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

Optimal. Leaf size=263 \[ -\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{64 b d^5 n \sqrt{d+e x}}{1155 e^4}+\frac{64 b d^4 n (d+e x)^{3/2}}{3465 e^4}+\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}-\frac{64 b d^{11/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{1155 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4} \]

[Out]

(64*b*d^5*n*Sqrt[d + e*x])/(1155*e^4) + (64*b*d^4*n*(d + e*x)^(3/2))/(3465*e^4) + (64*b*d^3*n*(d + e*x)^(5/2))
/(5775*e^4) - (172*b*d^2*n*(d + e*x)^(7/2))/(1617*e^4) + (32*b*d*n*(d + e*x)^(9/2))/(297*e^4) - (4*b*n*(d + e*
x)^(11/2))/(121*e^4) - (64*b*d^(11/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(1155*e^4) - (2*d^3*(d + e*x)^(5/2)*(a
+ b*Log[c*x^n]))/(5*e^4) + (6*d^2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4) - (2*d*(d + e*x)^(9/2)*(a + b*L
og[c*x^n]))/(3*e^4) + (2*(d + e*x)^(11/2)*(a + b*Log[c*x^n]))/(11*e^4)

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Rubi [A]  time = 0.240274, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{64 b d^5 n \sqrt{d+e x}}{1155 e^4}+\frac{64 b d^4 n (d+e x)^{3/2}}{3465 e^4}+\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}-\frac{64 b d^{11/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{1155 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*b*d^5*n*Sqrt[d + e*x])/(1155*e^4) + (64*b*d^4*n*(d + e*x)^(3/2))/(3465*e^4) + (64*b*d^3*n*(d + e*x)^(5/2))
/(5775*e^4) - (172*b*d^2*n*(d + e*x)^(7/2))/(1617*e^4) + (32*b*d*n*(d + e*x)^(9/2))/(297*e^4) - (4*b*n*(d + e*
x)^(11/2))/(121*e^4) - (64*b*d^(11/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(1155*e^4) - (2*d^3*(d + e*x)^(5/2)*(a
+ b*Log[c*x^n]))/(5*e^4) + (6*d^2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4) - (2*d*(d + e*x)^(9/2)*(a + b*L
og[c*x^n]))/(3*e^4) + (2*(d + e*x)^(11/2)*(a + b*Log[c*x^n]))/(11*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 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}-(b n) \int \frac{2 (d+e x)^{5/2} \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )}{1155 e^4 x} \, dx\\ &=-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}-\frac{(2 b n) \int \frac{(d+e x)^{5/2} \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )}{x} \, dx}{1155 e^4}\\ &=-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}-\frac{(2 b n) \int \left (215 d^2 e (d+e x)^{5/2}-\frac{16 d^3 (d+e x)^{5/2}}{x}-280 d e (d+e x)^{7/2}+105 e (d+e x)^{9/2}\right ) \, dx}{1155 e^4}\\ &=-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4}-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{\left (32 b d^3 n\right ) \int \frac{(d+e x)^{5/2}}{x} \, dx}{1155 e^4}\\ &=\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4}-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{\left (32 b d^4 n\right ) \int \frac{(d+e x)^{3/2}}{x} \, dx}{1155 e^4}\\ &=\frac{64 b d^4 n (d+e x)^{3/2}}{3465 e^4}+\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4}-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{\left (32 b d^5 n\right ) \int \frac{\sqrt{d+e x}}{x} \, dx}{1155 e^4}\\ &=\frac{64 b d^5 n \sqrt{d+e x}}{1155 e^4}+\frac{64 b d^4 n (d+e x)^{3/2}}{3465 e^4}+\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4}-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{\left (32 b d^6 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx}{1155 e^4}\\ &=\frac{64 b d^5 n \sqrt{d+e x}}{1155 e^4}+\frac{64 b d^4 n (d+e x)^{3/2}}{3465 e^4}+\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4}-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}+\frac{\left (64 b d^6 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{1155 e^5}\\ &=\frac{64 b d^5 n \sqrt{d+e x}}{1155 e^4}+\frac{64 b d^4 n (d+e x)^{3/2}}{3465 e^4}+\frac{64 b d^3 n (d+e x)^{5/2}}{5775 e^4}-\frac{172 b d^2 n (d+e x)^{7/2}}{1617 e^4}+\frac{32 b d n (d+e x)^{9/2}}{297 e^4}-\frac{4 b n (d+e x)^{11/2}}{121 e^4}-\frac{64 b d^{11/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{1155 e^4}-\frac{2 d^3 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{6 d^2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{2 d (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{2 (d+e x)^{11/2} \left (a+b \log \left (c x^n\right )\right )}{11 e^4}\\ \end{align*}

Mathematica [A]  time = 0.328016, size = 187, normalized size = 0.71 \[ \frac{2 \sqrt{d+e x} \left (-3465 a \left (-40 d^2 e x+16 d^3+70 d e^2 x^2-105 e^3 x^3\right ) (d+e x)^2-3465 b \left (-40 d^2 e x+16 d^3+70 d e^2 x^2-105 e^3 x^3\right ) (d+e x)^2 \log \left (c x^n\right )+2 b n \left (7863 d^3 e^2 x^2-5975 d^2 e^3 x^3-12794 d^4 e x+53308 d^5-57575 d e^4 x^4-33075 e^5 x^5\right )\right )-221760 b d^{11/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4002075 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-221760*b*d^(11/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 2*Sqrt[d + e*x]*(-3465*a*(d + e*x)^2*(16*d^3 - 40*d^2*e
*x + 70*d*e^2*x^2 - 105*e^3*x^3) + 2*b*n*(53308*d^5 - 12794*d^4*e*x + 7863*d^3*e^2*x^2 - 5975*d^2*e^3*x^3 - 57
575*d*e^4*x^4 - 33075*e^5*x^5) - 3465*b*(d + e*x)^2*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3)*Log[c*x
^n]))/(4002075*e^4)

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Maple [F]  time = 0.572, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50707, size = 1503, normalized size = 5.71 \begin{align*} \left [\frac{2 \,{\left (55440 \, b d^{\frac{11}{2}} n \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) +{\left (106616 \, b d^{5} n - 55440 \, a d^{5} - 33075 \,{\left (2 \, b e^{5} n - 11 \, a e^{5}\right )} x^{5} - 2450 \,{\left (47 \, b d e^{4} n - 198 \, a d e^{4}\right )} x^{4} - 25 \,{\left (478 \, b d^{2} e^{3} n - 693 \, a d^{2} e^{3}\right )} x^{3} + 6 \,{\left (2621 \, b d^{3} e^{2} n - 3465 \, a d^{3} e^{2}\right )} x^{2} - 4 \,{\left (6397 \, b d^{4} e n - 6930 \, a d^{4} e\right )} x + 3465 \,{\left (105 \, b e^{5} x^{5} + 140 \, b d e^{4} x^{4} + 5 \, b d^{2} e^{3} x^{3} - 6 \, b d^{3} e^{2} x^{2} + 8 \, b d^{4} e x - 16 \, b d^{5}\right )} \log \left (c\right ) + 3465 \,{\left (105 \, b e^{5} n x^{5} + 140 \, b d e^{4} n x^{4} + 5 \, b d^{2} e^{3} n x^{3} - 6 \, b d^{3} e^{2} n x^{2} + 8 \, b d^{4} e n x - 16 \, b d^{5} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{4002075 \, e^{4}}, \frac{2 \,{\left (110880 \, b \sqrt{-d} d^{5} n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (106616 \, b d^{5} n - 55440 \, a d^{5} - 33075 \,{\left (2 \, b e^{5} n - 11 \, a e^{5}\right )} x^{5} - 2450 \,{\left (47 \, b d e^{4} n - 198 \, a d e^{4}\right )} x^{4} - 25 \,{\left (478 \, b d^{2} e^{3} n - 693 \, a d^{2} e^{3}\right )} x^{3} + 6 \,{\left (2621 \, b d^{3} e^{2} n - 3465 \, a d^{3} e^{2}\right )} x^{2} - 4 \,{\left (6397 \, b d^{4} e n - 6930 \, a d^{4} e\right )} x + 3465 \,{\left (105 \, b e^{5} x^{5} + 140 \, b d e^{4} x^{4} + 5 \, b d^{2} e^{3} x^{3} - 6 \, b d^{3} e^{2} x^{2} + 8 \, b d^{4} e x - 16 \, b d^{5}\right )} \log \left (c\right ) + 3465 \,{\left (105 \, b e^{5} n x^{5} + 140 \, b d e^{4} n x^{4} + 5 \, b d^{2} e^{3} n x^{3} - 6 \, b d^{3} e^{2} n x^{2} + 8 \, b d^{4} e n x - 16 \, b d^{5} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{4002075 \, e^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/4002075*(55440*b*d^(11/2)*n*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + (106616*b*d^5*n - 55440*a*d^5 -
33075*(2*b*e^5*n - 11*a*e^5)*x^5 - 2450*(47*b*d*e^4*n - 198*a*d*e^4)*x^4 - 25*(478*b*d^2*e^3*n - 693*a*d^2*e^3
)*x^3 + 6*(2621*b*d^3*e^2*n - 3465*a*d^3*e^2)*x^2 - 4*(6397*b*d^4*e*n - 6930*a*d^4*e)*x + 3465*(105*b*e^5*x^5
+ 140*b*d*e^4*x^4 + 5*b*d^2*e^3*x^3 - 6*b*d^3*e^2*x^2 + 8*b*d^4*e*x - 16*b*d^5)*log(c) + 3465*(105*b*e^5*n*x^5
+ 140*b*d*e^4*n*x^4 + 5*b*d^2*e^3*n*x^3 - 6*b*d^3*e^2*n*x^2 + 8*b*d^4*e*n*x - 16*b*d^5*n)*log(x))*sqrt(e*x +
d))/e^4, 2/4002075*(110880*b*sqrt(-d)*d^5*n*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (106616*b*d^5*n - 55440*a*d^5 -
33075*(2*b*e^5*n - 11*a*e^5)*x^5 - 2450*(47*b*d*e^4*n - 198*a*d*e^4)*x^4 - 25*(478*b*d^2*e^3*n - 693*a*d^2*e^
3)*x^3 + 6*(2621*b*d^3*e^2*n - 3465*a*d^3*e^2)*x^2 - 4*(6397*b*d^4*e*n - 6930*a*d^4*e)*x + 3465*(105*b*e^5*x^5
+ 140*b*d*e^4*x^4 + 5*b*d^2*e^3*x^3 - 6*b*d^3*e^2*x^2 + 8*b*d^4*e*x - 16*b*d^5)*log(c) + 3465*(105*b*e^5*n*x^
5 + 140*b*d*e^4*n*x^4 + 5*b*d^2*e^3*n*x^3 - 6*b*d^3*e^2*n*x^2 + 8*b*d^4*e*n*x - 16*b*d^5*n)*log(x))*sqrt(e*x +
d))/e^4]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{3}{2}}{\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*(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^3, x)