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

Optimal. Leaf size=730 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.3364, antiderivative size = 730, normalized size of antiderivative = 1., 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 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 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 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 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]

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*}

Mathematica [A]  time = 0.986439, size = 435, normalized size = 0.6 \[ \frac{2^{-3 p-2} 105^{-p} e^{-\frac{8 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \left (c^7 d^7 \left (-8^{p+1}\right ) 105^p e^{\frac{7 a}{b}} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )+c^6 d^6 15^p 28^{p+1} e^{\frac{6 a}{b}} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )-c^5 d^5 5^p 56^{p+1} e^{\frac{5 a}{b}} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )+c^4 d^4 3^p 70^{p+1} e^{\frac{4 a}{b}} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )-c^3 d^3 3^p 56^{p+1} e^{\frac{3 a}{b}} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )+c^2 d^2 5^p 28^{p+1} e^{\frac{2 a}{b}} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )-c d 8^{p+1} 15^p e^{a/b} \text{Gamma}\left (p+1,-\frac{7 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )+105^p \text{Gamma}\left (p+1,-\frac{8 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )\right )}{c^8 e^8} \]

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)

________________________________________________________________________________________

Maple [F]  time = 0.489, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) \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))))^p,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )} 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))))^p,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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))))**p,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )} 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))))^p,x, algorithm="giac")

[Out]

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