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

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

[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)

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Rubi [A]  time = 0.249717, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {266, 43, 2350, 12, 1251, 897, 1261, 208} \[ \frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac{8 b d^3 n \sqrt{d+e x^2}}{105 e^3}-\frac{8 b d^2 n \left (d+e x^2\right )^{3/2}}{315 e^3}+\frac{8 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{105 e^3}+\frac{9 b d n \left (d+e x^2\right )^{5/2}}{175 e^3}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^3} \]

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 266

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 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 1251

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 897

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 1261

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 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^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) \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 0.195219, size = 251, normalized size = 1.21 \[ \sqrt{d+e x^2} \left (-\frac{d^2 x^2 \left (420 a+420 b \left (\log \left (c x^n\right )-n \log (x)\right )-179 b n\right )}{11025 e^2}+\frac{2 d^3 \left (420 a+420 b \left (\log \left (c x^n\right )-n \log (x)\right )-389 b n\right )}{11025 e^3}+\frac{d x^4 \left (35 a+35 b \left (\log \left (c x^n\right )-n \log (x)\right )-12 b n\right )}{1225 e}+\frac{1}{49} x^6 \left (7 a+7 b \left (\log \left (c x^n\right )-n \log (x)\right )-b n\right )\right )+\frac{8 b d^{7/2} n \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )}{105 e^3}+\frac{b n \log (x) \sqrt{d+e x^2} \left (-4 d^2 e x^2+8 d^3+3 d e^2 x^4+15 e^3 x^6\right )}{105 e^3}-\frac{8 b d^{7/2} n \log (x)}{105 e^3} \]

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 = 0.488, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75575, size = 1006, normalized size = 4.84 \begin{align*} \left [\frac{420 \, b d^{\frac{7}{2}} n \log \left (-\frac{e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{d} + 2 \, d}{x^{2}}\right ) -{\left (225 \,{\left (b e^{3} n - 7 \, a e^{3}\right )} x^{6} + 778 \, b d^{3} n + 9 \,{\left (12 \, b d e^{2} n - 35 \, a d e^{2}\right )} x^{4} - 840 \, a d^{3} -{\left (179 \, b d^{2} e n - 420 \, a d^{2} e\right )} x^{2} - 105 \,{\left (15 \, b e^{3} x^{6} + 3 \, b d e^{2} x^{4} - 4 \, b d^{2} e x^{2} + 8 \, b d^{3}\right )} \log \left (c\right ) - 105 \,{\left (15 \, b e^{3} n x^{6} + 3 \, b d e^{2} n x^{4} - 4 \, b d^{2} e n x^{2} + 8 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{11025 \, e^{3}}, -\frac{840 \, b \sqrt{-d} d^{3} n \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) +{\left (225 \,{\left (b e^{3} n - 7 \, a e^{3}\right )} x^{6} + 778 \, b d^{3} n + 9 \,{\left (12 \, b d e^{2} n - 35 \, a d e^{2}\right )} x^{4} - 840 \, a d^{3} -{\left (179 \, b d^{2} e n - 420 \, a d^{2} e\right )} x^{2} - 105 \,{\left (15 \, b e^{3} x^{6} + 3 \, b d e^{2} x^{4} - 4 \, b d^{2} e x^{2} + 8 \, b d^{3}\right )} \log \left (c\right ) - 105 \,{\left (15 \, b e^{3} n x^{6} + 3 \, b d e^{2} n x^{4} - 4 \, b d^{2} e n x^{2} + 8 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{11025 \, e^{3}}\right ] \end{align*}

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

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

[Out]

Timed out

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Giac [A]  time = 1.4689, size = 400, normalized size = 1.92 \begin{align*} \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{align*}

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