3.2.37 \(\int x^3 (d+e x)^{3/2} (a+b \log (c x^n)) \, dx\) [137]

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

[Out]

64/3465*b*d^4*n*(e*x+d)^(3/2)/e^4+64/5775*b*d^3*n*(e*x+d)^(5/2)/e^4-172/1617*b*d^2*n*(e*x+d)^(7/2)/e^4+32/297*
b*d*n*(e*x+d)^(9/2)/e^4-4/121*b*n*(e*x+d)^(11/2)/e^4-64/1155*b*d^(11/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2))/e^4-2
/5*d^3*(e*x+d)^(5/2)*(a+b*ln(c*x^n))/e^4+6/7*d^2*(e*x+d)^(7/2)*(a+b*ln(c*x^n))/e^4-2/3*d*(e*x+d)^(9/2)*(a+b*ln
(c*x^n))/e^4+2/11*(e*x+d)^(11/2)*(a+b*ln(c*x^n))/e^4+64/1155*b*d^5*n*(e*x+d)^(1/2)/e^4

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 263, normalized size of antiderivative = 1.00, 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 = {45, 2392, 12, 1634, 52, 65, 214} \begin {gather*} -\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^{11/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{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} \end {gather*}

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 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 52

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 65

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 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 1634

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 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*} \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 ) \text {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 = 0.23, size = 187, normalized size = 0.71 \begin {gather*} \frac {-221760 b d^{11/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 \sqrt {d+e x} \left (-3465 a (d+e x)^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )+2 b n \left (53308 d^5-12794 d^4 e x+7863 d^3 e^2 x^2-5975 d^2 e^3 x^3-57575 d e^4 x^4-33075 e^5 x^5\right )-3465 b (d+e x)^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right ) \log \left (c x^n\right )\right )}{4002075 e^4} \end {gather*}

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)

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]

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)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 245, normalized size = 0.93 \begin {gather*} \frac {4}{4002075} \, {\left (27720 \, d^{\frac {11}{2}} e^{\left (-4\right )} \log \left (\frac {\sqrt {x e + d} - \sqrt {d}}{\sqrt {x e + d} + \sqrt {d}}\right ) - {\left (33075 \, {\left (x e + d\right )}^{\frac {11}{2}} - 107800 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 106425 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 11088 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} - 18480 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 55440 \, \sqrt {x e + d} d^{5}\right )} e^{\left (-4\right )}\right )} b n + \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} e^{\left (-4\right )} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d e^{\left (-4\right )} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} e^{\left (-4\right )} - 231 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} e^{\left (-4\right )}\right )} b \log \left (c x^{n}\right ) + \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} e^{\left (-4\right )} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d e^{\left (-4\right )} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} e^{\left (-4\right )} - 231 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} e^{\left (-4\right )}\right )} a \end {gather*}

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]

4/4002075*(27720*d^(11/2)*e^(-4)*log((sqrt(x*e + d) - sqrt(d))/(sqrt(x*e + d) + sqrt(d))) - (33075*(x*e + d)^(
11/2) - 107800*(x*e + d)^(9/2)*d + 106425*(x*e + d)^(7/2)*d^2 - 11088*(x*e + d)^(5/2)*d^3 - 18480*(x*e + d)^(3
/2)*d^4 - 55440*sqrt(x*e + d)*d^5)*e^(-4))*b*n + 2/1155*(105*(x*e + d)^(11/2)*e^(-4) - 385*(x*e + d)^(9/2)*d*e
^(-4) + 495*(x*e + d)^(7/2)*d^2*e^(-4) - 231*(x*e + d)^(5/2)*d^3*e^(-4))*b*log(c*x^n) + 2/1155*(105*(x*e + d)^
(11/2)*e^(-4) - 385*(x*e + d)^(9/2)*d*e^(-4) + 495*(x*e + d)^(7/2)*d^2*e^(-4) - 231*(x*e + d)^(5/2)*d^3*e^(-4)
)*a

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 554, normalized size = 2.11 \begin {gather*} \left [\frac {2}{4002075} \, {\left (55440 \, b d^{\frac {11}{2}} n \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (106616 \, b d^{5} n - 33075 \, {\left (2 \, b n - 11 \, a\right )} x^{5} e^{5} - 55440 \, a d^{5} - 2450 \, {\left (47 \, b d n - 198 \, a d\right )} x^{4} e^{4} - 25 \, {\left (478 \, b d^{2} n - 693 \, a d^{2}\right )} x^{3} e^{3} + 6 \, {\left (2621 \, b d^{3} n - 3465 \, a d^{3}\right )} x^{2} e^{2} - 4 \, {\left (6397 \, b d^{4} n - 6930 \, a d^{4}\right )} x e + 3465 \, {\left (105 \, b x^{5} e^{5} + 140 \, b d x^{4} e^{4} + 5 \, b d^{2} x^{3} e^{3} - 6 \, b d^{3} x^{2} e^{2} + 8 \, b d^{4} x e - 16 \, b d^{5}\right )} \log \left (c\right ) + 3465 \, {\left (105 \, b n x^{5} e^{5} + 140 \, b d n x^{4} e^{4} + 5 \, b d^{2} n x^{3} e^{3} - 6 \, b d^{3} n x^{2} e^{2} + 8 \, b d^{4} n x e - 16 \, b d^{5} n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )}, \frac {2}{4002075} \, {\left (110880 \, b \sqrt {-d} d^{5} n \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (106616 \, b d^{5} n - 33075 \, {\left (2 \, b n - 11 \, a\right )} x^{5} e^{5} - 55440 \, a d^{5} - 2450 \, {\left (47 \, b d n - 198 \, a d\right )} x^{4} e^{4} - 25 \, {\left (478 \, b d^{2} n - 693 \, a d^{2}\right )} x^{3} e^{3} + 6 \, {\left (2621 \, b d^{3} n - 3465 \, a d^{3}\right )} x^{2} e^{2} - 4 \, {\left (6397 \, b d^{4} n - 6930 \, a d^{4}\right )} x e + 3465 \, {\left (105 \, b x^{5} e^{5} + 140 \, b d x^{4} e^{4} + 5 \, b d^{2} x^{3} e^{3} - 6 \, b d^{3} x^{2} e^{2} + 8 \, b d^{4} x e - 16 \, b d^{5}\right )} \log \left (c\right ) + 3465 \, {\left (105 \, b n x^{5} e^{5} + 140 \, b d n x^{4} e^{4} + 5 \, b d^{2} n x^{3} e^{3} - 6 \, b d^{3} n x^{2} e^{2} + 8 \, b d^{4} n x e - 16 \, b d^{5} n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )}\right ] \end {gather*}

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((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + (106616*b*d^5*n - 33075*(2*b*n -
11*a)*x^5*e^5 - 55440*a*d^5 - 2450*(47*b*d*n - 198*a*d)*x^4*e^4 - 25*(478*b*d^2*n - 693*a*d^2)*x^3*e^3 + 6*(2
621*b*d^3*n - 3465*a*d^3)*x^2*e^2 - 4*(6397*b*d^4*n - 6930*a*d^4)*x*e + 3465*(105*b*x^5*e^5 + 140*b*d*x^4*e^4
+ 5*b*d^2*x^3*e^3 - 6*b*d^3*x^2*e^2 + 8*b*d^4*x*e - 16*b*d^5)*log(c) + 3465*(105*b*n*x^5*e^5 + 140*b*d*n*x^4*e
^4 + 5*b*d^2*n*x^3*e^3 - 6*b*d^3*n*x^2*e^2 + 8*b*d^4*n*x*e - 16*b*d^5*n)*log(x))*sqrt(x*e + d))*e^(-4), 2/4002
075*(110880*b*sqrt(-d)*d^5*n*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (106616*b*d^5*n - 33075*(2*b*n - 11*a)*x^5*e^5
- 55440*a*d^5 - 2450*(47*b*d*n - 198*a*d)*x^4*e^4 - 25*(478*b*d^2*n - 693*a*d^2)*x^3*e^3 + 6*(2621*b*d^3*n -
3465*a*d^3)*x^2*e^2 - 4*(6397*b*d^4*n - 6930*a*d^4)*x*e + 3465*(105*b*x^5*e^5 + 140*b*d*x^4*e^4 + 5*b*d^2*x^3*
e^3 - 6*b*d^3*x^2*e^2 + 8*b*d^4*x*e - 16*b*d^5)*log(c) + 3465*(105*b*n*x^5*e^5 + 140*b*d*n*x^4*e^4 + 5*b*d^2*n
*x^3*e^3 - 6*b*d^3*n*x^2*e^2 + 8*b*d^4*n*x*e - 16*b*d^5*n)*log(x))*sqrt(x*e + d))*e^(-4)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (267) = 534\).
time = 73.96, size = 1188, normalized size = 4.52 \begin {gather*} \frac {2 a d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {2 a \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {2 b d \left (- 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 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 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 ) + \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 )}{e^{4}} + \frac {2 b \left (d^{4} \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 ) - 4 d^{3} \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 ) + 6 d^{2} \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 ) - 4 d \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 ) + \frac {\left (d + e x\right )^{\frac {11}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{11} - \frac {2 n \left (\frac {d^{6} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d^{5} e \sqrt {d + e x} + \frac {d^{4} e \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {d^{3} e \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {d^{2} e \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {d e \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {e \left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{11 e}\right )}{e^{4}} \end {gather*}

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]

2*a*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**
4 + 2*a*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2
)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*b*d*(-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*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*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)) + (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 + 2*b*(d**4*((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)) - 4*d**3*((d + e*x)**(5/2)*log(c*(-d/e + (d + e*x)/e)**n)/5 - 2*n*(d**3*e*atan(sq
rt(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))
+ 6*d**2*((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)) -
4*d*((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)) + (d + e*x)**(11/2)*log(c*(-d/e + (d + e*x)/e)**n)/11 - 2*n*(d**6*e*atan(sqrt(d + e*x)/
sqrt(-d))/sqrt(-d) + d**5*e*sqrt(d + e*x) + d**4*e*(d + e*x)**(3/2)/3 + d**3*e*(d + e*x)**(5/2)/5 + d**2*e*(d
+ e*x)**(7/2)/7 + d*e*(d + e*x)**(9/2)/9 + e*(d + e*x)**(11/2)/11)/(11*e))/e**4

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________