Integrand size = 22, antiderivative size = 730 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx =\text {Too large to display} \] Output:
2^(-2-3*p)*GAMMA(p+1,(-8*a-8*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-7*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-6 *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-5*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-4*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-3*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-2*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)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx \] 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]
Time = 2.87 (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))
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]
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])
\[\int x^{3} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )\right )\right )^{p}d x\]
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)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \] 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)
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
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \] 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)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \] 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)
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)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x^3*(a+b*log(c*(d+e*x^(1/2))))^p,x)
Output:
(840*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**7*e*p**2 + 840*sqrt(x) *(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**7*e*p + 280*sqrt(x)*(log(sqrt(x)*c *e + c*d)*b + a)**p*b*d**5*e**3*p**2*x + 280*sqrt(x)*(log(sqrt(x)*c*e + c* d)*b + a)**p*b*d**5*e**3*p*x + 168*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a)* *p*b*d**3*e**5*p**2*x**2 + 168*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a)**p*b *d**3*e**5*p*x**2 + 120*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d*e**7 *p**2*x**3 + 120*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d*e**7*p*x**3 - 840*(log(sqrt(x)*c*e + c*d)*b + a)**p*log(sqrt(x)*c*e + c*d)*b*d**8*p - 840*(log(sqrt(x)*c*e + c*d)*b + a)**p*a*d**8*p + 840*(log(sqrt(x)*c*e + c *d)*b + a)**p*a*e**8*p*x**4 + 840*(log(sqrt(x)*c*e + c*d)*b + a)**p*a*e**8 *x**4 - 420*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**6*e**2*p**2*x - 420*(lo g(sqrt(x)*c*e + c*d)*b + a)**p*b*d**6*e**2*p*x - 210*(log(sqrt(x)*c*e + c* d)*b + a)**p*b*d**4*e**4*p**2*x**2 - 210*(log(sqrt(x)*c*e + c*d)*b + a)**p *b*d**4*e**4*p*x**2 - 140*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**2*e**6*p* *2*x**3 - 140*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**2*e**6*p*x**3 - 3360* int((log(sqrt(x)*c*e + c*d)*b + a)**p/(8*sqrt(x)*log(sqrt(x)*c*e + c*d)*a* b*e + sqrt(x)*log(sqrt(x)*c*e + c*d)*b**2*e*p + 8*sqrt(x)*a**2*e + sqrt(x) *a*b*e*p + 8*log(sqrt(x)*c*e + c*d)*a*b*d + log(sqrt(x)*c*e + c*d)*b**2*d* p + 8*a**2*d + a*b*d*p),x)*a*b**2*d**7*e**2*p**3 - 3360*int((log(sqrt(x)*c *e + c*d)*b + a)**p/(8*sqrt(x)*log(sqrt(x)*c*e + c*d)*a*b*e + sqrt(x)*l...