d e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 3 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 b d e \left (\begin {cases} \frac {x^{r - 3}}{r - 3} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {3}{2} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{2} \left (\begin {cases} \frac {x^{2 r - 3}}{2 r - 3} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)**2*(a+b*ln(c*x**n))/x**4,x)

[Out]

-a*d**2/(3*x**3) + 2*a*d*e*Piecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)), (log(x), True)) + a*e**2*Piecewise((x
**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (log(x), True)) - b*d**2*n/(9*x**3) - b*d**2*log(c*x**n)/(3*x**3) -
2*b*d*e*n*Piecewise((Piecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)), (log(x), True))/(r - 3), (r > -oo) & (r < o
o) & Ne(r, 3)), (log(x)**2/2, True)) + 2*b*d*e*Piecewise((x**(r - 3)/(r - 3), Ne(r, 3)), (log(x), True))*log(c
*x**n) - b*e**2*n*Piecewise((Piecewise((x**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (log(x), True))/(2*r - 3),
(r > -oo) & (r < oo) & Ne(r, 3/2)), (log(x)**2/2, True)) + b*e**2*Piecewise((x**(2*r - 3)/(2*r - 3), Ne(r, 3/2
)), (log(x), True))*log(c*x**n)

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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((d+e*x^r)^2*(a+b*log(c*x^n))/x^4,x, algorithm="giac")

[Out]

integrate((x^r*e + d)^2*(b*log(c*x^n) + a)/x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^4,x)

[Out]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^4, x)

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