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

Optimal. Leaf size=242 \[ -\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}-\frac {64 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^4}+\frac {64 b d^4 n \sqrt {d+e x}}{315 e^4}+\frac {64 b d^3 n (d+e x)^{3/2}}{945 e^4}-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4} \]

[Out]

64/945*b*d^3*n*(e*x+d)^(3/2)/e^4-356/1575*b*d^2*n*(e*x+d)^(5/2)/e^4+80/441*b*d*n*(e*x+d)^(7/2)/e^4-4/81*b*n*(e
*x+d)^(9/2)/e^4-64/315*b*d^(9/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2))/e^4-2/3*d^3*(e*x+d)^(3/2)*(a+b*ln(c*x^n))/e^
4+6/5*d^2*(e*x+d)^(5/2)*(a+b*ln(c*x^n))/e^4-6/7*d*(e*x+d)^(7/2)*(a+b*ln(c*x^n))/e^4+2/9*(e*x+d)^(9/2)*(a+b*ln(
c*x^n))/e^4+64/315*b*d^4*n*(e*x+d)^(1/2)/e^4

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {43, 2350, 12, 1620, 50, 63, 208} \[ -\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}+\frac {64 b d^4 n \sqrt {d+e x}}{315 e^4}+\frac {64 b d^3 n (d+e x)^{3/2}}{945 e^4}-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}-\frac {64 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*b*d^4*n*Sqrt[d + e*x])/(315*e^4) + (64*b*d^3*n*(d + e*x)^(3/2))/(945*e^4) - (356*b*d^2*n*(d + e*x)^(5/2))/
(1575*e^4) + (80*b*d*n*(d + e*x)^(7/2))/(441*e^4) - (4*b*n*(d + e*x)^(9/2))/(81*e^4) - (64*b*d^(9/2)*n*ArcTanh
[Sqrt[d + e*x]/Sqrt[d]])/(315*e^4) - (2*d^3*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^4) + (6*d^2*(d + e*x)^(5/
2)*(a + b*Log[c*x^n]))/(5*e^4) - (6*d*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4) + (2*(d + e*x)^(9/2)*(a + b*
Log[c*x^n]))/(9*e^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps

\begin {align*} \int x^3 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}-(b n) \int \frac {2 (d+e x)^{3/2} \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )}{315 e^4 x} \, dx\\ &=-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}-\frac {(2 b n) \int \frac {(d+e x)^{3/2} \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )}{x} \, dx}{315 e^4}\\ &=-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}-\frac {(2 b n) \int \left (89 d^2 e (d+e x)^{3/2}-\frac {16 d^3 (d+e x)^{3/2}}{x}-100 d e (d+e x)^{5/2}+35 e (d+e x)^{7/2}\right ) \, dx}{315 e^4}\\ &=-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4}-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}+\frac {\left (32 b d^3 n\right ) \int \frac {(d+e x)^{3/2}}{x} \, dx}{315 e^4}\\ &=\frac {64 b d^3 n (d+e x)^{3/2}}{945 e^4}-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4}-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}+\frac {\left (32 b d^4 n\right ) \int \frac {\sqrt {d+e x}}{x} \, dx}{315 e^4}\\ &=\frac {64 b d^4 n \sqrt {d+e x}}{315 e^4}+\frac {64 b d^3 n (d+e x)^{3/2}}{945 e^4}-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4}-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}+\frac {\left (32 b d^5 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx}{315 e^4}\\ &=\frac {64 b d^4 n \sqrt {d+e x}}{315 e^4}+\frac {64 b d^3 n (d+e x)^{3/2}}{945 e^4}-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4}-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}+\frac {\left (64 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{315 e^5}\\ &=\frac {64 b d^4 n \sqrt {d+e x}}{315 e^4}+\frac {64 b d^3 n (d+e x)^{3/2}}{945 e^4}-\frac {356 b d^2 n (d+e x)^{5/2}}{1575 e^4}+\frac {80 b d n (d+e x)^{7/2}}{441 e^4}-\frac {4 b n (d+e x)^{9/2}}{81 e^4}-\frac {64 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^4}-\frac {2 d^3 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {6 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}-\frac {6 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.47, size = 183, normalized size = 0.76 \[ -\frac {2 \left (\sqrt {d+e x} \left (315 a \left (16 d^4-8 d^3 e x+6 d^2 e^2 x^2-5 d e^3 x^3-35 e^4 x^4\right )+315 b \left (16 d^4-8 d^3 e x+6 d^2 e^2 x^2-5 d e^3 x^3-35 e^4 x^4\right ) \log \left (c x^n\right )+2 b n \left (-4388 d^4+934 d^3 e x-543 d^2 e^2 x^2+400 d e^3 x^3+1225 e^4 x^4\right )\right )+10080 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{99225 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(10080*b*d^(9/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + Sqrt[d + e*x]*(315*a*(16*d^4 - 8*d^3*e*x + 6*d^2*e^2*x
^2 - 5*d*e^3*x^3 - 35*e^4*x^4) + 2*b*n*(-4388*d^4 + 934*d^3*e*x - 543*d^2*e^2*x^2 + 400*d*e^3*x^3 + 1225*e^4*x
^4) + 315*b*(16*d^4 - 8*d^3*e*x + 6*d^2*e^2*x^2 - 5*d*e^3*x^3 - 35*e^4*x^4)*Log[c*x^n])))/(99225*e^4)

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 495, normalized size = 2.05 \[ \left [\frac {2 \, {\left (5040 \, b d^{\frac {9}{2}} n \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (8776 \, b d^{4} n - 5040 \, a d^{4} - 1225 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} - 25 \, {\left (32 \, b d e^{3} n - 63 \, a d e^{3}\right )} x^{3} + 6 \, {\left (181 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 4 \, {\left (467 \, b d^{3} e n - 630 \, a d^{3} e\right )} x + 315 \, {\left (35 \, b e^{4} x^{4} + 5 \, b d e^{3} x^{3} - 6 \, b d^{2} e^{2} x^{2} + 8 \, b d^{3} e x - 16 \, b d^{4}\right )} \log \relax (c) + 315 \, {\left (35 \, b e^{4} n x^{4} + 5 \, b d e^{3} n x^{3} - 6 \, b d^{2} e^{2} n x^{2} + 8 \, b d^{3} e n x - 16 \, b d^{4} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{99225 \, e^{4}}, \frac {2 \, {\left (10080 \, b \sqrt {-d} d^{4} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (8776 \, b d^{4} n - 5040 \, a d^{4} - 1225 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} - 25 \, {\left (32 \, b d e^{3} n - 63 \, a d e^{3}\right )} x^{3} + 6 \, {\left (181 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 4 \, {\left (467 \, b d^{3} e n - 630 \, a d^{3} e\right )} x + 315 \, {\left (35 \, b e^{4} x^{4} + 5 \, b d e^{3} x^{3} - 6 \, b d^{2} e^{2} x^{2} + 8 \, b d^{3} e x - 16 \, b d^{4}\right )} \log \relax (c) + 315 \, {\left (35 \, b e^{4} n x^{4} + 5 \, b d e^{3} n x^{3} - 6 \, b d^{2} e^{2} n x^{2} + 8 \, b d^{3} e n x - 16 \, b d^{4} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{99225 \, e^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/99225*(5040*b*d^(9/2)*n*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + (8776*b*d^4*n - 5040*a*d^4 - 1225*(2
*b*e^4*n - 9*a*e^4)*x^4 - 25*(32*b*d*e^3*n - 63*a*d*e^3)*x^3 + 6*(181*b*d^2*e^2*n - 315*a*d^2*e^2)*x^2 - 4*(46
7*b*d^3*e*n - 630*a*d^3*e)*x + 315*(35*b*e^4*x^4 + 5*b*d*e^3*x^3 - 6*b*d^2*e^2*x^2 + 8*b*d^3*e*x - 16*b*d^4)*l
og(c) + 315*(35*b*e^4*n*x^4 + 5*b*d*e^3*n*x^3 - 6*b*d^2*e^2*n*x^2 + 8*b*d^3*e*n*x - 16*b*d^4*n)*log(x))*sqrt(e
*x + d))/e^4, 2/99225*(10080*b*sqrt(-d)*d^4*n*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (8776*b*d^4*n - 5040*a*d^4 -
1225*(2*b*e^4*n - 9*a*e^4)*x^4 - 25*(32*b*d*e^3*n - 63*a*d*e^3)*x^3 + 6*(181*b*d^2*e^2*n - 315*a*d^2*e^2)*x^2
- 4*(467*b*d^3*e*n - 630*a*d^3*e)*x + 315*(35*b*e^4*x^4 + 5*b*d*e^3*x^3 - 6*b*d^2*e^2*x^2 + 8*b*d^3*e*x - 16*b
*d^4)*log(c) + 315*(35*b*e^4*n*x^4 + 5*b*d*e^3*n*x^3 - 6*b*d^2*e^2*n*x^2 + 8*b*d^3*e*n*x - 16*b*d^4*n)*log(x))
*sqrt(e*x + d))/e^4]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.49, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e x +d}\, x^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 227, normalized size = 0.94 \[ \frac {4}{99225} \, {\left (\frac {2520 \, d^{\frac {9}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{4}} - \frac {1225 \, {\left (e x + d\right )}^{\frac {9}{2}} - 4500 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 5607 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 1680 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} - 5040 \, \sqrt {e x + d} d^{4}}{e^{4}}\right )} b n + \frac {2}{315} \, b {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}}}{e^{4}} - \frac {135 \, {\left (e x + d\right )}^{\frac {7}{2}} d}{e^{4}} + \frac {189 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2}}{e^{4}} - \frac {105 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3}}{e^{4}}\right )} \log \left (c x^{n}\right ) + \frac {2}{315} \, a {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}}}{e^{4}} - \frac {135 \, {\left (e x + d\right )}^{\frac {7}{2}} d}{e^{4}} + \frac {189 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2}}{e^{4}} - \frac {105 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3}}{e^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/99225*(2520*d^(9/2)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d)))/e^4 - (1225*(e*x + d)^(9/2) - 4
500*(e*x + d)^(7/2)*d + 5607*(e*x + d)^(5/2)*d^2 - 1680*(e*x + d)^(3/2)*d^3 - 5040*sqrt(e*x + d)*d^4)/e^4)*b*n
+ 2/315*b*(35*(e*x + d)^(9/2)/e^4 - 135*(e*x + d)^(7/2)*d/e^4 + 189*(e*x + d)^(5/2)*d^2/e^4 - 105*(e*x + d)^(
3/2)*d^3/e^4)*log(c*x^n) + 2/315*a*(35*(e*x + d)^(9/2)/e^4 - 135*(e*x + d)^(7/2)*d/e^4 + 189*(e*x + d)^(5/2)*d
^2/e^4 - 105*(e*x + d)^(3/2)*d^3/e^4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 16.49, size = 518, normalized size = 2.14 \[ \frac {2 \left (- \frac {a d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 a d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 a d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {a \left (d + e x\right )^{\frac {9}{2}}}{9} - b d^{3} \left (\frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (\frac {d^{2} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d e \sqrt {d + e x} + \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right ) + 3 b d^{2} \left (\frac {\left (d + e x\right )^{\frac {5}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{5} - \frac {2 n \left (\frac {d^{3} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d^{2} e \sqrt {d + e x} + \frac {d e \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {e \left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{5 e}\right ) - 3 b d \left (\frac {\left (d + e x\right )^{\frac {7}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{7} - \frac {2 n \left (\frac {d^{4} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d^{3} e \sqrt {d + e x} + \frac {d^{2} e \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {d e \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {e \left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{7 e}\right ) + b \left (\frac {\left (d + e x\right )^{\frac {9}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{9} - \frac {2 n \left (\frac {d^{5} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d^{4} e \sqrt {d + e x} + \frac {d^{3} e \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {d^{2} e \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {d e \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {e \left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{9 e}\right )\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-a*d**3*(d + e*x)**(3/2)/3 + 3*a*d**2*(d + e*x)**(5/2)/5 - 3*a*d*(d + e*x)**(7/2)/7 + a*(d + e*x)**(9/2)/9
- b*d**3*((d + e*x)**(3/2)*log(c*(-d/e + (d + e*x)/e)**n)/3 - 2*n*(d**2*e*atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d
) + d*e*sqrt(d + e*x) + e*(d + e*x)**(3/2)/3)/(3*e)) + 3*b*d**2*((d + e*x)**(5/2)*log(c*(-d/e + (d + e*x)/e)**
n)/5 - 2*n*(d**3*e*atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d) + d**2*e*sqrt(d + e*x) + d*e*(d + e*x)**(3/2)/3 + e*(
d + e*x)**(5/2)/5)/(5*e)) - 3*b*d*((d + e*x)**(7/2)*log(c*(-d/e + (d + e*x)/e)**n)/7 - 2*n*(d**4*e*atan(sqrt(d
+ e*x)/sqrt(-d))/sqrt(-d) + d**3*e*sqrt(d + e*x) + d**2*e*(d + e*x)**(3/2)/3 + d*e*(d + e*x)**(5/2)/5 + e*(d
+ e*x)**(7/2)/7)/(7*e)) + b*((d + e*x)**(9/2)*log(c*(-d/e + (d + e*x)/e)**n)/9 - 2*n*(d**5*e*atan(sqrt(d + e*x
)/sqrt(-d))/sqrt(-d) + d**4*e*sqrt(d + e*x) + d**3*e*(d + e*x)**(3/2)/3 + d**2*e*(d + e*x)**(5/2)/5 + d*e*(d +
e*x)**(7/2)/7 + e*(d + e*x)**(9/2)/9)/(9*e)))/e**4

________________________________________________________________________________________