\(\int x^3 (a+b \log (c (d+e \sqrt {x})^2))^p \, dx\) [538]

Optimal result

Integrand size = 24, antiderivative size = 907 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx =\text {Too large to display} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 3.04 (sec) , antiderivative size = 896, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*log(c*e^2*x + 2*c*d*e*sqrt(x) + c*d^2) + a)^p*x^3, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:

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

Output:

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