3.6.32 \(\int x^3 (a+b \log (c (d+e \sqrt {x})))^p \, dx\) [532]

3.6.32.1 Optimal result

Integrand size = 22, antiderivative size = 730 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-2-3 p} e^{-\frac {8 a}{b}} \Gamma \left (1+p,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^8 e^8}-\frac {2\ 7^{-p} d e^{-\frac {7 a}{b}} \Gamma \left (1+p,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^7 e^8}+\frac {7\ 6^{-p} d^2 e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^6 e^8}-\frac {14\ 5^{-p} d^3 e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^5 e^8}+\frac {35\ 2^{-1-2 p} d^4 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^8}-\frac {14\ 3^{-p} d^5 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^3 e^8}+\frac {7\ 2^{-p} d^6 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^8}-\frac {2 d^7 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^8} \]

output

2^(-2-3*p)*GAMMA(p+1,-8*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2 
))))^p/c^8/e^8/exp(8*a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)-2*d*GAMMA(p+1 
,-7*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(7^p)/c^7/e^8 
/exp(7*a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)+7*d^2*GAMMA(p+1,-6*(a+b*ln( 
c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(6^p)/c^6/e^8/exp(6*a/b)/ 
(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)-14*d^3*GAMMA(p+1,-5*(a+b*ln(c*(d+e*x^(1 
/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(5^p)/c^5/e^8/exp(5*a/b)/(((-a-b*ln( 
c*(d+e*x^(1/2))))/b)^p)+35*2^(-1-2*p)*d^4*GAMMA(p+1,-4*(a+b*ln(c*(d+e*x^(1 
/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/c^4/e^8/exp(4*a/b)/(((-a-b*ln(c*(d+e 
*x^(1/2))))/b)^p)-14*d^5*GAMMA(p+1,-3*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln 
(c*(d+e*x^(1/2))))^p/(3^p)/c^3/e^8/exp(3*a/b)/(((-a-b*ln(c*(d+e*x^(1/2)))) 
/b)^p)+7*d^6*GAMMA(p+1,-2*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1 
/2))))^p/(2^p)/c^2/e^8/exp(2*a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)-2*d^7 
*GAMMA(p+1,(-a-b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/c/e^8 
/exp(a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)
 

3.6.32.2 Mathematica [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx \]

input

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

output

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

3.6.32.3 Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2904, 2848, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 2 \int x^{7/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^pd\sqrt {x}\)

\(\Big \downarrow \) 2848

\(\displaystyle 2 \int \left (\frac {\left (d+e \sqrt {x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}-\frac {7 d \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}+\frac {21 d^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}-\frac {35 d^3 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}+\frac {35 d^4 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}-\frac {21 d^5 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}+\frac {7 d^6 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}-\frac {d^7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{e^7}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {8^{-p-1} e^{-\frac {8 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^8 e^8}-\frac {d 7^{-p} e^{-\frac {7 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^7 e^8}+\frac {7 d^2 2^{-p-1} 3^{-p} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^6 e^8}-\frac {7 d^3 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^5 e^8}+\frac {35 d^4 4^{-p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^4 e^8}-\frac {7 d^5 3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^3 e^8}+\frac {7 d^6 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^2 e^8}-\frac {d^7 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )}{c e^8}\right )\)

input

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

output

2*((8^(-1 - p)*Gamma[1 + p, (-8*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b* 
Log[c*(d + e*Sqrt[x])])^p)/(c^8*e^8*E^((8*a)/b)*(-((a + b*Log[c*(d + e*Sqr 
t[x])])/b))^p) - (d*Gamma[1 + p, (-7*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a 
 + b*Log[c*(d + e*Sqrt[x])])^p)/(7^p*c^7*e^8*E^((7*a)/b)*(-((a + b*Log[c*( 
d + e*Sqrt[x])])/b))^p) + (7*2^(-1 - p)*d^2*Gamma[1 + p, (-6*(a + b*Log[c* 
(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(3^p*c^6*e^8*E^((6 
*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (7*d^3*Gamma[1 + p, (-5* 
(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(5^p* 
c^5*e^8*E^((5*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) + (35*4^(-1 - 
 p)*d^4*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*( 
d + e*Sqrt[x])])^p)/(c^4*e^8*E^((4*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])]) 
/b))^p) - (7*d^5*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + 
b*Log[c*(d + e*Sqrt[x])])^p)/(3^p*c^3*e^8*E^((3*a)/b)*(-((a + b*Log[c*(d + 
 e*Sqrt[x])])/b))^p) + (7*2^(-1 - p)*d^6*Gamma[1 + p, (-2*(a + b*Log[c*(d 
+ e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(c^2*e^8*E^((2*a)/b)* 
(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (d^7*Gamma[1 + p, -((a + b*Log[ 
c*(d + e*Sqrt[x])])/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(c*e^8*E^(a/b)*( 
-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p))
 

3.6.32.3.1 Defintions of rubi rules used

rule 2009

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

rule 2848

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

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])
 

3.6.32.4 Maple [F]

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

input

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

output

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

3.6.32.5 Fricas [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]

input

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

output

integral((b*log(c*e*sqrt(x) + c*d) + a)^p*x^3, x)
 

3.6.32.6 Sympy [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.6.32.7 Maxima [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]

input

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

output

integrate((b*log((e*sqrt(x) + d)*c) + a)^p*x^3, x)
 

3.6.32.8 Giac [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]

input

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

output

integrate((b*log((e*sqrt(x) + d)*c) + a)^p*x^3, x)
 

3.6.32.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,\left (d+e\,\sqrt {x}\right )\right )\right )}^p \,d x \]

input

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

output

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