3.2.34 \(\int \sqrt {x} \log (d (e+f \sqrt {x})^k) (a+b \log (c x^n)) \, dx\) [134]

Optimal. Leaf size=283 \[ \frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^3} \]

[Out]

-5/9*b*e*k*n*x/f+8/27*b*k*n*x^(3/2)+1/3*e*k*x*(a+b*ln(c*x^n))/f-2/9*k*x^(3/2)*(a+b*ln(c*x^n))-4/9*b*e^3*k*n*ln
(e+f*x^(1/2))/f^3+2/3*e^3*k*(a+b*ln(c*x^n))*ln(e+f*x^(1/2))/f^3-4/3*b*e^3*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))
/f^3-4/9*b*n*x^(3/2)*ln(d*(e+f*x^(1/2))^k)+2/3*x^(3/2)*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)-4/3*b*e^3*k*n*pol
ylog(2,1+f*x^(1/2)/e)/f^3+16/9*b*e^2*k*n*x^(1/2)/f^2-2/3*e^2*k*(a+b*ln(c*x^n))*x^(1/2)/f^2

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Rubi [A]
time = 0.15, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2504, 2442, 45, 2423, 2441, 2352} \begin {gather*} -\frac {4 b e^3 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{3 f^3}+\frac {2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}+\frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b*e^2*k*n*Sqrt[x])/(9*f^2) - (5*b*e*k*n*x)/(9*f) + (8*b*k*n*x^(3/2))/27 - (4*b*e^3*k*n*Log[e + f*Sqrt[x]])
/(9*f^3) - (4*b*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k])/9 - (4*b*e^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])
/(3*f^3) - (2*e^2*k*Sqrt[x]*(a + b*Log[c*x^n]))/(3*f^2) + (e*k*x*(a + b*Log[c*x^n]))/(3*f) - (2*k*x^(3/2)*(a +
b*Log[c*x^n]))/9 + (2*e^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^3) + (2*x^(3/2)*Log[d*(e + f*Sqrt[x])
^k]*(a + b*Log[c*x^n]))/3 - (4*b*e^3*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^3)

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 \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \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 k}{3 f}-\frac {2 e^2 k}{3 f^2 \sqrt {x}}-\frac {2 k \sqrt {x}}{9}+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right )}{3 f^3 x}+\frac {2}{3} \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right ) \, dx\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (2 b n) \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx-\frac {\left (2 b e^3 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{3 f^3}\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (4 b n) \text {Subst}\left (\int x^2 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt {x}\right )-\frac {\left (4 b e^3 k n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{3 f^3}\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\left (4 b e^3 k n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{3 f^2}+\frac {1}{9} (4 b f k n) \text {Subst}\left (\int \frac {x^3}{e+f x} \, dx,x,\sqrt {x}\right )\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^3}+\frac {1}{9} (4 b f k n) \text {Subst}\left (\int \left (\frac {e^2}{f^3}-\frac {e x}{f^2}+\frac {x^2}{f}-\frac {e^3}{f^3 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^3}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 296, normalized size = 1.05 \begin {gather*} \frac {-18 a e^2 f k \sqrt {x}+48 b e^2 f k n \sqrt {x}+9 a e f^2 k x-15 b e f^2 k n x-6 a f^3 k x^{3/2}+8 b f^3 k n x^{3/2}+18 a f^3 x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-12 b f^3 n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b e^3 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-18 b e^2 f k \sqrt {x} \log \left (c x^n\right )+9 b e f^2 k x \log \left (c x^n\right )-6 b f^3 k x^{3/2} \log \left (c x^n\right )+18 b f^3 x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+6 e^3 k \log \left (e+f \sqrt {x}\right ) \left (3 a-2 b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+36 b e^3 k n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{27 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*a*e^2*f*k*Sqrt[x] + 48*b*e^2*f*k*n*Sqrt[x] + 9*a*e*f^2*k*x - 15*b*e*f^2*k*n*x - 6*a*f^3*k*x^(3/2) + 8*b*f
^3*k*n*x^(3/2) + 18*a*f^3*x^(3/2)*Log[d*(e + f*Sqrt[x])^k] - 12*b*f^3*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k] + 18*
b*e^3*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 18*b*e^2*f*k*Sqrt[x]*Log[c*x^n] + 9*b*e*f^2*k*x*Log[c*x^n] - 6*b*f^3
*k*x^(3/2)*Log[c*x^n] + 18*b*f^3*x^(3/2)*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 6*e^3*k*Log[e + f*Sqrt[x]]*(3*a
- 2*b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + 36*b*e^3*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/(27*f^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral((b*sqrt(x)*log(c*x^n) + a*sqrt(x))*log((f*sqrt(x) + e)^k*d), 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(x**(1/2)*(a+b*ln(c*x**n))*ln(d*(e+f*x**(1/2))**k),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(x^(1/2)*(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)*sqrt(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 \sqrt {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^(1/2)*log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)),x)

[Out]

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

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