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

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

[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)

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Rubi [A]  time = 0.219638, antiderivative size = 242, normalized size of antiderivative = 1., 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 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 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])

Rule 12

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

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 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]

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 = 0.406325, size = 183, normalized size = 0.76 \[ -\frac{2 \left (\sqrt{d+e x} \left (315 a \left (6 d^2 e^2 x^2-8 d^3 e x+16 d^4-5 d e^3 x^3-35 e^4 x^4\right )+315 b \left (6 d^2 e^2 x^2-8 d^3 e x+16 d^4-5 d e^3 x^3-35 e^4 x^4\right ) \log \left (c x^n\right )+2 b n \left (-543 d^2 e^2 x^2+934 d^3 e x-4388 d^4+400 d e^3 x^3+1225 e^4 x^4\right )\right )+10080 b d^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{99225 e^4} \]

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)

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Maple [F]  time = 0.606, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{ex+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.48266, size = 1220, normalized size = 5.04 \begin{align*} \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 \left (c\right ) + 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 \left (x\right )\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 \left (c\right ) + 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 \left (x\right )\right )} \sqrt{e x + d}\right )}}{99225 \, e^{4}}\right ] \end{align*}

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]

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Sympy [B]  time = 26.5006, size = 518, normalized size = 2.14 \begin{align*} \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}} \end{align*}

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

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

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)