3.3.51 \(\int x^5 \sqrt {d+e x^2} (a+b \log (c x^n)) \, dx\) [251]

Optimal. Leaf size=208 \[ -\frac {8 b d^3 n \sqrt {d+e x^2}}{105 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{3/2}}{315 e^3}+\frac {9 b d n \left (d+e x^2\right )^{5/2}}{175 e^3}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 e^3}+\frac {8 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{105 e^3}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3} \]

[Out]

-8/315*b*d^2*n*(e*x^2+d)^(3/2)/e^3+9/175*b*d*n*(e*x^2+d)^(5/2)/e^3-1/49*b*n*(e*x^2+d)^(7/2)/e^3+8/105*b*d^(7/2
)*n*arctanh((e*x^2+d)^(1/2)/d^(1/2))/e^3+1/3*d^2*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/e^3-2/5*d*(e*x^2+d)^(5/2)*(a+
b*ln(c*x^n))/e^3+1/7*(e*x^2+d)^(7/2)*(a+b*ln(c*x^n))/e^3-8/105*b*d^3*n*(e*x^2+d)^(1/2)/e^3

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Rubi [A]
time = 0.17, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {272, 45, 2392, 12, 1265, 911, 1275, 214} \begin {gather*} \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {8 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{105 e^3}-\frac {8 b d^3 n \sqrt {d+e x^2}}{105 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{3/2}}{315 e^3}+\frac {9 b d n \left (d+e x^2\right )^{5/2}}{175 e^3}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]),x]

[Out]

(-8*b*d^3*n*Sqrt[d + e*x^2])/(105*e^3) - (8*b*d^2*n*(d + e*x^2)^(3/2))/(315*e^3) + (9*b*d*n*(d + e*x^2)^(5/2))
/(175*e^3) - (b*n*(d + e*x^2)^(7/2))/(49*e^3) + (8*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(105*e^3) + (
d^2*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3) + ((d +
e*x^2)^(7/2)*(a + b*Log[c*x^n]))/(7*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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

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*} \int x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{105 e^3 x} \, dx\\ &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{x} \, dx}{105 e^3}\\ &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(b n) \text {Subst}\left (\int \frac {(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x} \, dx,x,x^2\right )}{210 e^3}\\ &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(b n) \text {Subst}\left (\int \frac {x^4 \left (35 d^2-42 d x^2+15 x^4\right )}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{105 e^4}\\ &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(b n) \text {Subst}\left (\int \left (8 d^3 e+8 d^2 e x^2-27 d e x^4+15 e x^6+\frac {8 d^4}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{105 e^4}\\ &=-\frac {8 b d^3 n \sqrt {d+e x^2}}{105 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{3/2}}{315 e^3}+\frac {9 b d n \left (d+e x^2\right )^{5/2}}{175 e^3}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 e^3}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {\left (8 b d^4 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{105 e^4}\\ &=-\frac {8 b d^3 n \sqrt {d+e x^2}}{105 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{3/2}}{315 e^3}+\frac {9 b d n \left (d+e x^2\right )^{5/2}}{175 e^3}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 e^3}+\frac {8 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{105 e^3}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 251, normalized size = 1.21 \begin {gather*} -\frac {8 b d^{7/2} n \log (x)}{105 e^3}+\frac {b n \sqrt {d+e x^2} \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right ) \log (x)}{105 e^3}+\sqrt {d+e x^2} \left (\frac {1}{49} x^6 \left (7 a-b n+7 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+\frac {d x^4 \left (35 a-12 b n+35 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{1225 e}+\frac {2 d^3 \left (420 a-389 b n+420 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{11025 e^3}-\frac {d^2 x^2 \left (420 a-179 b n+420 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{11025 e^2}\right )+\frac {8 b d^{7/2} n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{105 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]),x]

[Out]

(-8*b*d^(7/2)*n*Log[x])/(105*e^3) + (b*n*Sqrt[d + e*x^2]*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*Log[
x])/(105*e^3) + Sqrt[d + e*x^2]*((x^6*(7*a - b*n + 7*b*(-(n*Log[x]) + Log[c*x^n])))/49 + (d*x^4*(35*a - 12*b*n
+ 35*b*(-(n*Log[x]) + Log[c*x^n])))/(1225*e) + (2*d^3*(420*a - 389*b*n + 420*b*(-(n*Log[x]) + Log[c*x^n])))/(
11025*e^3) - (d^2*x^2*(420*a - 179*b*n + 420*b*(-(n*Log[x]) + Log[c*x^n])))/(11025*e^2)) + (8*b*d^(7/2)*n*Log[
d + Sqrt[d]*Sqrt[d + e*x^2]])/(105*e^3)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{5} \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2),x)

[Out]

int(x^5*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2),x)

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Maxima [A]
time = 0.50, size = 224, normalized size = 1.08 \begin {gather*} -\frac {1}{11025} \, {\left (420 \, d^{\frac {7}{2}} e^{\left (-3\right )} \log \left (\frac {\sqrt {x^{2} e + d} - \sqrt {d}}{\sqrt {x^{2} e + d} + \sqrt {d}}\right ) + {\left (225 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}} - 567 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} d + 280 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{2} + 840 \, \sqrt {x^{2} e + d} d^{3}\right )} e^{\left (-3\right )}\right )} b n + \frac {1}{105} \, {\left (15 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} x^{4} e^{\left (-1\right )} - 12 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d x^{2} e^{\left (-2\right )} + 8 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{2} e^{\left (-3\right )}\right )} b \log \left (c x^{n}\right ) + \frac {1}{105} \, {\left (15 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} x^{4} e^{\left (-1\right )} - 12 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d x^{2} e^{\left (-2\right )} + 8 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{2} e^{\left (-3\right )}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))*(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/11025*(420*d^(7/2)*e^(-3)*log((sqrt(x^2*e + d) - sqrt(d))/(sqrt(x^2*e + d) + sqrt(d))) + (225*(x^2*e + d)^(
7/2) - 567*(x^2*e + d)^(5/2)*d + 280*(x^2*e + d)^(3/2)*d^2 + 840*sqrt(x^2*e + d)*d^3)*e^(-3))*b*n + 1/105*(15*
(x^2*e + d)^(3/2)*x^4*e^(-1) - 12*(x^2*e + d)^(3/2)*d*x^2*e^(-2) + 8*(x^2*e + d)^(3/2)*d^2*e^(-3))*b*log(c*x^n
) + 1/105*(15*(x^2*e + d)^(3/2)*x^4*e^(-1) - 12*(x^2*e + d)^(3/2)*d*x^2*e^(-2) + 8*(x^2*e + d)^(3/2)*d^2*e^(-3
))*a

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Fricas [A]
time = 0.43, size = 397, normalized size = 1.91 \begin {gather*} \left [\frac {1}{11025} \, {\left (420 \, b d^{\frac {7}{2}} n \log \left (-\frac {x^{2} e + 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (225 \, {\left (b n - 7 \, a\right )} x^{6} e^{3} + 9 \, {\left (12 \, b d n - 35 \, a d\right )} x^{4} e^{2} + 778 \, b d^{3} n - 840 \, a d^{3} - {\left (179 \, b d^{2} n - 420 \, a d^{2}\right )} x^{2} e - 105 \, {\left (15 \, b x^{6} e^{3} + 3 \, b d x^{4} e^{2} - 4 \, b d^{2} x^{2} e + 8 \, b d^{3}\right )} \log \left (c\right ) - 105 \, {\left (15 \, b n x^{6} e^{3} + 3 \, b d n x^{4} e^{2} - 4 \, b d^{2} n x^{2} e + 8 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-3\right )}, -\frac {1}{11025} \, {\left (840 \, b \sqrt {-d} d^{3} n \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + {\left (225 \, {\left (b n - 7 \, a\right )} x^{6} e^{3} + 9 \, {\left (12 \, b d n - 35 \, a d\right )} x^{4} e^{2} + 778 \, b d^{3} n - 840 \, a d^{3} - {\left (179 \, b d^{2} n - 420 \, a d^{2}\right )} x^{2} e - 105 \, {\left (15 \, b x^{6} e^{3} + 3 \, b d x^{4} e^{2} - 4 \, b d^{2} x^{2} e + 8 \, b d^{3}\right )} \log \left (c\right ) - 105 \, {\left (15 \, b n x^{6} e^{3} + 3 \, b d n x^{4} e^{2} - 4 \, b d^{2} n x^{2} e + 8 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-3\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))*(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/11025*(420*b*d^(7/2)*n*log(-(x^2*e + 2*sqrt(x^2*e + d)*sqrt(d) + 2*d)/x^2) - (225*(b*n - 7*a)*x^6*e^3 + 9*(
12*b*d*n - 35*a*d)*x^4*e^2 + 778*b*d^3*n - 840*a*d^3 - (179*b*d^2*n - 420*a*d^2)*x^2*e - 105*(15*b*x^6*e^3 + 3
*b*d*x^4*e^2 - 4*b*d^2*x^2*e + 8*b*d^3)*log(c) - 105*(15*b*n*x^6*e^3 + 3*b*d*n*x^4*e^2 - 4*b*d^2*n*x^2*e + 8*b
*d^3*n)*log(x))*sqrt(x^2*e + d))*e^(-3), -1/11025*(840*b*sqrt(-d)*d^3*n*arctan(sqrt(-d)/sqrt(x^2*e + d)) + (22
5*(b*n - 7*a)*x^6*e^3 + 9*(12*b*d*n - 35*a*d)*x^4*e^2 + 778*b*d^3*n - 840*a*d^3 - (179*b*d^2*n - 420*a*d^2)*x^
2*e - 105*(15*b*x^6*e^3 + 3*b*d*x^4*e^2 - 4*b*d^2*x^2*e + 8*b*d^3)*log(c) - 105*(15*b*n*x^6*e^3 + 3*b*d*n*x^4*
e^2 - 4*b*d^2*n*x^2*e + 8*b*d^3*n)*log(x))*sqrt(x^2*e + d))*e^(-3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*ln(c*x**n))*(e*x**2+d)**(1/2),x)

[Out]

Integral(x**5*(a + b*log(c*x**n))*sqrt(d + e*x**2), x)

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Giac [A]
time = 2.84, size = 296, normalized size = 1.42 \begin {gather*} \frac {1}{7} \, \sqrt {x^{2} e + d} b x^{6} \log \left (c\right ) + \frac {1}{35} \, \sqrt {x^{2} e + d} b d x^{4} e^{\left (-1\right )} \log \left (c\right ) + \frac {1}{7} \, \sqrt {x^{2} e + d} a x^{6} + \frac {1}{35} \, \sqrt {x^{2} e + d} a d x^{4} e^{\left (-1\right )} - \frac {4}{105} \, \sqrt {x^{2} e + d} b d^{2} x^{2} e^{\left (-2\right )} \log \left (c\right ) - \frac {4}{105} \, \sqrt {x^{2} e + d} a d^{2} x^{2} e^{\left (-2\right )} + \frac {8}{105} \, \sqrt {x^{2} e + d} b d^{3} e^{\left (-3\right )} \log \left (c\right ) + \frac {8}{105} \, \sqrt {x^{2} e + d} a d^{3} e^{\left (-3\right )} + \frac {1}{11025} \, {\left (105 \, {\left (15 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}} - 42 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{2}\right )} e^{\left (-3\right )} \log \left (x\right ) - {\left (\frac {840 \, d^{4} \arctan \left (\frac {\sqrt {x^{2} e + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} + 225 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}} - 567 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} d + 280 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{2} + 840 \, \sqrt {x^{2} e + d} d^{3}\right )} e^{\left (-3\right )}\right )} b n \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))*(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

1/7*sqrt(x^2*e + d)*b*x^6*log(c) + 1/35*sqrt(x^2*e + d)*b*d*x^4*e^(-1)*log(c) + 1/7*sqrt(x^2*e + d)*a*x^6 + 1/
35*sqrt(x^2*e + d)*a*d*x^4*e^(-1) - 4/105*sqrt(x^2*e + d)*b*d^2*x^2*e^(-2)*log(c) - 4/105*sqrt(x^2*e + d)*a*d^
2*x^2*e^(-2) + 8/105*sqrt(x^2*e + d)*b*d^3*e^(-3)*log(c) + 8/105*sqrt(x^2*e + d)*a*d^3*e^(-3) + 1/11025*(105*(
15*(x^2*e + d)^(7/2) - 42*(x^2*e + d)^(5/2)*d + 35*(x^2*e + d)^(3/2)*d^2)*e^(-3)*log(x) - (840*d^4*arctan(sqrt
(x^2*e + d)/sqrt(-d))/sqrt(-d) + 225*(x^2*e + d)^(7/2) - 567*(x^2*e + d)^(5/2)*d + 280*(x^2*e + d)^(3/2)*d^2 +
840*sqrt(x^2*e + d)*d^3)*e^(-3))*b*n

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^5\,\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d + e*x^2)^(1/2)*(a + b*log(c*x^n)),x)

[Out]

int(x^5*(d + e*x^2)^(1/2)*(a + b*log(c*x^n)), x)

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