\(\int \frac {a+b \log (c x^n)}{x^6 (d+e x^2)^{3/2}} \, dx\) [296]

Optimal result

Integrand size = 25, antiderivative size = 236 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {148 b e^2 n \sqrt {d+e x^2}}{75 d^4 x}+\frac {16 b e^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d^4}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}} \]

[Out]

16/5*b*e^(5/2)*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/d^4+1/5*(-a-b*ln(c*x^n))/d/x^5/(e*x^2+d)^(1/2)+2/5*e*(a+b*
ln(c*x^n))/d^2/x^3/(e*x^2+d)^(1/2)-8/5*e^2*(a+b*ln(c*x^n))/d^3/x/(e*x^2+d)^(1/2)-16/5*e^3*x*(a+b*ln(c*x^n))/d^
4/(e*x^2+d)^(1/2)-1/25*b*n*(e*x^2+d)^(1/2)/d^2/x^5+14/75*b*e*n*(e*x^2+d)^(1/2)/d^3/x^3-148/75*b*e^2*n*(e*x^2+d
)^(1/2)/d^4/x

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {277, 197, 2392, 12, 1821, 1599, 1279, 462, 223, 212} \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {16 b e^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d^4}-\frac {148 b e^2 n \sqrt {d+e x^2}}{75 d^4 x}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5} \]

[In]

Int[(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)),x]

[Out]

-1/25*(b*n*Sqrt[d + e*x^2])/(d^2*x^5) + (14*b*e*n*Sqrt[d + e*x^2])/(75*d^3*x^3) - (148*b*e^2*n*Sqrt[d + e*x^2]
)/(75*d^4*x) + (16*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(5*d^4) - (a + b*Log[c*x^n])/(5*d*x^5*Sqr
t[d + e*x^2]) + (2*e*(a + b*Log[c*x^n]))/(5*d^2*x^3*Sqrt[d + e*x^2]) - (8*e^2*(a + b*Log[c*x^n]))/(5*d^3*x*Sqr
t[d + e*x^2]) - (16*e^3*x*(a + b*Log[c*x^n]))/(5*d^4*Sqrt[d + e*x^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 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 277

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

Rule 462

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

Rule 1279

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 1599

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 2392

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*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] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-(b n) \int \frac {-d^3+2 d^2 e x^2-8 d e^2 x^4-16 e^3 x^6}{5 d^4 x^6 \sqrt {d+e x^2}} \, dx \\ & = -\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-d^3+2 d^2 e x^2-8 d e^2 x^4-16 e^3 x^6}{x^6 \sqrt {d+e x^2}} \, dx}{5 d^4} \\ & = -\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}+\frac {(b n) \int \frac {-14 d^3 e x+40 d^2 e^2 x^3+80 d e^3 x^5}{x^5 \sqrt {d+e x^2}} \, dx}{25 d^5} \\ & = -\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}+\frac {(b n) \int \frac {-14 d^3 e+40 d^2 e^2 x^2+80 d e^3 x^4}{x^4 \sqrt {d+e x^2}} \, dx}{25 d^5} \\ & = -\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-148 d^3 e^2-240 d^2 e^3 x^2}{x^2 \sqrt {d+e x^2}} \, dx}{75 d^6} \\ & = -\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {148 b e^2 n \sqrt {d+e x^2}}{75 d^4 x}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}+\frac {\left (16 b e^3 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{5 d^4} \\ & = -\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {148 b e^2 n \sqrt {d+e x^2}}{75 d^4 x}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}+\frac {\left (16 b e^3 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{5 d^4} \\ & = -\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {148 b e^2 n \sqrt {d+e x^2}}{75 d^4 x}+\frac {16 b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d^4}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-15 a d^3-3 b d^3 n+30 a d^2 e x^2+11 b d^2 e n x^2-120 a d e^2 x^4-134 b d e^2 n x^4-240 a e^3 x^6-148 b e^3 n x^6-15 b \left (d^3-2 d^2 e x^2+8 d e^2 x^4+16 e^3 x^6\right ) \log \left (c x^n\right )+240 b e^{5/2} n x^5 \sqrt {d+e x^2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{75 d^4 x^5 \sqrt {d+e x^2}} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)),x]

[Out]

(-15*a*d^3 - 3*b*d^3*n + 30*a*d^2*e*x^2 + 11*b*d^2*e*n*x^2 - 120*a*d*e^2*x^4 - 134*b*d*e^2*n*x^4 - 240*a*e^3*x
^6 - 148*b*e^3*n*x^6 - 15*b*(d^3 - 2*d^2*e*x^2 + 8*d*e^2*x^4 + 16*e^3*x^6)*Log[c*x^n] + 240*b*e^(5/2)*n*x^5*Sq
rt[d + e*x^2]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(75*d^4*x^5*Sqrt[d + e*x^2])

Maple [F]

\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*ln(c*x^n))/x^6/(e*x^2+d)^(3/2),x)

[Out]

int((a+b*ln(c*x^n))/x^6/(e*x^2+d)^(3/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {120 \, {\left (b e^{3} n x^{7} + b d e^{2} n x^{5}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (4 \, {\left (37 \, b e^{3} n + 60 \, a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 2 \, {\left (67 \, b d e^{2} n + 60 \, a d e^{2}\right )} x^{4} + 15 \, a d^{3} - {\left (11 \, b d^{2} e n + 30 \, a d^{2} e\right )} x^{2} + 15 \, {\left (16 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} - 2 \, b d^{2} e x^{2} + b d^{3}\right )} \log \left (c\right ) + 15 \, {\left (16 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} - 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{75 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac {240 \, {\left (b e^{3} n x^{7} + b d e^{2} n x^{5}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (4 \, {\left (37 \, b e^{3} n + 60 \, a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 2 \, {\left (67 \, b d e^{2} n + 60 \, a d e^{2}\right )} x^{4} + 15 \, a d^{3} - {\left (11 \, b d^{2} e n + 30 \, a d^{2} e\right )} x^{2} + 15 \, {\left (16 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} - 2 \, b d^{2} e x^{2} + b d^{3}\right )} \log \left (c\right ) + 15 \, {\left (16 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} - 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{75 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\right ] \]

[In]

integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

[1/75*(120*(b*e^3*n*x^7 + b*d*e^2*n*x^5)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - (4*(37*b*e^
3*n + 60*a*e^3)*x^6 + 3*b*d^3*n + 2*(67*b*d*e^2*n + 60*a*d*e^2)*x^4 + 15*a*d^3 - (11*b*d^2*e*n + 30*a*d^2*e)*x
^2 + 15*(16*b*e^3*x^6 + 8*b*d*e^2*x^4 - 2*b*d^2*e*x^2 + b*d^3)*log(c) + 15*(16*b*e^3*n*x^6 + 8*b*d*e^2*n*x^4 -
2*b*d^2*e*n*x^2 + b*d^3*n)*log(x))*sqrt(e*x^2 + d))/(d^4*e*x^7 + d^5*x^5), -1/75*(240*(b*e^3*n*x^7 + b*d*e^2*
n*x^5)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (4*(37*b*e^3*n + 60*a*e^3)*x^6 + 3*b*d^3*n + 2*(67*b*d*e^
2*n + 60*a*d*e^2)*x^4 + 15*a*d^3 - (11*b*d^2*e*n + 30*a*d^2*e)*x^2 + 15*(16*b*e^3*x^6 + 8*b*d*e^2*x^4 - 2*b*d^
2*e*x^2 + b*d^3)*log(c) + 15*(16*b*e^3*n*x^6 + 8*b*d*e^2*n*x^4 - 2*b*d^2*e*n*x^2 + b*d^3*n)*log(x))*sqrt(e*x^2
+ d))/(d^4*e*x^7 + d^5*x^5)]

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*x**n))/x**6/(e*x**2+d)**(3/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),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(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{6}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^6\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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