3.5.12 \(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^2}{x^2} \, dx\) [412]

Optimal. Leaf size=155 \[ -\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {2 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {b^2 e^2 n^2 \log (x)}{d^2}+\frac {2 b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31} \begin {gather*} \frac {2 b^2 e^2 n^2 \text {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^2}-\frac {2 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {b^2 e^2 n^2 \log (x)}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*e*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(d^2*Sqrt[x]) - (2*b*e^2*n*Log[1 - d/(d + e*Sqrt[x
])]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/d^2 - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/x + (b^2*e^2*n^2*Log[x])/d^2 +
(2*b^2*e^2*n^2*PolyLog[2, d/(d + e*Sqrt[x])])/d^2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2445

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

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{d}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt {x}\right )}{d}\\ &=-\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (2 b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^2}\\ &=-\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}+\frac {e^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}-\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2}\\ &=-\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}+\frac {e^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}-\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2}-\frac {2 b^2 e^2 n^2 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 188, normalized size = 1.21 \begin {gather*} 2 \left (-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x}+b e n \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{d \sqrt {x}}+\frac {e \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 b d^2 n}-\frac {e \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {b e n \left (-\frac {\log \left (d+e \sqrt {x}\right )}{d}+\frac {\log (x)}{2 d}\right )}{d}-\frac {b e n \text {Li}_2\left (\frac {d+e \sqrt {x}}{d}\right )}{d^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-1/2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x + b*e*n*(-((a + b*Log[c*(d + e*Sqrt[x])^n])/(d*Sqrt[x])) + (e*(a
+ b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*b*d^2*n) - (e*(a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)])/d^2
+ (b*e*n*(-(Log[d + e*Sqrt[x]]/d) + Log[x]/(2*d)))/d - (b*e*n*PolyLog[2, (d + e*Sqrt[x])/d])/d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(log(sqrt(x))*log(e^(1/2*log(x) + 1)/d + 1) + dilog(-e^(1/2*log(x) + 1)/d))*b^2*n^2*e^2/d^2 - 2*((n^2 - n*lo
g(c))*b^2 - a*b*n)*e^2*log(d + e^(1/2*log(x) + 1))/d^2 - 2*(b^2*n*log(c) + a*b*n)*e^2*log(sqrt(x))/d^2 + integ
rate((b^2*d^2*n^2*e^2 + b^2*n^2*x*e^4)/x, x)/d^4 - 1/3*(6*b^2*d^2*n^2*e^(1/2*log(x) + 3)*log(sqrt(x)) - 12*b^2
*d^2*n^2*e^(1/2*log(x) + 3) + 3*b^2*d*n^2*e^(log(x) + 4) - 2*b^2*n^2*e^(3/2*log(x) + 5))/d^5 - 1/3*(3*b^2*d^5*
n^2*sqrt(x)*log(sqrt(x)*e + d)^2 + 3*b^2*d^3*n^2*x^(3/2)*e^2*log(sqrt(x)*e + d)^2 - 3*b^2*d^2*n^2*x^2*e^3*log(
x) + 12*b^2*d^2*n^2*x^2*e^3 + 2*b^2*n^2*x^3*e^5 + 6*(b^2*d^4*n*log(c) + a*b*d^4*n)*x*e - 3*(2*b^2*d^3*n*x^(3/2
)*e^2*log(sqrt(x)*e + d) - 2*b^2*d^4*n*x*e - (b^2*d^3*n*x*e^2*log(x) + 2*b^2*d^5*log(c) + 2*a*b*d^5)*sqrt(x))*
n*log(sqrt(x)*e + d))/(d^5*x^(3/2))

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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