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

Optimal. Leaf size=730 \[ \frac {2^{-3 p-2} 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 {2 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 6^{-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 {14 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 2^{-2 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 {14 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} 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 {2 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} \]

[Out]

2^(-2-3*p)*GAMMA(1+p,-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(1+p,-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(1+p,-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(1+p,-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(1+p,-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(1+p,-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(1+p,-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(1+p,(-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)

________________________________________________________________________________________

Rubi [A]  time = 1.34, antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2454, 2401, 2389, 2299, 2181, 2390, 2309} \[ \frac {7 d^2 6^{-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} \text {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 {14 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} \text {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 2^{-2 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} \text {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 {14 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} \text {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} 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} \text {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 {2^{-3 p-2} 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} \text {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 {2 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} \text {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 {2 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} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )}{c e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-2 - 3*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*Sqrt[x])])/b))^p) - (2*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*d^2*
Gamma[1 + p, (-6*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(6^p*c^6*e^8*E^((6*a)/b)
*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (14*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*2^(-1 - 2*
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) - (14*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*d^6*Gamm
a[1 + p, (-2*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*c^2*e^8*E^((2*a)/b)*(-(
(a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (2*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)

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2390

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

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[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])

Rubi steps

\begin {align*} \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx &=2 \operatorname {Subst}\left (\int x^7 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {d^7 (a+b \log (c (d+e x)))^p}{e^7}+\frac {7 d^6 (d+e x) (a+b \log (c (d+e x)))^p}{e^7}-\frac {21 d^5 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^7}+\frac {35 d^4 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^7}-\frac {35 d^3 (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^7}+\frac {21 d^2 (d+e x)^5 (a+b \log (c (d+e x)))^p}{e^7}-\frac {7 d (d+e x)^6 (a+b \log (c (d+e x)))^p}{e^7}+\frac {(d+e x)^7 (a+b \log (c (d+e x)))^p}{e^7}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \operatorname {Subst}\left (\int (d+e x)^7 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {(14 d) \operatorname {Subst}\left (\int (d+e x)^6 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}+\frac {\left (42 d^2\right ) \operatorname {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {\left (70 d^3\right ) \operatorname {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}+\frac {\left (70 d^4\right ) \operatorname {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {\left (42 d^5\right ) \operatorname {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}+\frac {\left (14 d^6\right ) \operatorname {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {\left (2 d^7\right ) \operatorname {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}\\ &=\frac {2 \operatorname {Subst}\left (\int x^7 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {(14 d) \operatorname {Subst}\left (\int x^6 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}+\frac {\left (42 d^2\right ) \operatorname {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {\left (70 d^3\right ) \operatorname {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}+\frac {\left (70 d^4\right ) \operatorname {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {\left (42 d^5\right ) \operatorname {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}+\frac {\left (14 d^6\right ) \operatorname {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {\left (2 d^7\right ) \operatorname {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}\\ &=\frac {2 \operatorname {Subst}\left (\int e^{8 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^8 e^8}-\frac {(14 d) \operatorname {Subst}\left (\int e^{7 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^7 e^8}+\frac {\left (42 d^2\right ) \operatorname {Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^6 e^8}-\frac {\left (70 d^3\right ) \operatorname {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^5 e^8}+\frac {\left (70 d^4\right ) \operatorname {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^4 e^8}-\frac {\left (42 d^5\right ) \operatorname {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^3 e^8}+\frac {\left (14 d^6\right ) \operatorname {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^2 e^8}-\frac {\left (2 d^7\right ) \operatorname {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c e^8}\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 1.02, size = 435, normalized size = 0.60 \[ \frac {2^{-3 p-2} 105^{-p} e^{-\frac {8 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \left (c^7 d^7 \left (-8^{p+1}\right ) 105^p e^{\frac {7 a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )+c^6 d^6 15^p 28^{p+1} e^{\frac {6 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )-c^5 d^5 5^p 56^{p+1} e^{\frac {5 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )+c^4 d^4 3^p 70^{p+1} e^{\frac {4 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )-c^3 d^3 3^p 56^{p+1} e^{\frac {3 a}{b}} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )+c^2 d^2 5^p 28^{p+1} e^{\frac {2 a}{b}} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )-c d 8^{p+1} 15^p e^{a/b} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )+105^p \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )\right )}{c^8 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-2 - 3*p)*(105^p*Gamma[1 + p, (-8*(a + b*Log[c*(d + e*Sqrt[x])]))/b] - 8^(1 + p)*15^p*c*d*E^(a/b)*Gamma[1
+ p, (-7*(a + b*Log[c*(d + e*Sqrt[x])]))/b] + 5^p*28^(1 + p)*c^2*d^2*E^((2*a)/b)*Gamma[1 + p, (-6*(a + b*Log[c
*(d + e*Sqrt[x])]))/b] - 3^p*56^(1 + p)*c^3*d^3*E^((3*a)/b)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*Sqrt[x])]))/b
] + 3^p*70^(1 + p)*c^4*d^4*E^((4*a)/b)*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*Sqrt[x])]))/b] - 5^p*56^(1 + p)*c^
5*d^5*E^((5*a)/b)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*Sqrt[x])]))/b] + 15^p*28^(1 + p)*c^6*d^6*E^((6*a)/b)*Ga
mma[1 + p, (-2*(a + b*Log[c*(d + e*Sqrt[x])]))/b] - 8^(1 + p)*105^p*c^7*d^7*E^((7*a)/b)*Gamma[1 + p, -((a + b*
Log[c*(d + e*Sqrt[x])])/b)])*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(105^p*c^8*e^8*E^((8*a)/b)*(-((a + b*Log[c*(d +
e*Sqrt[x])])/b))^p)

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c e \sqrt {x} + c d\right ) + a\right )}^{p} x^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int x^{3} \left (b \ln \left (\left (e \sqrt {x}+d \right ) c \right )+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\ln \left (c\,\left (d+e\,\sqrt {x}\right )\right )\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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