3.2.16 \(\int x \log (d (e+f \sqrt {x})^k) (a+b \log (c x^n)) \, dx\) [116]

Optimal. Leaf size=313 \[ -\frac {5 b e^3 k n \sqrt {x}}{4 f^3}+\frac {3 b e^2 k n x}{8 f^2}-\frac {7 b e k n x^{3/2}}{36 f}+\frac {1}{8} b k n x^2+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right )}{4 f^4}-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{f^4} \]

[Out]

3/8*b*e^2*k*n*x/f^2-7/36*b*e*k*n*x^(3/2)/f+1/8*b*k*n*x^2-1/4*e^2*k*x*(a+b*ln(c*x^n))/f^2+1/6*e*k*x^(3/2)*(a+b*
ln(c*x^n))/f-1/8*k*x^2*(a+b*ln(c*x^n))+1/4*b*e^4*k*n*ln(e+f*x^(1/2))/f^4-1/2*e^4*k*(a+b*ln(c*x^n))*ln(e+f*x^(1
/2))/f^4+b*e^4*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f^4-1/4*b*n*x^2*ln(d*(e+f*x^(1/2))^k)+1/2*x^2*(a+b*ln(c*x^
n))*ln(d*(e+f*x^(1/2))^k)+b*e^4*k*n*polylog(2,1+f*x^(1/2)/e)/f^4-5/4*b*e^3*k*n*x^(1/2)/f^3+1/2*e^3*k*(a+b*ln(c
*x^n))*x^(1/2)/f^3

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Rubi [A]
time = 0.17, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2504, 2442, 45, 2423, 2441, 2352} \begin {gather*} \frac {b e^4 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{f^4}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right )}{4 f^4}+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {5 b e^3 k n \sqrt {x}}{4 f^3}+\frac {3 b e^2 k n x}{8 f^2}-\frac {7 b e k n x^{3/2}}{36 f}+\frac {1}{8} b k n x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*e^3*k*n*Sqrt[x])/(4*f^3) + (3*b*e^2*k*n*x)/(8*f^2) - (7*b*e*k*n*x^(3/2))/(36*f) + (b*k*n*x^2)/8 + (b*e^4
*k*n*Log[e + f*Sqrt[x]])/(4*f^4) - (b*n*x^2*Log[d*(e + f*Sqrt[x])^k])/4 + (b*e^4*k*n*Log[e + f*Sqrt[x]]*Log[-(
(f*Sqrt[x])/e)])/f^4 + (e^3*k*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) - (e^2*k*x*(a + b*Log[c*x^n]))/(4*f^2) + (e*
k*x^(3/2)*(a + b*Log[c*x^n]))/(6*f) - (k*x^2*(a + b*Log[c*x^n]))/8 - (e^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^
n]))/(2*f^4) + (x^2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/2 + (b*e^4*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])
/f^4

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 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 x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {e^2 k}{4 f^2}+\frac {e^3 k}{2 f^3 \sqrt {x}}+\frac {e k \sqrt {x}}{6 f}-\frac {k x}{8}-\frac {e^4 k \log \left (e+f \sqrt {x}\right )}{2 f^4 x}+\frac {1}{2} x \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right ) \, dx\\ &=-\frac {b e^3 k n \sqrt {x}}{f^3}+\frac {b e^2 k n x}{4 f^2}-\frac {b e k n x^{3/2}}{9 f}+\frac {1}{16} b k n x^2+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx+\frac {\left (b e^4 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{2 f^4}\\ &=-\frac {b e^3 k n \sqrt {x}}{f^3}+\frac {b e^2 k n x}{4 f^2}-\frac {b e k n x^{3/2}}{9 f}+\frac {1}{16} b k n x^2+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \text {Subst}\left (\int x^3 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt {x}\right )+\frac {\left (b e^4 k n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{f^4}\\ &=-\frac {b e^3 k n \sqrt {x}}{f^3}+\frac {b e^2 k n x}{4 f^2}-\frac {b e k n x^{3/2}}{9 f}+\frac {1}{16} b k n x^2-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (b e^4 k n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{f^3}+\frac {1}{4} (b f k n) \text {Subst}\left (\int \frac {x^4}{e+f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b e^3 k n \sqrt {x}}{f^3}+\frac {b e^2 k n x}{4 f^2}-\frac {b e k n x^{3/2}}{9 f}+\frac {1}{16} b k n x^2-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {1}{4} (b f k n) \text {Subst}\left (\int \left (-\frac {e^3}{f^4}+\frac {e^2 x}{f^3}-\frac {e x^2}{f^2}+\frac {x^3}{f}+\frac {e^4}{f^4 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {5 b e^3 k n \sqrt {x}}{4 f^3}+\frac {3 b e^2 k n x}{8 f^2}-\frac {7 b e k n x^{3/2}}{36 f}+\frac {1}{8} b k n x^2+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right )}{4 f^4}-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{f^4}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 336, normalized size = 1.07 \begin {gather*} -\frac {-36 a e^3 f k \sqrt {x}+90 b e^3 f k n \sqrt {x}+18 a e^2 f^2 k x-27 b e^2 f^2 k n x-12 a e f^3 k x^{3/2}+14 b e f^3 k n x^{3/2}+9 a f^4 k x^2-9 b f^4 k n x^2-36 a f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+36 b e^4 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-36 b e^3 f k \sqrt {x} \log \left (c x^n\right )+18 b e^2 f^2 k x \log \left (c x^n\right )-12 b e f^3 k x^{3/2} \log \left (c x^n\right )+9 b f^4 k x^2 \log \left (c x^n\right )-36 b f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+18 e^4 k \log \left (e+f \sqrt {x}\right ) \left (2 a-b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right )+72 b e^4 k n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{72 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/72*(-36*a*e^3*f*k*Sqrt[x] + 90*b*e^3*f*k*n*Sqrt[x] + 18*a*e^2*f^2*k*x - 27*b*e^2*f^2*k*n*x - 12*a*e*f^3*k*x
^(3/2) + 14*b*e*f^3*k*n*x^(3/2) + 9*a*f^4*k*x^2 - 9*b*f^4*k*n*x^2 - 36*a*f^4*x^2*Log[d*(e + f*Sqrt[x])^k] + 18
*b*f^4*n*x^2*Log[d*(e + f*Sqrt[x])^k] + 36*b*e^4*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 36*b*e^3*f*k*Sqrt[x]*Log[
c*x^n] + 18*b*e^2*f^2*k*x*Log[c*x^n] - 12*b*e*f^3*k*x^(3/2)*Log[c*x^n] + 9*b*f^4*k*x^2*Log[c*x^n] - 36*b*f^4*x
^2*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 18*e^4*k*Log[e + f*Sqrt[x]]*(2*a - b*n - 2*b*n*Log[x] + 2*b*Log[c*x^n
]) + 72*b*e^4*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/f^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/100*(50*b*x^2*e*log(d)*log(x^n) - 25*((n*log(d) - 2*log(c)*log(d))*b - 2*a*log(d))*x^2*e + 25*(2*b*x^2*e*log
(x^n) - (b*(n - 2*log(c)) - 2*a)*x^2*e)*k*log(f*sqrt(x) + e) - (10*b*f*k*x^3*log(x^n) + (10*a*f*k - (9*f*k*n -
10*f*k*log(c))*b)*x^3)/sqrt(x))*e^(-1) + integrate(1/8*(2*b*f^2*k*x^2*log(x^n) + (2*a*f^2*k - (f^2*k*n - 2*f^
2*k*log(c))*b)*x^2)/(f*e^(1/2*log(x) + 1) + e^2), 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(x*(a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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

[Out]

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

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

[Out]

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

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