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

Optimal. Leaf size=907 \[ \text{result too large to display} \]

[Out]

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

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Rubi [A]  time = 1.40442, antiderivative size = 907, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2454, 2401, 2389, 2300, 2181, 2390, 2310} \[ \frac{2^{-2 (p+1)} e^{-\frac{4 a}{b}} \text{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{2^{p+1} 7^{-p} d e^{-\frac{7 a}{2 b}} \left (d+e \sqrt{x}\right )^7 \text{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}} \text{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}}{c^3 e^8}-\frac{7\ 2^{p+1} 5^{-p} d^3 e^{-\frac{5 a}{2 b}} \left (d+e \sqrt{x}\right )^5 \text{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-1} d^4 e^{-\frac{2 a}{b}} \text{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\ 2^{p+1} 3^{-p} d^5 e^{-\frac{3 a}{2 b}} \left (d+e \sqrt{x}\right )^3 \text{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}} \text{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}}{c e^8}-\frac{2^{p+1} d^7 e^{-\frac{a}{2 b}} \left (d+e \sqrt{x}\right ) \text{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}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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 2300

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

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 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 2310

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

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \, dx &=2 \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{d^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}+\frac{7 d^6 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}-\frac{21 d^5 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}+\frac{35 d^4 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}-\frac{35 d^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}+\frac{21 d^2 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}-\frac{7 d (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}+\frac{(d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^7}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}-\frac{(14 d) \operatorname{Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}+\frac{\left (42 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}-\frac{\left (70 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}+\frac{\left (70 d^4\right ) \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}-\frac{\left (42 d^5\right ) \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}+\frac{\left (14 d^6\right ) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}-\frac{\left (2 d^7\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^7}\\ &=\frac{2 \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}-\frac{(14 d) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}+\frac{\left (42 d^2\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}-\frac{\left (70 d^3\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}+\frac{\left (70 d^4\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}-\frac{\left (42 d^5\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}+\frac{\left (14 d^6\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}-\frac{\left (2 d^7\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^8}\\ &=\frac{\operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{c^4 e^8}+\frac{\left (21 d^2\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{c^3 e^8}+\frac{\left (35 d^4\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{c^2 e^8}+\frac{\left (7 d^6\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{c e^8}-\frac{\left (7 d \left (d+e \sqrt{x}\right )^7\right ) \operatorname{Subst}\left (\int e^{7 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^8 \left (c \left (d+e \sqrt{x}\right )^2\right )^{7/2}}-\frac{\left (35 d^3 \left (d+e \sqrt{x}\right )^5\right ) \operatorname{Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^8 \left (c \left (d+e \sqrt{x}\right )^2\right )^{5/2}}-\frac{\left (21 d^5 \left (d+e \sqrt{x}\right )^3\right ) \operatorname{Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^8 \left (c \left (d+e \sqrt{x}\right )^2\right )^{3/2}}-\frac{\left (d^7 \left (d+e \sqrt{x}\right )\right ) \operatorname{Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^8 \sqrt{c \left (d+e \sqrt{x}\right )^2}}\\ &=\frac{4^{-1-p} e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\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{2^{1+p} 7^{-p} d e^{-\frac{7 a}{2 b}} \left (d+e \sqrt{x}\right )^7 \Gamma \left (1+p,-\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 (1+p,-\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}}{c^3 e^8}-\frac{7\ 2^{1+p} 5^{-p} d^3 e^{-\frac{5 a}{2 b}} \left (d+e \sqrt{x}\right )^5 \Gamma \left (1+p,-\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^{-1-p} d^4 e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\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\ 2^{1+p} 3^{-p} d^5 e^{-\frac{3 a}{2 b}} \left (d+e \sqrt{x}\right )^3 \Gamma \left (1+p,-\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 (1+p,-\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}}{c e^8}-\frac{2^{1+p} d^7 e^{-\frac{a}{2 b}} \left (d+e \sqrt{x}\right ) \Gamma \left (1+p,-\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}}\\ \end{align*}

Mathematica [F]  time = 0.473294, size = 0, normalized size = 0. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.571, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{2} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c e^{2} x + 2 \, c d e \sqrt{x} + c d^{2}\right ) + a\right )}^{p} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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