\(\int x^2 (d+e x)^{3/2} (a+b \log (c x^n)) \, dx\) [138]

Optimal result

Integrand size = 23, antiderivative size = 213 \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3}+\frac {32 b d^{9/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^3}+\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3} \]

[Out]

-32/945*b*d^3*n*(e*x+d)^(3/2)/e^3-32/1575*b*d^2*n*(e*x+d)^(5/2)/e^3+44/441*b*d*n*(e*x+d)^(7/2)/e^3-4/81*b*n*(e
*x+d)^(9/2)/e^3+32/315*b*d^(9/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2))/e^3+2/5*d^2*(e*x+d)^(5/2)*(a+b*ln(c*x^n))/e^
3-4/7*d*(e*x+d)^(7/2)*(a+b*ln(c*x^n))/e^3+2/9*(e*x+d)^(9/2)*(a+b*ln(c*x^n))/e^3-32/315*b*d^4*n*(e*x+d)^(1/2)/e
^3

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {45, 2392, 12, 911, 1275, 214} \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {32 b d^{9/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^3}-\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3} \]

[In]

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

[Out]

(-32*b*d^4*n*Sqrt[d + e*x])/(315*e^3) - (32*b*d^3*n*(d + e*x)^(3/2))/(945*e^3) - (32*b*d^2*n*(d + e*x)^(5/2))/
(1575*e^3) + (44*b*d*n*(d + e*x)^(7/2))/(441*e^3) - (4*b*n*(d + e*x)^(9/2))/(81*e^3) + (32*b*d^(9/2)*n*ArcTanh
[Sqrt[d + e*x]/Sqrt[d]])/(315*e^3) + (2*d^2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3) - (4*d*(d + e*x)^(7/2)
*(a + b*Log[c*x^n]))/(7*e^3) + (2*(d + e*x)^(9/2)*(a + b*Log[c*x^n]))/(9*e^3)

Rule 12

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

Rule 45

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 214

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 2392

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*} \text {integral}& = \frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-(b n) \int \frac {2 (d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{315 e^3 x} \, dx \\ & = \frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(2 b n) \int \frac {(d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{x} \, dx}{315 e^3} \\ & = \frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(4 b n) \text {Subst}\left (\int \frac {x^6 \left (63 d^2-90 d x^2+35 x^4\right )}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{315 e^4} \\ & = \frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(4 b n) \text {Subst}\left (\int \left (8 d^4 e+8 d^3 e x^2+8 d^2 e x^4-55 d e x^6+35 e x^8+\frac {8 d^5}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x}\right )}{315 e^4} \\ & = -\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3}+\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {\left (32 b d^5 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{315 e^4} \\ & = -\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3}+\frac {32 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^3}+\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.72 \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \left (5040 b d^{9/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\sqrt {d+e x} \left (315 a (d+e x)^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )-2 b n \left (2614 d^4-677 d^3 e x+429 d^2 e^2 x^2+2425 d e^3 x^3+1225 e^4 x^4\right )+315 b (d+e x)^2 \left (8 d^2-20 d e x+35 e^2 x^2\right ) \log \left (c x^n\right )\right )\right )}{99225 e^3} \]

[In]

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

[Out]

(2*(5040*b*d^(9/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + Sqrt[d + e*x]*(315*a*(d + e*x)^2*(8*d^2 - 20*d*e*x + 35*
e^2*x^2) - 2*b*n*(2614*d^4 - 677*d^3*e*x + 429*d^2*e^2*x^2 + 2425*d*e^3*x^3 + 1225*e^4*x^4) + 315*b*(d + e*x)^
2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)*Log[c*x^n])))/(99225*e^3)

Maple [F]

\[\int x^{2} \left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.33 \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\left [\frac {2 \, {\left (2520 \, b d^{\frac {9}{2}} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (5228 \, b d^{4} n - 2520 \, a d^{4} + 1225 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} + 50 \, {\left (97 \, b d e^{3} n - 315 \, a d e^{3}\right )} x^{3} + 3 \, {\left (286 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 2 \, {\left (677 \, b d^{3} e n - 630 \, a d^{3} e\right )} x - 315 \, {\left (35 \, b e^{4} x^{4} + 50 \, b d e^{3} x^{3} + 3 \, b d^{2} e^{2} x^{2} - 4 \, b d^{3} e x + 8 \, b d^{4}\right )} \log \left (c\right ) - 315 \, {\left (35 \, b e^{4} n x^{4} + 50 \, b d e^{3} n x^{3} + 3 \, b d^{2} e^{2} n x^{2} - 4 \, b d^{3} e n x + 8 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{99225 \, e^{3}}, -\frac {2 \, {\left (5040 \, b \sqrt {-d} d^{4} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (5228 \, b d^{4} n - 2520 \, a d^{4} + 1225 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} + 50 \, {\left (97 \, b d e^{3} n - 315 \, a d e^{3}\right )} x^{3} + 3 \, {\left (286 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 2 \, {\left (677 \, b d^{3} e n - 630 \, a d^{3} e\right )} x - 315 \, {\left (35 \, b e^{4} x^{4} + 50 \, b d e^{3} x^{3} + 3 \, b d^{2} e^{2} x^{2} - 4 \, b d^{3} e x + 8 \, b d^{4}\right )} \log \left (c\right ) - 315 \, {\left (35 \, b e^{4} n x^{4} + 50 \, b d e^{3} n x^{3} + 3 \, b d^{2} e^{2} n x^{2} - 4 \, b d^{3} e n x + 8 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{99225 \, e^{3}}\right ] \]

[In]

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

[Out]

[2/99225*(2520*b*d^(9/2)*n*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - (5228*b*d^4*n - 2520*a*d^4 + 1225*(2
*b*e^4*n - 9*a*e^4)*x^4 + 50*(97*b*d*e^3*n - 315*a*d*e^3)*x^3 + 3*(286*b*d^2*e^2*n - 315*a*d^2*e^2)*x^2 - 2*(6
77*b*d^3*e*n - 630*a*d^3*e)*x - 315*(35*b*e^4*x^4 + 50*b*d*e^3*x^3 + 3*b*d^2*e^2*x^2 - 4*b*d^3*e*x + 8*b*d^4)*
log(c) - 315*(35*b*e^4*n*x^4 + 50*b*d*e^3*n*x^3 + 3*b*d^2*e^2*n*x^2 - 4*b*d^3*e*n*x + 8*b*d^4*n)*log(x))*sqrt(
e*x + d))/e^3, -2/99225*(5040*b*sqrt(-d)*d^4*n*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (5228*b*d^4*n - 2520*a*d^4 +
1225*(2*b*e^4*n - 9*a*e^4)*x^4 + 50*(97*b*d*e^3*n - 315*a*d*e^3)*x^3 + 3*(286*b*d^2*e^2*n - 315*a*d^2*e^2)*x^
2 - 2*(677*b*d^3*e*n - 630*a*d^3*e)*x - 315*(35*b*e^4*x^4 + 50*b*d*e^3*x^3 + 3*b*d^2*e^2*x^2 - 4*b*d^3*e*x + 8
*b*d^4)*log(c) - 315*(35*b*e^4*n*x^4 + 50*b*d*e^3*n*x^3 + 3*b*d^2*e^2*n*x^2 - 4*b*d^3*e*n*x + 8*b*d^4*n)*log(x
))*sqrt(e*x + d))/e^3]

Sympy [A] (verification not implemented)

Time = 129.07 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.02 \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=a d \left (\begin {cases} \frac {2 d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} - \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{3}} + \frac {2 \left (d + e x\right )^{\frac {7}{2}}}{7 e^{3}} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} x^{3}}{3} & \text {otherwise} \end {cases}\right ) + a e \left (\begin {cases} - \frac {2 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3 e^{4}} + \frac {6 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} - \frac {6 d \left (d + e x\right )^{\frac {7}{2}}}{7 e^{4}} + \frac {2 \left (d + e x\right )^{\frac {9}{2}}}{9 e^{4}} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} x^{4}}{4} & \text {otherwise} \end {cases}\right ) - b d n \left (\begin {cases} \frac {3112 d^{\frac {7}{2}} \sqrt {1 + \frac {e x}{d}}}{11025 e^{3}} + \frac {16 d^{\frac {7}{2}} \log {\left (\frac {e x}{d} \right )}}{105 e^{3}} - \frac {32 d^{\frac {7}{2}} \log {\left (\sqrt {1 + \frac {e x}{d}} + 1 \right )}}{105 e^{3}} - \frac {716 d^{\frac {5}{2}} x \sqrt {1 + \frac {e x}{d}}}{11025 e^{2}} + \frac {48 d^{\frac {3}{2}} x^{2} \sqrt {1 + \frac {e x}{d}}}{1225 e} + \frac {4 \sqrt {d} x^{3} \sqrt {1 + \frac {e x}{d}}}{49} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {\sqrt {d} x^{3}}{9} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} \frac {2 d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} - \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{3}} + \frac {2 \left (d + e x\right )^{\frac {7}{2}}}{7 e^{3}} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e n \left (\begin {cases} - \frac {17552 d^{\frac {9}{2}} \sqrt {1 + \frac {e x}{d}}}{99225 e^{4}} - \frac {32 d^{\frac {9}{2}} \log {\left (\frac {e x}{d} \right )}}{315 e^{4}} + \frac {64 d^{\frac {9}{2}} \log {\left (\sqrt {1 + \frac {e x}{d}} + 1 \right )}}{315 e^{4}} + \frac {3736 d^{\frac {7}{2}} x \sqrt {1 + \frac {e x}{d}}}{99225 e^{3}} - \frac {724 d^{\frac {5}{2}} x^{2} \sqrt {1 + \frac {e x}{d}}}{33075 e^{2}} + \frac {64 d^{\frac {3}{2}} x^{3} \sqrt {1 + \frac {e x}{d}}}{3969 e} + \frac {4 \sqrt {d} x^{4} \sqrt {1 + \frac {e x}{d}}}{81} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {\sqrt {d} x^{4}}{16} & \text {otherwise} \end {cases}\right ) + b e \left (\begin {cases} - \frac {2 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3 e^{4}} + \frac {6 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} - \frac {6 d \left (d + e x\right )^{\frac {7}{2}}}{7 e^{4}} + \frac {2 \left (d + e x\right )^{\frac {9}{2}}}{9 e^{4}} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]

[In]

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

[Out]

a*d*Piecewise((2*d**2*(d + e*x)**(3/2)/(3*e**3) - 4*d*(d + e*x)**(5/2)/(5*e**3) + 2*(d + e*x)**(7/2)/(7*e**3),
Ne(e, 0)), (sqrt(d)*x**3/3, True)) + a*e*Piecewise((-2*d**3*(d + e*x)**(3/2)/(3*e**4) + 6*d**2*(d + e*x)**(5/
2)/(5*e**4) - 6*d*(d + e*x)**(7/2)/(7*e**4) + 2*(d + e*x)**(9/2)/(9*e**4), Ne(e, 0)), (sqrt(d)*x**4/4, True))
- b*d*n*Piecewise((3112*d**(7/2)*sqrt(1 + e*x/d)/(11025*e**3) + 16*d**(7/2)*log(e*x/d)/(105*e**3) - 32*d**(7/2
)*log(sqrt(1 + e*x/d) + 1)/(105*e**3) - 716*d**(5/2)*x*sqrt(1 + e*x/d)/(11025*e**2) + 48*d**(3/2)*x**2*sqrt(1
+ e*x/d)/(1225*e) + 4*sqrt(d)*x**3*sqrt(1 + e*x/d)/49, (e > -oo) & (e < oo) & Ne(e, 0)), (sqrt(d)*x**3/9, True
)) + b*d*Piecewise((2*d**2*(d + e*x)**(3/2)/(3*e**3) - 4*d*(d + e*x)**(5/2)/(5*e**3) + 2*(d + e*x)**(7/2)/(7*e
**3), Ne(e, 0)), (sqrt(d)*x**3/3, True))*log(c*x**n) - b*e*n*Piecewise((-17552*d**(9/2)*sqrt(1 + e*x/d)/(99225
*e**4) - 32*d**(9/2)*log(e*x/d)/(315*e**4) + 64*d**(9/2)*log(sqrt(1 + e*x/d) + 1)/(315*e**4) + 3736*d**(7/2)*x
*sqrt(1 + e*x/d)/(99225*e**3) - 724*d**(5/2)*x**2*sqrt(1 + e*x/d)/(33075*e**2) + 64*d**(3/2)*x**3*sqrt(1 + e*x
/d)/(3969*e) + 4*sqrt(d)*x**4*sqrt(1 + e*x/d)/81, (e > -oo) & (e < oo) & Ne(e, 0)), (sqrt(d)*x**4/16, True)) +
b*e*Piecewise((-2*d**3*(d + e*x)**(3/2)/(3*e**4) + 6*d**2*(d + e*x)**(5/2)/(5*e**4) - 6*d*(d + e*x)**(7/2)/(7
*e**4) + 2*(d + e*x)**(9/2)/(9*e**4), Ne(e, 0)), (sqrt(d)*x**4/4, True))*log(c*x**n)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.92 \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4}{99225} \, {\left (\frac {1260 \, d^{\frac {9}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{3}} + \frac {1225 \, {\left (e x + d\right )}^{\frac {9}{2}} - 2475 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 504 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} + 840 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 2520 \, \sqrt {e x + d} d^{4}}{e^{3}}\right )} b n + \frac {2}{315} \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}}}{e^{3}} - \frac {90 \, {\left (e x + d\right )}^{\frac {7}{2}} d}{e^{3}} + \frac {63 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2}}{e^{3}}\right )} b \log \left (c x^{n}\right ) + \frac {2}{315} \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}}}{e^{3}} - \frac {90 \, {\left (e x + d\right )}^{\frac {7}{2}} d}{e^{3}} + \frac {63 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2}}{e^{3}}\right )} a \]

[In]

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

[Out]

-4/99225*(1260*d^(9/2)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d)))/e^3 + (1225*(e*x + d)^(9/2) -
2475*(e*x + d)^(7/2)*d + 504*(e*x + d)^(5/2)*d^2 + 840*(e*x + d)^(3/2)*d^3 + 2520*sqrt(e*x + d)*d^4)/e^3)*b*n
+ 2/315*(35*(e*x + d)^(9/2)/e^3 - 90*(e*x + d)^(7/2)*d/e^3 + 63*(e*x + d)^(5/2)*d^2/e^3)*b*log(c*x^n) + 2/315*
(35*(e*x + d)^(9/2)/e^3 - 90*(e*x + d)^(7/2)*d/e^3 + 63*(e*x + d)^(5/2)*d^2/e^3)*a

Giac [F]

\[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]

[In]

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

[Out]

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