\(\int \frac {a+b \log (c x^n)}{x^3 (d+e x^2)^{3/2}} \, dx\) [291]

Optimal result

Integrand size = 25, antiderivative size = 287 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {d+e x^2}}{4 d^2 x^2}-\frac {5 b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{4 d^{5/2}}-\frac {3 b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{4 d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{2 d x^2 \sqrt {d+e x^2}}+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac {3 b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{2 d^{5/2}}+\frac {3 b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{4 d^{5/2}} \] Output:

-1/4*b*n*(e*x^2+d)^(1/2)/d^2/x^2-5/4*b*e*n*arctanh((e*x^2+d)^(1/2)/d^(1/2) 
)/d^(5/2)-3/4*b*e*n*arctanh((e*x^2+d)^(1/2)/d^(1/2))^2/d^(5/2)-3/2*e*(a+b* 
ln(c*x^n))/d^2/(e*x^2+d)^(1/2)-1/2*(a+b*ln(c*x^n))/d/x^2/(e*x^2+d)^(1/2)+3 
/2*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))*(a+b*ln(c*x^n))/d^(5/2)+3/2*b*e*n*ar 
ctanh((e*x^2+d)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(e*x^2+d)^(1/2)))/d^( 
5/2)+3/4*b*e*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(e*x^2+d)^(1/2)))/d^(5/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.40 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\frac {3 b d^{5/2} n \sqrt {1+\frac {d}{e x^2}} \, _3F_2\left (\frac {5}{2},\frac {5}{2},\frac {5}{2};\frac {7}{2},\frac {7}{2};-\frac {d}{e x^2}\right )-5 b d^{5/2} n \sqrt {1+\frac {d}{e x^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},-\frac {d}{e x^2}\right ) (1+2 \log (x))-25 e x^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\sqrt {d} \left (d+3 e x^2\right )+3 e x^2 \sqrt {d+e x^2} \log (x)-3 e x^2 \sqrt {d+e x^2} \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )}{50 d^{5/2} e x^4 \sqrt {d+e x^2}} \] Input:

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

Output:

(3*b*d^(5/2)*n*Sqrt[1 + d/(e*x^2)]*HypergeometricPFQ[{5/2, 5/2, 5/2}, {7/2 
, 7/2}, -(d/(e*x^2))] - 5*b*d^(5/2)*n*Sqrt[1 + d/(e*x^2)]*Hypergeometric2F 
1[3/2, 5/2, 7/2, -(d/(e*x^2))]*(1 + 2*Log[x]) - 25*e*x^2*(a - b*n*Log[x] + 
 b*Log[c*x^n])*(Sqrt[d]*(d + 3*e*x^2) + 3*e*x^2*Sqrt[d + e*x^2]*Log[x] - 3 
*e*x^2*Sqrt[d + e*x^2]*Log[d + Sqrt[d]*Sqrt[d + e*x^2]]))/(50*d^(5/2)*e*x^ 
4*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2792, 2009}

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.

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

\(\Big \downarrow \) 2792

\(\displaystyle -b n \int \left (\frac {3 e \text {arctanh}\left (\frac {\sqrt {e x^2+d}}{\sqrt {d}}\right )}{2 d^{5/2} x}-\frac {3 e x^2+d}{2 d^2 x^3 \sqrt {e x^2+d}}\right )dx+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{2 d x^2 \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{2 d x^2 \sqrt {d+e x^2}}-b n \left (\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{4 d^{5/2}}+\frac {5 e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{4 d^{5/2}}-\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{2 d^{5/2}}-\frac {3 e \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^2+d}}\right )}{4 d^{5/2}}+\frac {\sqrt {d+e x^2}}{4 d^2 x^2}\right )\)

Input:

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

Output:

(-3*e*(a + b*Log[c*x^n]))/(2*d^2*Sqrt[d + e*x^2]) - (a + b*Log[c*x^n])/(2* 
d*x^2*Sqrt[d + e*x^2]) + (3*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[ 
c*x^n]))/(2*d^(5/2)) - b*n*(Sqrt[d + e*x^2]/(4*d^2*x^2) + (5*e*ArcTanh[Sqr 
t[d + e*x^2]/Sqrt[d]])/(4*d^(5/2)) + (3*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] 
^2)/(4*d^(5/2)) - (3*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(S 
qrt[d] - Sqrt[d + e*x^2])])/(2*d^(5/2)) - (3*e*PolyLog[2, 1 - (2*Sqrt[d])/ 
(Sqrt[d] - Sqrt[d + e*x^2])])/(4*d^(5/2)))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((sqrt(e*x^2 + d)*b*log(c*x^n) + sqrt(e*x^2 + d)*a)/(e^2*x^7 + 2*d 
*e*x^5 + d^2*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a+b*log(c*x^n))/x^3/(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, a \,d^{2}-3 \sqrt {e \,x^{2}+d}\, a d e \,x^{2}-3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a d e \,x^{2}-3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a \,e^{2} x^{4}+3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a \,e^{2} x^{4}+2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{\sqrt {e \,x^{2}+d}\, d \,x^{3}+\sqrt {e \,x^{2}+d}\, e \,x^{5}}d x \right ) b \,d^{4} x^{2}+2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{\sqrt {e \,x^{2}+d}\, d \,x^{3}+\sqrt {e \,x^{2}+d}\, e \,x^{5}}d x \right ) b \,d^{3} e \,x^{4}}{2 d^{3} x^{2} \left (e \,x^{2}+d \right )} \] Input:

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

Output:

( - sqrt(d + e*x**2)*a*d**2 - 3*sqrt(d + e*x**2)*a*d*e*x**2 - 3*sqrt(d)*lo 
g((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*a*d*e*x**2 - 3*sqrt(d) 
*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*a*e**2*x**4 + 3*sqr 
t(d)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*a*d*e*x**2 + 3* 
sqrt(d)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*a*e**2*x**4 
+ 2*int(log(x**n*c)/(sqrt(d + e*x**2)*d*x**3 + sqrt(d + e*x**2)*e*x**5),x) 
*b*d**4*x**2 + 2*int(log(x**n*c)/(sqrt(d + e*x**2)*d*x**3 + sqrt(d + e*x** 
2)*e*x**5),x)*b*d**3*e*x**4)/(2*d**3*x**2*(d + e*x**2))