3.3.37 \(\int \frac {x^2 (a+b \log (c x^n))}{(d+e x^2)^3} \, dx\) [237]

Optimal. Leaf size=187 \[ \frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}} \]

[Out]

1/8*b*n*x/d/e/(e*x^2+d)-1/4*x*(a+b*ln(c*x^n))/e/(e*x^2+d)^2+1/8*x*(a+b*ln(c*x^n))/d/e/(e*x^2+d)+1/8*arctan(x*e
^(1/2)/d^(1/2))*(a+b*ln(c*x^n))/d^(3/2)/e^(3/2)-1/16*I*b*n*polylog(2,-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)+1/1
6*I*b*n*polylog(2,I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)

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Rubi [A]
time = 0.26, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {294, 205, 211, 2393, 2360, 2361, 12, 4940, 2438} \begin {gather*} -\frac {i b n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {\text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {b n x}{8 d e \left (d+e x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n*x)/(8*d*e*(d + e*x^2)) - (x*(a + b*Log[c*x^n]))/(4*e*(d + e*x^2)^2) + (x*(a + b*Log[c*x^n]))/(8*d*e*(d +
e*x^2)) + (ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*d^(3/2)*e^(3/2)) - ((I/16)*b*n*PolyLog[2, ((-I)*
Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*e^(3/2)) + ((I/16)*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*e^(3/2))

Rule 12

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(q +
1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[
c*x^n]), x], x] + Dist[b*(n/(2*d*(q + 1))), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &
& LtQ[q, -1]

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 2393

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

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]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^3}+\frac {a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )^2}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx}{e}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d e \left (d+e x^2\right )}-\frac {3 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{4 e}+\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d e}+\frac {(b n) \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{4 e}-\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{2 d e}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} e^{3/2}}-\frac {3 \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d e}+\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{8 d e}+\frac {(3 b n) \int \frac {1}{d+e x^2} \, dx}{8 d e}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 d e}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 d^{3/2} e^{3/2}}+\frac {(3 b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d e}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {(i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(3 b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{3/2} e^{3/2}}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} e^{3/2}}+\frac {(3 i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}-\frac {(3 i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(497\) vs. \(2(187)=374\).
time = 0.65, size = 497, normalized size = 2.66 \begin {gather*} \frac {\frac {d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d-d \sqrt {e} x}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d+d \sqrt {e} x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{3/2}}+\frac {b n \left (d+\left (d-\sqrt {-d} \sqrt {e} x\right ) \log (x)+\left (-d+\sqrt {-d} \sqrt {e} x\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2}}-\frac {b n \left (d+\left (d+\sqrt {-d} \sqrt {e} x\right ) \log (x)-\left (d+\sqrt {-d} \sqrt {e} x\right ) \log \left ((-d)^{3/2}+d \sqrt {e} x\right )\right )}{d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}+\frac {b d n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2}}+\frac {b n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}}{16 e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(a + b*Log[c*x^n]))/((-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (a + b*Log[c*x^n])/(Sqrt[-d]*(Sqrt[-d] + Sqrt[
e]*x)^2) - (a + b*Log[c*x^n])/(Sqrt[-d]*d - d*Sqrt[e]*x) + (a + b*Log[c*x^n])/(Sqrt[-d]*d + d*Sqrt[e]*x) + (b*
d*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(-d)^(5/2) + (b*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/(-d)^(3/2) +
(b*n*(d + (d - Sqrt[-d]*Sqrt[e]*x)*Log[x] + (-d + Sqrt[-d]*Sqrt[e]*x)*Log[Sqrt[-d] + Sqrt[e]*x]))/(d^2*(Sqrt[
-d] + Sqrt[e]*x)) + ((a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3/2) - (b*n*(d + (d + Sqrt[-d]*Sq
rt[e]*x)*Log[x] - (d + Sqrt[-d]*Sqrt[e]*x)*Log[(-d)^(3/2) + d*Sqrt[e]*x]))/(d^2*(Sqrt[-d] - Sqrt[e]*x)) + (d*(
a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(5/2) + (b*d*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d
)^(5/2) + (b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(3/2))/(16*e^(3/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 1247, normalized size = 6.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/16*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/(e*x^2+d)^2/d*x^3-1/16*I*b*Pi*csgn(I*c*x^n)^3/(e*x^2+d)^2/d*x^3+1/8*a/
e/d/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+1/16*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e/d/(e*d)^(1/2)*arctan(x*e/(e*
d)^(1/2))+1/8*b/e/d/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))*ln(x^n)-1/8*b*ln(c)/(e*x^2+d)^2/e*x+1/8*b*ln(c)/(e*x^2
+d)^2/d*x^3-1/16*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/(e*x^2+d)^2/e*x+1/8*a/(e*x^2+d)^2/d*x^3-1/8*a/(e*x^2+d)^2/
e*x+1/16*b*n/d/e/(-e*d)^(1/2)*dilog((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-1/16*b*n/d/e/(-e*d)^(1/2)*dilog((e*x+(-e
*d)^(1/2))/(-e*d)^(1/2))-1/2*b*n/d*ln(x)/(e*x^2+d)^2*x^3-1/2*b*n/e*ln(x)/(e*x^2+d)^2*x+1/8*b/(e*x^2+d)^2/d*x^3
*ln(x^n)-1/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/(e*x^2+d)^2/e*x+1/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/(e*x^2+d)
^2/d*x^3-1/16*I*b*Pi*csgn(I*c*x^n)^3/e/d/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))-3/16*b*n/d*e*ln(x)/(e*x^2+d)^2/(-
e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^4+3/16*b*n/d*e*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-e*d)
^(1/2))/(-e*d)^(1/2))*x^4+1/8*b*ln(c)/e/d/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+1/16*I*b*Pi*csgn(I*c*x^n)^3/(e*x
^2+d)^2/e*x+1/4*b*n/e*ln(x)/(e*x^2+d)/(-e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-1/4*b*n/e*ln(x)/(e*x^2
+d)/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))+1/2*b*n/e*ln(x)*x/d/(e*x^2+d)-1/8*b/e/d/(e*d)^(1/2)*arcta
n(x*e/(e*d)^(1/2))*n*ln(x)-3/8*b*n*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^2+3/8
*b*n*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^2-3/16*b*n*d/e*ln(x)/(e*x^2+d)^2/(-e
*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))+3/16*b*n*d/e*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2
))/(-e*d)^(1/2))+1/4*b*n*ln(x)/d/(e*x^2+d)/(-e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^2-1/4*b*n*ln(x)
/d/(e*x^2+d)/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^2+1/8*b*n*x/d/e/(e*x^2+d)-1/8*b/(e*x^2+d)^2/e*
x*ln(x^n)-1/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/e/d/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))-1/16*I*b*Pi*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/(e*x^2+d)^2/d*x^3+1/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/(e*x^2+d
)^2/e*x+1/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e/d/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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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))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*x^2*log(c*x^n) + a*x^2)/(x^6*e^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e + d^3), 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 )}{\left (d + e x^{2}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))/(e*x**2+d)**3,x)

[Out]

Integral(x**2*(a + b*log(c*x**n))/(d + e*x**2)**3, 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))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^2/(x^2*e + d)^3, 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 )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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