\(\int x^3 (a+b \log (c (d+\frac {e}{\sqrt {x}})^n)) \, dx\) [421]

Optimal result

Integrand size = 22, antiderivative size = 171 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {b e^7 n \sqrt {x}}{4 d^7}-\frac {b e^6 n x}{8 d^6}+\frac {b e^5 n x^{3/2}}{12 d^5}-\frac {b e^4 n x^2}{16 d^4}+\frac {b e^3 n x^{5/2}}{20 d^3}-\frac {b e^2 n x^3}{24 d^2}+\frac {b e n x^{7/2}}{28 d}-\frac {b e^8 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{4 d^8}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^8 n \log (x)}{8 d^8} \] Output:

1/4*b*e^7*n*x^(1/2)/d^7-1/8*b*e^6*n*x/d^6+1/12*b*e^5*n*x^(3/2)/d^5-1/16*b* 
e^4*n*x^2/d^4+1/20*b*e^3*n*x^(5/2)/d^3-1/24*b*e^2*n*x^3/d^2+1/28*b*e*n*x^( 
7/2)/d-1/4*b*e^8*n*ln(d+e/x^(1/2))/d^8+1/4*x^4*(a+b*ln(c*(d+e/x^(1/2))^n)) 
-1/8*b*e^8*n*ln(x)/d^8
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^4}{4}+\frac {1}{4} b x^4 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+\frac {1}{8} b e n \left (\frac {2 e^6 \sqrt {x}}{d^7}-\frac {e^5 x}{d^6}+\frac {2 e^4 x^{3/2}}{3 d^5}-\frac {e^3 x^2}{2 d^4}+\frac {2 e^2 x^{5/2}}{5 d^3}-\frac {e x^3}{3 d^2}+\frac {2 x^{7/2}}{7 d}-\frac {2 e^7 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^8}-\frac {e^7 \log (x)}{d^8}\right ) \] Input:

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

Output:

(a*x^4)/4 + (b*x^4*Log[c*(d + e/Sqrt[x])^n])/4 + (b*e*n*((2*e^6*Sqrt[x])/d 
^7 - (e^5*x)/d^6 + (2*e^4*x^(3/2))/(3*d^5) - (e^3*x^2)/(2*d^4) + (2*e^2*x^ 
(5/2))/(5*d^3) - (e*x^3)/(3*d^2) + (2*x^(7/2))/(7*d) - (2*e^7*Log[d + e/Sq 
rt[x]])/d^8 - (e^7*Log[x])/d^8))/8
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2904, 2842, 54, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle -2 \int x^{9/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2842

\(\displaystyle -2 \left (\frac {1}{8} b e n \int \frac {x^4}{d+\frac {e}{\sqrt {x}}}d\frac {1}{\sqrt {x}}-\frac {1}{8} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\)

\(\Big \downarrow \) 54

\(\displaystyle -2 \left (\frac {1}{8} b e n \int \left (\frac {e^8}{d^8 \left (d+\frac {e}{\sqrt {x}}\right )}-\frac {\sqrt {x} e^7}{d^8}+\frac {x e^6}{d^7}-\frac {x^{3/2} e^5}{d^6}+\frac {x^2 e^4}{d^5}-\frac {x^{5/2} e^3}{d^4}+\frac {x^3 e^2}{d^3}-\frac {x^{7/2} e}{d^2}+\frac {x^4}{d}\right )d\frac {1}{\sqrt {x}}-\frac {1}{8} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {1}{8} b e n \left (\frac {e^7 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^8}-\frac {e^7 \log \left (\frac {1}{\sqrt {x}}\right )}{d^8}-\frac {e^6 \sqrt {x}}{d^7}+\frac {e^5 x}{2 d^6}-\frac {e^4 x^{3/2}}{3 d^5}+\frac {e^3 x^2}{4 d^4}-\frac {e^2 x^{5/2}}{5 d^3}+\frac {e x^3}{6 d^2}-\frac {x^{7/2}}{7 d}\right )-\frac {1}{8} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\)

Input:

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

Output:

-2*(-1/8*(x^4*(a + b*Log[c*(d + e/Sqrt[x])^n])) + (b*e*n*(-((e^6*Sqrt[x])/ 
d^7) + (e^5*x)/(2*d^6) - (e^4*x^(3/2))/(3*d^5) + (e^3*x^2)/(4*d^4) - (e^2* 
x^(5/2))/(5*d^3) + (e*x^3)/(6*d^2) - x^(7/2)/(7*d) + (e^7*Log[d + e/Sqrt[x 
]])/d^8 - (e^7*Log[1/Sqrt[x]])/d^8))/8)
 

Defintions of rubi rules used

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

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

rule 2842

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ 
))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( 
g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1)))   Int[(f + g*x)^(q + 1)/(d + e*x 
), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && 
NeQ[q, -1]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
 

Maple [F]

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

Input:

int(x^3*(a+b*ln(c*(d+e/x^(1/2))^n)),x)
 

Output:

int(x^3*(a+b*ln(c*(d+e/x^(1/2))^n)),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.05 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {420 \, b d^{8} x^{4} \log \left (c\right ) - 70 \, b d^{6} e^{2} n x^{3} + 420 \, a d^{8} x^{4} - 105 \, b d^{4} e^{4} n x^{2} - 210 \, b d^{2} e^{6} n x - 420 \, b d^{8} n \log \left (\sqrt {x}\right ) + 420 \, {\left (b d^{8} - b e^{8}\right )} n \log \left (d \sqrt {x} + e\right ) + 420 \, {\left (b d^{8} n x^{4} - b d^{8} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) + 4 \, {\left (15 \, b d^{7} e n x^{3} + 21 \, b d^{5} e^{3} n x^{2} + 35 \, b d^{3} e^{5} n x + 105 \, b d e^{7} n\right )} \sqrt {x}}{1680 \, d^{8}} \] Input:

integrate(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="fricas")
 

Output:

1/1680*(420*b*d^8*x^4*log(c) - 70*b*d^6*e^2*n*x^3 + 420*a*d^8*x^4 - 105*b* 
d^4*e^4*n*x^2 - 210*b*d^2*e^6*n*x - 420*b*d^8*n*log(sqrt(x)) + 420*(b*d^8 
- b*e^8)*n*log(d*sqrt(x) + e) + 420*(b*d^8*n*x^4 - b*d^8*n)*log((d*x + e*s 
qrt(x))/x) + 4*(15*b*d^7*e*n*x^3 + 21*b*d^5*e^3*n*x^2 + 35*b*d^3*e^5*n*x + 
 105*b*d*e^7*n)*sqrt(x))/d^8
 

Sympy [A] (verification not implemented)

Time = 46.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^{4}}{4} + b \left (\frac {e n \left (\frac {2 x^{\frac {7}{2}}}{7 d} - \frac {e x^{3}}{3 d^{2}} + \frac {2 e^{2} x^{\frac {5}{2}}}{5 d^{3}} - \frac {e^{3} x^{2}}{2 d^{4}} + \frac {2 e^{4} x^{\frac {3}{2}}}{3 d^{5}} - \frac {e^{5} x}{d^{6}} - \frac {2 e^{7} \left (\begin {cases} \frac {\sqrt {x}}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d \sqrt {x} + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{7}} + \frac {2 e^{6} \sqrt {x}}{d^{7}}\right )}{8} + \frac {x^{4} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{4}\right ) \] Input:

integrate(x**3*(a+b*ln(c*(d+e/x**(1/2))**n)),x)
 

Output:

a*x**4/4 + b*(e*n*(2*x**(7/2)/(7*d) - e*x**3/(3*d**2) + 2*e**2*x**(5/2)/(5 
*d**3) - e**3*x**2/(2*d**4) + 2*e**4*x**(3/2)/(3*d**5) - e**5*x/d**6 - 2*e 
**7*Piecewise((sqrt(x)/e, Eq(d, 0)), (log(d*sqrt(x) + e)/d, True))/d**7 + 
2*e**6*sqrt(x)/d**7)/8 + x**4*log(c*(d + e/sqrt(x))**n)/4)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{1680} \, b e n {\left (\frac {420 \, e^{7} \log \left (d \sqrt {x} + e\right )}{d^{8}} - \frac {60 \, d^{6} x^{\frac {7}{2}} - 70 \, d^{5} e x^{3} + 84 \, d^{4} e^{2} x^{\frac {5}{2}} - 105 \, d^{3} e^{3} x^{2} + 140 \, d^{2} e^{4} x^{\frac {3}{2}} - 210 \, d e^{5} x + 420 \, e^{6} \sqrt {x}}{d^{7}}\right )} \] Input:

integrate(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="maxima")
 

Output:

1/4*b*x^4*log(c*(d + e/sqrt(x))^n) + 1/4*a*x^4 - 1/1680*b*e*n*(420*e^7*log 
(d*sqrt(x) + e)/d^8 - (60*d^6*x^(7/2) - 70*d^5*e*x^3 + 84*d^4*e^2*x^(5/2) 
- 105*d^3*e^3*x^2 + 140*d^2*e^4*x^(3/2) - 210*d*e^5*x + 420*e^6*sqrt(x))/d 
^7)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.56 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left (c\right ) + \frac {1}{4} \, a x^{4} - \frac {{\left (e^{9} {\left (\frac {420 \, \log \left (\frac {{\left | d \sqrt {x} + e \right |}}{\sqrt {{\left | x \right |}}}\right )}{d^{8}} - \frac {420 \, \log \left ({\left | -d + \frac {d \sqrt {x} + e}{\sqrt {x}} \right |}\right )}{d^{8}} + \frac {1089 \, d^{7} - \frac {4683 \, {\left (d \sqrt {x} + e\right )} d^{6}}{\sqrt {x}} + \frac {9639 \, {\left (d \sqrt {x} + e\right )}^{2} d^{5}}{x} - \frac {11165 \, {\left (d \sqrt {x} + e\right )}^{3} d^{4}}{x^{\frac {3}{2}}} + \frac {7490 \, {\left (d \sqrt {x} + e\right )}^{4} d^{3}}{x^{2}} - \frac {2730 \, {\left (d \sqrt {x} + e\right )}^{5} d^{2}}{x^{\frac {5}{2}}} + \frac {420 \, {\left (d \sqrt {x} + e\right )}^{6} d}{x^{3}}}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{7} d^{8}}\right )} - \frac {420 \, e^{9} \log \left (-{\left (e - \frac {d}{\frac {d}{e} - \frac {d \sqrt {x} + e}{e \sqrt {x}}}\right )} {\left (\frac {d}{e} - \frac {d \sqrt {x} + e}{e \sqrt {x}}\right )}\right )}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{8}}\right )} b n}{1680 \, e} \] Input:

integrate(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="giac")
 

Output:

1/4*b*x^4*log(c) + 1/4*a*x^4 - 1/1680*(e^9*(420*log(abs(d*sqrt(x) + e)/sqr 
t(abs(x)))/d^8 - 420*log(abs(-d + (d*sqrt(x) + e)/sqrt(x)))/d^8 + (1089*d^ 
7 - 4683*(d*sqrt(x) + e)*d^6/sqrt(x) + 9639*(d*sqrt(x) + e)^2*d^5/x - 1116 
5*(d*sqrt(x) + e)^3*d^4/x^(3/2) + 7490*(d*sqrt(x) + e)^4*d^3/x^2 - 2730*(d 
*sqrt(x) + e)^5*d^2/x^(5/2) + 420*(d*sqrt(x) + e)^6*d/x^3)/((d - (d*sqrt(x 
) + e)/sqrt(x))^7*d^8)) - 420*e^9*log(-(e - d/(d/e - (d*sqrt(x) + e)/(e*sq 
rt(x))))*(d/e - (d*sqrt(x) + e)/(e*sqrt(x))))/(d - (d*sqrt(x) + e)/sqrt(x) 
)^8)*b*n/e
 

Mupad [B] (verification not implemented)

Time = 15.32 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {\frac {b\,d\,e^7\,n\,\sqrt {x}}{4}-\frac {b\,d^2\,e^6\,n\,x}{8}+\frac {b\,d^7\,e\,n\,x^{7/2}}{28}-\frac {b\,d^4\,e^4\,n\,x^2}{16}-\frac {b\,d^6\,e^2\,n\,x^3}{24}+\frac {b\,d^3\,e^5\,n\,x^{3/2}}{12}+\frac {b\,d^5\,e^3\,n\,x^{5/2}}{20}+\frac {b\,e^8\,n\,\mathrm {atan}\left (\frac {d\,1{}\mathrm {i}+\frac {e\,2{}\mathrm {i}}{\sqrt {x}}}{d}\right )\,1{}\mathrm {i}}{2}}{d^8}+\frac {a\,x^4}{4}+\frac {b\,x^4\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{4} \] Input:

int(x^3*(a + b*log(c*(d + e/x^(1/2))^n)),x)
 

Output:

((b*e^8*n*atan((d*1i + (e*2i)/x^(1/2))/d)*1i)/2 - (b*d^2*e^6*n*x)/8 + (b*d 
*e^7*n*x^(1/2))/4 + (b*d^7*e*n*x^(7/2))/28 - (b*d^4*e^4*n*x^2)/16 - (b*d^6 
*e^2*n*x^3)/24 + (b*d^3*e^5*n*x^(3/2))/12 + (b*d^5*e^3*n*x^(5/2))/20)/d^8 
+ (a*x^4)/4 + (b*x^4*log(c*(d + e/x^(1/2))^n))/4
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {60 \sqrt {x}\, b \,d^{7} e n \,x^{3}+84 \sqrt {x}\, b \,d^{5} e^{3} n \,x^{2}+140 \sqrt {x}\, b \,d^{3} e^{5} n x +420 \sqrt {x}\, b d \,e^{7} n +420 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,d^{8} x^{4}-420 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,e^{8}-420 \,\mathrm {log}\left (\sqrt {x}\right ) b \,e^{8} n +420 a \,d^{8} x^{4}-70 b \,d^{6} e^{2} n \,x^{3}-105 b \,d^{4} e^{4} n \,x^{2}-210 b \,d^{2} e^{6} n x}{1680 d^{8}} \] Input:

int(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x)
 

Output:

(60*sqrt(x)*b*d**7*e*n*x**3 + 84*sqrt(x)*b*d**5*e**3*n*x**2 + 140*sqrt(x)* 
b*d**3*e**5*n*x + 420*sqrt(x)*b*d*e**7*n + 420*log(((sqrt(x)*d + e)**n*c)/ 
x**(n/2))*b*d**8*x**4 - 420*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b*e**8 - 
420*log(sqrt(x))*b*e**8*n + 420*a*d**8*x**4 - 70*b*d**6*e**2*n*x**3 - 105* 
b*d**4*e**4*n*x**2 - 210*b*d**2*e**6*n*x)/(1680*d**8)