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

Optimal. Leaf size=251 \[ -\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b d^4 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

(2*b*d^2*n*(d^2 - e^2*x^2))/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*n*(d^2 - e^2*x^2)^2)/(9*e^4*Sqrt[d - e*x]
*Sqrt[d + e*x]) - (2*b*d^4*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(3*e^4*Sqrt[d - e*x]*Sq
rt[d + e*x]) - (d^2*(d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) + ((d^2 - e^2*x^2)^2
*(a + b*Log[c*x^n]))/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi [A]  time = 0.519758, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {2342, 266, 43, 2350, 12, 446, 80, 50, 63, 208} \[ -\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b d^4 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*d^2*n*(d^2 - e^2*x^2))/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*n*(d^2 - e^2*x^2)^2)/(9*e^4*Sqrt[d - e*x]
*Sqrt[d + e*x]) - (2*b*d^4*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(3*e^4*Sqrt[d - e*x]*Sq
rt[d + e*x]) - (d^2*(d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) + ((d^2 - e^2*x^2)^2
*(a + b*Log[c*x^n]))/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x])

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(q_)*((d2_) + (e2_.)*(x_))^(q_), x_
Symbol] :> Dist[((d1 + e1*x)^q*(d2 + e2*x)^q)/(1 + (e1*e2*x^2)/(d1*d2))^q, Int[x^m*(1 + (e1*e2*x^2)/(d1*d2))^q
*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m]
&& IntegerQ[q - 1/2]

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{d^2 \left (-2 d^2-e^2 x^2\right ) \sqrt{1-\frac{e^2 x^2}{d^2}}}{3 e^4 x} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{\left (-2 d^2-e^2 x^2\right ) \sqrt{1-\frac{e^2 x^2}{d^2}}}{x} \, dx}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\left (-2 d^2-e^2 x\right ) \sqrt{1-\frac{e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{6 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b d^4 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b d^4 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{e^2 x}{d^2}}} \, dx,x,x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (2 b d^6 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{d^2 x^2}{e^2}} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{3 e^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b d^4 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.391462, size = 163, normalized size = 0.65 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (d^2 \left (6 a+6 b \log \left (c x^n\right )-6 b n \log (x)-5 b n\right )+e^2 x^2 \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)-b n\right )\right )+3 b n \log (x) \sqrt{d-e x} \sqrt{d+e x} \left (2 d^2+e^2 x^2\right )+6 b d^3 n \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right )-6 b d^3 n \log (x)}{9 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-6*b*d^3*n*Log[x] + 3*b*n*Sqrt[d - e*x]*Sqrt[d + e*x]*(2*d^2 + e^2*x^2)*Log[x] + Sqrt[d - e*x]*Sqrt[d + e*x]
*(e^2*x^2*(3*a - b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + d^2*(6*a - 5*b*n - 6*b*n*Log[x] + 6*b*Log[c*x^n])) + 6
*b*d^3*n*Log[d + Sqrt[d - e*x]*Sqrt[d + e*x]])/(9*e^4)

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Maple [F]  time = 0.707, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\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)/(e*x+d)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.89477, size = 269, normalized size = 1.07 \begin{align*} -\frac{1}{9} \, b n{\left (\frac{3 \, d^{3} \log \left (d + \sqrt{-e^{2} x^{2} + d^{2}}\right )}{e^{4}} - \frac{3 \, d^{3} \log \left (-d + \sqrt{-e^{2} x^{2} + d^{2}}\right )}{e^{4}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} -{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}}{e^{4}}\right )} - \frac{1}{3} \, b{\left (\frac{\sqrt{-e^{2} x^{2} + d^{2}} x^{2}}{e^{2}} + \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{e^{4}}\right )} \log \left (c x^{n}\right ) - \frac{1}{3} \, a{\left (\frac{\sqrt{-e^{2} x^{2} + d^{2}} x^{2}}{e^{2}} + \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{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)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-1/9*b*n*(3*d^3*log(d + sqrt(-e^2*x^2 + d^2))/e^4 - 3*d^3*log(-d + sqrt(-e^2*x^2 + d^2))/e^4 - (6*sqrt(-e^2*x^
2 + d^2)*d^2 - (-e^2*x^2 + d^2)^(3/2))/e^4) - 1/3*b*(sqrt(-e^2*x^2 + d^2)*x^2/e^2 + 2*sqrt(-e^2*x^2 + d^2)*d^2
/e^4)*log(c*x^n) - 1/3*a*(sqrt(-e^2*x^2 + d^2)*x^2/e^2 + 2*sqrt(-e^2*x^2 + d^2)*d^2/e^4)

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Fricas [A]  time = 1.44386, size = 286, normalized size = 1.14 \begin{align*} \frac{6 \, b d^{3} n \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right ) +{\left (5 \, b d^{2} n - 6 \, a d^{2} +{\left (b e^{2} n - 3 \, a e^{2}\right )} x^{2} - 3 \,{\left (b e^{2} x^{2} + 2 \, b d^{2}\right )} \log \left (c\right ) - 3 \,{\left (b e^{2} n x^{2} + 2 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x + d} \sqrt{-e x + d}}{9 \, e^{4}} \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)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

1/9*(6*b*d^3*n*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x) + (5*b*d^2*n - 6*a*d^2 + (b*e^2*n - 3*a*e^2)*x^2 - 3*
(b*e^2*x^2 + 2*b*d^2)*log(c) - 3*(b*e^2*n*x^2 + 2*b*d^2*n)*log(x))*sqrt(e*x + d)*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*(a+b*ln(c*x**n))/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{\sqrt{e x + d} \sqrt{-e x + d}}\,{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)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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