Integrand size = 25, antiderivative size = 232 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {d+e x^2}}{25 d^2 x^5}+\frac {14 b e n \sqrt {d+e x^2}}{75 d^3 x^3}-\frac {148 b e^2 n \sqrt {d+e x^2}}{75 d^4 x}+\frac {16 b e^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d^4}+\frac {a+b \log \left (c x^n\right )}{d x^5 \sqrt {d+e x^2}}-\frac {6 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^5}+\frac {8 e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x^3}-\frac {16 e^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d^4 x} \] Output:
-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+16/5*b*e^(5/2)*n*arctanh(e^(1/2)*x/(e*x^2 +d)^(1/2))/d^4+(a+b*ln(c*x^n))/d/x^5/(e*x^2+d)^(1/2)-6/5*(e*x^2+d)^(1/2)*( a+b*ln(c*x^n))/d^2/x^5+8/5*e*(e*x^2+d)^(1/2)*(a+b*ln(c*x^n))/d^3/x^3-16/5* e^2*(e*x^2+d)^(1/2)*(a+b*ln(c*x^n))/d^4/x
Time = 0.24 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-15 a d^3-3 b d^3 n+30 a d^2 e x^2+11 b d^2 e n x^2-120 a d e^2 x^4-134 b d e^2 n x^4-240 a e^3 x^6-148 b e^3 n x^6-15 b \left (d^3-2 d^2 e x^2+8 d e^2 x^4+16 e^3 x^6\right ) \log \left (c x^n\right )+240 b e^{5/2} n x^5 \sqrt {d+e x^2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{75 d^4 x^5 \sqrt {d+e x^2}} \] Input:
Integrate[(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)),x]
Output:
(-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*Sqrt [d + e*x^2]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(75*d^4*x^5*Sqrt[d + e*x^2 ])
Time = 0.64 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2792, 27, 2338, 9, 27, 1588, 27, 358, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2792 |
\(\displaystyle -b n \int -\frac {16 e^3 x^6+8 d e^2 x^4-2 d^2 e x^2+d^3}{5 d^4 x^6 \sqrt {e x^2+d}}dx-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b n \int \frac {16 e^3 x^6+8 d e^2 x^4-2 d^2 e x^2+d^3}{x^6 \sqrt {e x^2+d}}dx}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \frac {b n \left (-\frac {\int \frac {2 \left (-40 d e^3 x^5-20 d^2 e^2 x^3+7 d^3 e x\right )}{x^5 \sqrt {e x^2+d}}dx}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \frac {b n \left (-\frac {\int \frac {2 \left (-40 d e^3 x^4-20 d^2 e^2 x^2+7 d^3 e\right )}{x^4 \sqrt {e x^2+d}}dx}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b n \left (-\frac {2 \int \frac {-40 d e^3 x^4-20 d^2 e^2 x^2+7 d^3 e}{x^4 \sqrt {e x^2+d}}dx}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 1588 |
\(\displaystyle \frac {b n \left (-\frac {2 \left (-\frac {\int \frac {2 d^2 e^2 \left (60 e x^2+37 d\right )}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {7 d^2 e \sqrt {d+e x^2}}{3 x^3}\right )}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b n \left (-\frac {2 \left (-\frac {2}{3} d e^2 \int \frac {60 e x^2+37 d}{x^2 \sqrt {e x^2+d}}dx-\frac {7 d^2 e \sqrt {d+e x^2}}{3 x^3}\right )}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 358 |
\(\displaystyle \frac {b n \left (-\frac {2 \left (-\frac {2}{3} d e^2 \left (60 e \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {37 \sqrt {d+e x^2}}{x}\right )-\frac {7 d^2 e \sqrt {d+e x^2}}{3 x^3}\right )}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {b n \left (-\frac {2 \left (-\frac {2}{3} d e^2 \left (60 e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {37 \sqrt {d+e x^2}}{x}\right )-\frac {7 d^2 e \sqrt {d+e x^2}}{3 x^3}\right )}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}-\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt {d+e x^2}}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt {d+e x^2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{5 d x^5 \sqrt {d+e x^2}}+\frac {b n \left (-\frac {2 \left (-\frac {2}{3} d e^2 \left (60 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {37 \sqrt {d+e x^2}}{x}\right )-\frac {7 d^2 e \sqrt {d+e x^2}}{3 x^3}\right )}{5 d}-\frac {d^2 \sqrt {d+e x^2}}{5 x^5}\right )}{5 d^4}\) |
Input:
Int[(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)),x]
Output:
(b*n*(-1/5*(d^2*Sqrt[d + e*x^2])/x^5 - (2*((-7*d^2*e*Sqrt[d + e*x^2])/(3*x ^3) - (2*d*e^2*((-37*Sqrt[d + e*x^2])/x + 60*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/S qrt[d + e*x^2]]))/3))/(5*d)))/(5*d^4) - (a + b*Log[c*x^n])/(5*d*x^5*Sqrt[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*Sqrt[d + e*x^2]) - (16*e^3*x*(a + b*Log[c*x^ n]))/(5*d^4*Sqrt[d + e*x^2])
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S imp[d/e^2 Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{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] + Simp[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] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
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]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[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] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] }, Simp[(a + b*Log[c*x^n]) u, x] - Simp[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])
\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int((a+b*ln(c*x^n))/x^6/(e*x^2+d)^(3/2),x)
Output:
int((a+b*ln(c*x^n))/x^6/(e*x^2+d)^(3/2),x)
Time = 0.14 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.04 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {120 \, {\left (b e^{3} n x^{7} + b d e^{2} n x^{5}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (4 \, {\left (37 \, b e^{3} n + 60 \, a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 2 \, {\left (67 \, b d e^{2} n + 60 \, a d e^{2}\right )} x^{4} + 15 \, a d^{3} - {\left (11 \, b d^{2} e n + 30 \, a d^{2} e\right )} x^{2} + 15 \, {\left (16 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} - 2 \, b d^{2} e x^{2} + b d^{3}\right )} \log \left (c\right ) + 15 \, {\left (16 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} - 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{75 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac {240 \, {\left (b e^{3} n x^{7} + b d e^{2} n x^{5}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (4 \, {\left (37 \, b e^{3} n + 60 \, a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 2 \, {\left (67 \, b d e^{2} n + 60 \, a d e^{2}\right )} x^{4} + 15 \, a d^{3} - {\left (11 \, b d^{2} e n + 30 \, a d^{2} e\right )} x^{2} + 15 \, {\left (16 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} - 2 \, b d^{2} e x^{2} + b d^{3}\right )} \log \left (c\right ) + 15 \, {\left (16 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} - 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{75 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\right ] \] Input:
integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),x, algorithm="fricas")
Output:
[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*(6 7*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)]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*x**n))/x**6/(e*x**2+d)**(3/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:
integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)/((e*x^2 + d)^(3/2)*x^6), x)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^6\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((a + b*log(c*x^n))/(x^6*(d + e*x^2)^(3/2)),x)
Output:
int((a + b*log(c*x^n))/(x^6*(d + e*x^2)^(3/2)), x)
Time = 0.20 (sec) , antiderivative size = 756, normalized size of antiderivative = 3.26 \[ \int \frac {a+b \log \left (c x^n\right )}{x^6 \left (d+e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((a+b*log(c*x^n))/x^6/(e*x^2+d)^(3/2),x)
Output:
( - 15*sqrt(d + e*x**2)*log(((2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n* c)/(e**(n/2)*(sqrt(d + e*x**2) + sqrt(e)*x)**n*2**n))*b*d**3 + 30*sqrt(d + e*x**2)*log(((2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(s qrt(d + e*x**2) + sqrt(e)*x)**n*2**n))*b*d**2*e*x**2 - 120*sqrt(d + e*x**2 )*log(((2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(sqrt(d + e*x**2) + sqrt(e)*x)**n*2**n))*b*d*e**2*x**4 - 240*sqrt(d + e*x**2)*log(( (2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(sqrt(d + e*x**2 ) + sqrt(e)*x)**n*2**n))*b*e**3*x**6 - 15*sqrt(d + e*x**2)*a*d**3 + 30*sqr t(d + e*x**2)*a*d**2*e*x**2 - 120*sqrt(d + e*x**2)*a*d*e**2*x**4 - 240*sqr t(d + e*x**2)*a*e**3*x**6 - 3*sqrt(d + e*x**2)*b*d**3*n + 11*sqrt(d + e*x* *2)*b*d**2*e*n*x**2 - 134*sqrt(d + e*x**2)*b*d*e**2*n*x**4 - 148*sqrt(d + e*x**2)*b*e**3*n*x**6 + 240*sqrt(e)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt (e)*x)/sqrt(d))*b*d*e**2*n*x**5 + 240*sqrt(e)*log((sqrt(d + e*x**2) - sqrt (d) + sqrt(e)*x)/sqrt(d))*b*e**3*n*x**7 + 240*sqrt(e)*log((sqrt(d + e*x**2 ) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d*e**2*n*x**5 + 240*sqrt(e)*log((sqrt( d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*e**3*n*x**7 - 240*sqrt(e)*lo g(((2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(sqrt(d + e*x **2) + sqrt(e)*x)**n*2**n))*b*d*e**2*x**5 - 240*sqrt(e)*log(((2*sqrt(e)*sq rt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(sqrt(d + e*x**2) + sqrt(e)*x )**n*2**n))*b*e**3*x**7 + 240*sqrt(e)*a*d*e**2*x**5 + 240*sqrt(e)*a*e**...