3.2.15 \(\int \frac {x^2 (a+b \log (c x^n))^2}{(d+e x)^4} \, dx\) [115]

Optimal. Leaf size=161 \[ \frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {b n x \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}-\frac {b n \left (2 a+3 b n+2 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d e^3}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d e^3} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2381, 2384, 2354, 2438} \begin {gather*} -\frac {2 b^2 n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d e^3}-\frac {b n \log \left (\frac {e x}{d}+1\right ) \left (2 a+2 b \log \left (c x^n\right )+3 b n\right )}{3 d e^3}+\frac {b n x \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{3 d e^2 (d+e x)}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n*x^2*(a + b*Log[c*x^n]))/(3*d*e*(d + e*x)^2) + (x^3*(a + b*Log[c*x^n])^2)/(3*d*(d + e*x)^3) + (b*n*x*(2*a
+ b*n + 2*b*Log[c*x^n]))/(3*d*e^2*(d + e*x)) - (b*n*(2*a + 3*b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d*e^3)
- (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/(3*d*e^3)

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^4}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^2}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^2}-\frac {(2 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^2}+\frac {d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e^2}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 (d+e x)^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e^2 (d+e x)}-\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^3}+\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d e^2}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 (d+e x)^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e^2 (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^3}+\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^3}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^2}-\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e^2}+\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^3}\\ &=\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{d e^2 (d+e x)}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 (d+e x)^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e^2 (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^3}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^3}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 e^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d e^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 e^2}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d e^2}-\frac {\left (b^2 d n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 e^3}-\frac {\left (2 b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{d e^2}\\ &=\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)^2}+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 (d+e x)^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e^2 (d+e x)}-\frac {2 b^2 n^2 \log (d+e x)}{d e^3}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^3}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{3 d e^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d e^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^3}-\frac {\left (b^2 d n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 e^3}+\frac {\left (2 b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{3 d e^2}\\ &=-\frac {b^2 n^2}{3 e^3 (d+e x)}-\frac {b^2 n^2 \log (x)}{3 d e^3}+\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)^2}+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 (d+e x)^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e^2 (d+e x)}-\frac {b^2 n^2 \log (d+e x)}{d e^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d e^3}+\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{3 d e^3}\\ &=-\frac {b^2 n^2}{3 e^3 (d+e x)}-\frac {b^2 n^2 \log (x)}{3 d e^3}+\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)^2}+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 (d+e x)^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e^2 (d+e x)}-\frac {b^2 n^2 \log (d+e x)}{d e^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d e^3}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d e^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(371\) vs. \(2(161)=322\).
time = 0.32, size = 371, normalized size = 2.30 \begin {gather*} -\frac {-\frac {a^2}{d}+\frac {a^2 d^2}{(d+e x)^3}-\frac {3 a^2 d}{(d+e x)^2}-\frac {a b d n}{(d+e x)^2}+\frac {3 a^2}{d+e x}+\frac {4 a b n}{d+e x}+\frac {b^2 n^2}{d+e x}-\frac {3 b^2 n^2 \log (x)}{d}-\frac {2 a b \log \left (c x^n\right )}{d}+\frac {2 a b d^2 \log \left (c x^n\right )}{(d+e x)^3}-\frac {6 a b d \log \left (c x^n\right )}{(d+e x)^2}-\frac {b^2 d n \log \left (c x^n\right )}{(d+e x)^2}+\frac {6 a b \log \left (c x^n\right )}{d+e x}+\frac {4 b^2 n \log \left (c x^n\right )}{d+e x}-\frac {b^2 \log ^2\left (c x^n\right )}{d}+\frac {b^2 d^2 \log ^2\left (c x^n\right )}{(d+e x)^3}-\frac {3 b^2 d \log ^2\left (c x^n\right )}{(d+e x)^2}+\frac {3 b^2 \log ^2\left (c x^n\right )}{d+e x}+\frac {3 b^2 n^2 \log (d+e x)}{d}+\frac {2 a b n \log \left (1+\frac {e x}{d}\right )}{d}+\frac {2 b^2 n \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )}{d}+\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d}}{3 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(-(a^2/d) + (a^2*d^2)/(d + e*x)^3 - (3*a^2*d)/(d + e*x)^2 - (a*b*d*n)/(d + e*x)^2 + (3*a^2)/(d + e*x) + (
4*a*b*n)/(d + e*x) + (b^2*n^2)/(d + e*x) - (3*b^2*n^2*Log[x])/d - (2*a*b*Log[c*x^n])/d + (2*a*b*d^2*Log[c*x^n]
)/(d + e*x)^3 - (6*a*b*d*Log[c*x^n])/(d + e*x)^2 - (b^2*d*n*Log[c*x^n])/(d + e*x)^2 + (6*a*b*Log[c*x^n])/(d +
e*x) + (4*b^2*n*Log[c*x^n])/(d + e*x) - (b^2*Log[c*x^n]^2)/d + (b^2*d^2*Log[c*x^n]^2)/(d + e*x)^3 - (3*b^2*d*L
og[c*x^n]^2)/(d + e*x)^2 + (3*b^2*Log[c*x^n]^2)/(d + e*x) + (3*b^2*n^2*Log[d + e*x])/d + (2*a*b*n*Log[1 + (e*x
)/d])/d + (2*b^2*n*Log[c*x^n]*Log[1 + (e*x)/d])/d + (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/d)/e^3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 1658, normalized size = 10.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1658\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/e^3*n/d*ln(e*x+d)*b^2*ln(c)+2/3/e^3*n/d*ln(x)*b^2*ln(c)-4/3/e^3*n/(e*x+d)*b^2*ln(c)-2/3*ln(x^n)*d^2/e^3/(
e*x+d)^3*b^2*ln(c)+2/e^3*ln(x^n)*d/(e*x+d)^2*b^2*ln(c)-2/3*b*ln(x^n)*d^2/e^3/(e*x+d)^3*a+2*b/e^3*ln(x^n)*d/(e*
x+d)^2*a-1/3*b^2/e^3*n^2/(e*x+d)-1/3*b^2/e^3*n^2/d*ln(x)^2+2/3*b^2/e^3*n^2/d*dilog(-e*x/d)-b^2/e^3*n^2/d*ln(e*
x+d)+b^2/e^3*n^2/d*ln(x)+1/3*b/e^3*n*d/(e*x+d)^2*a+2/3*b/e^3*n/d*ln(x)*a-2/3*b/e^3*n/d*ln(e*x+d)*a+1/3/e^3*n*d
/(e*x+d)^2*b^2*ln(c)-b^2*ln(x^n)^2/e^3/(e*x+d)-I/e^3*ln(x^n)/(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I/e^3*ln
(x^n)*d/(e*x+d)^2*b^2*Pi*csgn(I*c*x^n)^3-2/3*I/e^3*n/(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/4*(-I*b*Pi*csg
n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*cs
gn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*(-1/3*d^2/e^3/(e*x+d)^3-1/e^3/(e*x+d)+d/e^3/(e*x+d)^2)+1/3*I/e^3*n/d*ln(e*x+d)*
b^2*Pi*csgn(I*c*x^n)^3-1/6*I/e^3*n*d/(e*x+d)^2*b^2*Pi*csgn(I*c*x^n)^3-2/3*I/e^3*n/(e*x+d)*b^2*Pi*csgn(I*x^n)*c
sgn(I*c*x^n)^2-1/3*I/e^3*n/d*ln(x)*b^2*Pi*csgn(I*c*x^n)^3-I/e^3*ln(x^n)/(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^
n)^2+1/3*I*ln(x^n)*d^2/e^3/(e*x+d)^3*b^2*Pi*csgn(I*c*x^n)^3+1/3*I/e^3*n/d*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)+1/3*I*ln(x^n)*d^2/e^3/(e*x+d)^3*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2/3*b^2*n/e^3*ln(x
^n)/d*ln(x)+1/3*I/e^3*n/d*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/3*I/e^3*n/d*ln(x)*b^2*Pi*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)-1/3*I*ln(x^n)*d^2/e^3/(e*x+d)^3*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I/e^3*n*d/(e*x+d)^2
*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+2/3*I/e^3*n/(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2/3*b^2/e^3
*n^2/d*ln(e*x+d)*ln(-e*x/d)+b^2*ln(x^n)^2*d/e^3/(e*x+d)^2-4/3*b^2*n/e^3*ln(x^n)/(e*x+d)+I/e^3*ln(x^n)*d/(e*x+d
)^2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I/e^3*ln(x^n)*d/(e*x+d)^2*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I/e^3*ln(x^n
)/(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/3*I/e^3*n/d*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-
4/3*b/e^3*n/(e*x+d)*a-1/3*I*ln(x^n)*d^2/e^3/(e*x+d)^3*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-2/3*b^2*n/e^3*ln(x^n)/d
*ln(e*x+d)+1/3*b^2*n/e^3*ln(x^n)*d/(e*x+d)^2-2*b/e^3*ln(x^n)/(e*x+d)*a-1/3*b^2*ln(x^n)^2*d^2/e^3/(e*x+d)^3-2/e
^3*ln(x^n)/(e*x+d)*b^2*ln(c)+1/6*I/e^3*n*d/(e*x+d)^2*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I/e^3*ln(x^n)*d/(e*x+d)^
2*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I/e^3*n*d/(e*x+d)^2*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)+2/3*I/e^3*n/(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3+I/e^3*ln(x^n)/(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3+1/3*I/e^3*n/d*ln(x)*
b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/3*I/e^3*n/d*ln(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2

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

[Out]

-1/3*a*b*n*(2*e^(-3)*log(x*e + d)/d - 2*e^(-3)*log(x)/d + (4*x*e + 3*d)/(x^2*e^5 + 2*d*x*e^4 + d^2*e^3)) - 1/3
*((3*x^2*e^2 + 3*d*x*e + d^2)*log(x^n)^2/(x^3*e^6 + 3*d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3) - 3*integrate(1/3*(3*
x^3*e^3*log(c)^2 + 2*(3*(n + log(c))*x^3*e^3 + 6*d*n*x^2*e^2 + 4*d^2*n*x*e + d^3*n)*log(x^n))/(x^5*e^7 + 4*d*x
^4*e^6 + 6*d^2*x^3*e^5 + 4*d^3*x^2*e^4 + d^4*x*e^3), x))*b^2 - 2/3*(3*x^2*e^2 + 3*d*x*e + d^2)*a*b*log(c*x^n)/
(x^3*e^6 + 3*d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3) - 1/3*(3*x^2*e^2 + 3*d*x*e + d^2)*a^2/(x^3*e^6 + 3*d*x^2*e^5 +
3*d^2*x*e^4 + d^3*e^3)

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

[Out]

integral((b^2*x^2*log(c*x^n)^2 + 2*a*b*x^2*log(c*x^n) + a^2*x^2)/(x^4*e^4 + 4*d*x^3*e^3 + 6*d^2*x^2*e^2 + 4*d^
3*x*e + d^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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