3.2.35 \(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^{3/2}} \, dx\) [135]

Optimal. Leaf size=199 \[ -\frac {4 b f k n \log \left (e+f \sqrt {x}\right )}{e}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}+\frac {4 b f k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e}+\frac {2 b f k n \log (x)}{e}-\frac {b f k n \log ^2(x)}{2 e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {4 b f k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e} \]

[Out]

2*b*f*k*n*ln(x)/e-1/2*b*f*k*n*ln(x)^2/e+f*k*ln(x)*(a+b*ln(c*x^n))/e-4*b*f*k*n*ln(e+f*x^(1/2))/e-2*f*k*(a+b*ln(
c*x^n))*ln(e+f*x^(1/2))/e+4*b*f*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/e+4*b*f*k*n*polylog(2,1+f*x^(1/2)/e)/e-4*
b*n*ln(d*(e+f*x^(1/2))^k)/x^(1/2)-2*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2504, 2442, 36, 29, 31, 2423, 2441, 2352, 2338} \begin {gather*} \frac {4 b f k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{e}-\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}-\frac {b f k n \log ^2(x)}{2 e}+\frac {2 b f k n \log (x)}{e}-\frac {4 b f k n \log \left (e+f \sqrt {x}\right )}{e}+\frac {4 b f k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(3/2),x]

[Out]

(-4*b*f*k*n*Log[e + f*Sqrt[x]])/e - (4*b*n*Log[d*(e + f*Sqrt[x])^k])/Sqrt[x] + (4*b*f*k*n*Log[e + f*Sqrt[x]]*L
og[-((f*Sqrt[x])/e)])/e + (2*b*f*k*n*Log[x])/e - (b*f*k*n*Log[x]^2)/(2*e) - (2*f*k*Log[e + f*Sqrt[x]]*(a + b*L
og[c*x^n]))/e - (2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/Sqrt[x] + (f*k*Log[x]*(a + b*Log[c*x^n]))/e +
(4*b*f*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/e

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2423

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2441

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

Rule 2442

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

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 {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx &=-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-(b n) \int \left (-\frac {2 f k \log \left (e+f \sqrt {x}\right )}{e x}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x^{3/2}}+\frac {f k \log (x)}{e x}\right ) \, dx\\ &=-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+(2 b n) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x^{3/2}} \, dx-\frac {(b f k n) \int \frac {\log (x)}{x} \, dx}{e}+\frac {(2 b f k n) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{e}\\ &=-\frac {b f k n \log ^2(x)}{2 e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+(4 b n) \text {Subst}\left (\int \frac {\log \left (d (e+f x)^k\right )}{x^2} \, dx,x,\sqrt {x}\right )+\frac {(4 b f k n) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{e}\\ &=-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}+\frac {4 b f k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e}-\frac {b f k n \log ^2(x)}{2 e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+(4 b f k n) \text {Subst}\left (\int \frac {1}{x (e+f x)} \, dx,x,\sqrt {x}\right )-\frac {\left (4 b f^2 k n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{e}\\ &=-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}+\frac {4 b f k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e}-\frac {b f k n \log ^2(x)}{2 e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {4 b f k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e}+\frac {(4 b f k n) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sqrt {x}\right )}{e}-\frac {\left (4 b f^2 k n\right ) \text {Subst}\left (\int \frac {1}{e+f x} \, dx,x,\sqrt {x}\right )}{e}\\ &=-\frac {4 b f k n \log \left (e+f \sqrt {x}\right )}{e}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}+\frac {4 b f k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e}+\frac {2 b f k n \log (x)}{e}-\frac {b f k n \log ^2(x)}{2 e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {4 b f k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 145, normalized size = 0.73 \begin {gather*} -\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+2 b n+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+2 b n-b n \log (x)+b \log \left (c x^n\right )\right )}{e}-\frac {f k \log (x) \left (4 b n \log \left (1+\frac {f \sqrt {x}}{e}\right )+b n \log (x)-2 \left (a+2 b n+b \log \left (c x^n\right )\right )\right )}{2 e}-\frac {4 b f k n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(3/2),x]

[Out]

(-2*Log[d*(e + f*Sqrt[x])^k]*(a + 2*b*n + b*Log[c*x^n]))/Sqrt[x] - (2*f*k*Log[e + f*Sqrt[x]]*(a + 2*b*n - b*n*
Log[x] + b*Log[c*x^n]))/e - (f*k*Log[x]*(4*b*n*Log[1 + (f*Sqrt[x])/e] + b*n*Log[x] - 2*(a + 2*b*n + b*Log[c*x^
n])))/(2*e) - (4*b*f*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^(3/2),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*x^n))*log(d*(e+f*x^(1/2))^k)/x^(3/2),x, algorithm="maxima")

[Out]

-1/9*(18*(b*x*e^4*log(x^n) + (b*(2*n + log(c)) + a)*x*e^4)*k*log(f*sqrt(x) + e)/x^(3/2) + 2*(3*b*f^4*k*x^2*log
(x^n) + (3*a*f^4*k + (4*f^4*k*n + 3*f^4*k*log(c))*b)*x^2)/sqrt(x) - 9*(b*f^3*k*x^2*e*log(x^n) + (a*f^3*k + (f^
3*k*n + f^3*k*log(c))*b)*x^2*e)/x + 18*((b*f^2*k*log(c) + a*f^2*k)*x^2*e^2 + ((2*n*log(d) + log(c)*log(d))*b +
a*log(d))*x*e^4 + (b*f^2*k*x^2*e^2 + b*x*e^4*log(d))*log(x^n))/x^(3/2))*e^(-4) + integrate((b*f*k*x*log(x^n)
+ (a*f*k + (2*f*k*n + f*k*log(c))*b)*x)*e^(-2*log(x) - 1), x) + integrate((b*f^5*k*x*log(x^n) + (a*f^5*k + (2*
f^5*k*n + f^5*k*log(c))*b)*x)/(f*sqrt(x)*e^4 + e^5), x)

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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*x^n))*log(d*(e+f*x^(1/2))^k)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*ln(d*(e+f*x**(1/2))**k)/x**(3/2),x)

[Out]

Timed out

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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*x^n))*log(d*(e+f*x^(1/2))^k)/x^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)))/x^(3/2),x)

[Out]

int((log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)))/x^(3/2), x)

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