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

Optimal result

Integrand size = 25, antiderivative size = 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
 

Mathematica [A] (verified)

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 
])
 

Rubi [A] (verified)

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])
 

Defintions of rubi rules used

rule 9

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]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 224

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 358

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]
 

rule 1588

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]
 

rule 2338

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])
 

rule 2792

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])
 

Maple [F]

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

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)
 

Fricas [A] (verification not implemented)

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)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

Giac [F]

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

Mupad [F(-1)]

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

Reduce [B] (verification not implemented)

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**...