\(\int \frac {a+b \log (c x^n)}{x^2 (d+e x^2)^{3/2}} \, dx\) [294]

Optimal result

Integrand size = 25, antiderivative size = 110 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {d+e x^2}}{d^2 x}+\frac {2 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d^2}+\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{d^2 x} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-a d-b d n-2 a e x^2-b e n x^2-b \left (d+2 e x^2\right ) \log \left (c x^n\right )+2 b \sqrt {e} n x \sqrt {d+e x^2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{d^2 x \sqrt {d+e x^2}} \] Input:

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

Output:

(-(a*d) - b*d*n - 2*a*e*x^2 - b*e*n*x^2 - b*(d + 2*e*x^2)*Log[c*x^n] + 2*b 
*Sqrt[e]*n*x*Sqrt[d + e*x^2]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(d^2*x*Sq 
rt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2792, 25, 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.

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

\(\Big \downarrow \) 2792

\(\displaystyle -b n \int -\frac {2 e x^2+d}{d^2 x^2 \sqrt {e x^2+d}}dx-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle b n \int \frac {2 e x^2+d}{d^2 x^2 \sqrt {e x^2+d}}dx-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b n \int \frac {2 e x^2+d}{x^2 \sqrt {e x^2+d}}dx}{d^2}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 358

\(\displaystyle \frac {b n \left (2 e \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {\sqrt {d+e x^2}}{x}\right )}{d^2}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b n \left (2 e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {\sqrt {d+e x^2}}{x}\right )}{d^2}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}+\frac {b n \left (2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {d+e x^2}}{x}\right )}{d^2}\)

Input:

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

Output:

(b*n*(-(Sqrt[d + e*x^2]/x) + 2*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2] 
]))/d^2 - (a + b*Log[c*x^n])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*Log[c*x 
^n]))/(d^2*Sqrt[d + e*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.19 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b e n x^{3} + b d n x\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (b d n + {\left (b e n + 2 \, a e\right )} x^{2} + a d + {\left (2 \, b e x^{2} + b d\right )} \log \left (c\right ) + {\left (2 \, b e n x^{2} + b d n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{d^{2} e x^{3} + d^{3} x}, -\frac {2 \, {\left (b e n x^{3} + b d n x\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (b d n + {\left (b e n + 2 \, a e\right )} x^{2} + a d + {\left (2 \, b e x^{2} + b d\right )} \log \left (c\right ) + {\left (2 \, b e n x^{2} + b d n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{d^{2} e x^{3} + d^{3} x}\right ] \] Input:

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

Output:

[((b*e*n*x^3 + b*d*n*x)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x 
 - d) - (b*d*n + (b*e*n + 2*a*e)*x^2 + a*d + (2*b*e*x^2 + b*d)*log(c) + (2 
*b*e*n*x^2 + b*d*n)*log(x))*sqrt(e*x^2 + d))/(d^2*e*x^3 + d^3*x), -(2*(b*e 
*n*x^3 + b*d*n*x)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (b*d*n + ( 
b*e*n + 2*a*e)*x^2 + a*d + (2*b*e*x^2 + b*d)*log(c) + (2*b*e*n*x^2 + b*d*n 
)*log(x))*sqrt(e*x^2 + d))/(d^2*e*x^3 + d^3*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a+b*log(c*x^n))/x^2/(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 490, normalized size of antiderivative = 4.45 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b d -2 \sqrt {e \,x^{2}+d}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b e \,x^{2}-\sqrt {e \,x^{2}+d}\, a d -2 \sqrt {e \,x^{2}+d}\, a e \,x^{2}-\sqrt {e \,x^{2}+d}\, b d n -\sqrt {e \,x^{2}+d}\, b e n \,x^{2}+2 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d n x +2 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b e n \,x^{3}+2 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d n x +2 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b e n \,x^{3}-2 \sqrt {e}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b d x -2 \sqrt {e}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b e \,x^{3}-2 \sqrt {e}\, a d x -2 \sqrt {e}\, a e \,x^{3}-\sqrt {e}\, b d n x -\sqrt {e}\, b e n \,x^{3}}{d^{2} x \left (e \,x^{2}+d \right )} \] Input:

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

Output:

( - 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 - 2*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*x**2 - sqrt(d + e*x**2)*a*d - 2*sqrt(d 
 + e*x**2)*a*e*x**2 - sqrt(d + e*x**2)*b*d*n - sqrt(d + e*x**2)*b*e*n*x**2 
 + 2*sqrt(e)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d*n*x 
 + 2*sqrt(e)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*b*e*n*x 
**3 + 2*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d* 
n*x + 2*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*e* 
n*x**3 - 2*sqrt(e)*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*x - 2*sqrt(e)*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*x**3 - 2*sqrt(e)*a*d*x - 2*sqrt(e)*a*e*x**3 - s 
qrt(e)*b*d*n*x - sqrt(e)*b*e*n*x**3)/(d**2*x*(d + e*x**2))