\(\int \sqrt {d+e x} (a+b \log (c x^n)) \, dx\) [133]

Optimal result

Integrand size = 20, antiderivative size = 94 \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e}+\frac {4 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 e}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e} \]

[Out]

-4/9*b*n*(e*x+d)^(3/2)/e+4/3*b*d^(3/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2))/e+2/3*(e*x+d)^(3/2)*(a+b*ln(c*x^n))/e-
4/3*b*d*n*(e*x+d)^(1/2)/e

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2356, 52, 65, 214} \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {4 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 e}-\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e} \]

[In]

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

[Out]

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

Rule 52

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 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2356

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.82 \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \left (6 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\sqrt {d+e x} \left (3 a (d+e x)-2 b n (4 d+e x)+3 b (d+e x) \log \left (c x^n\right )\right )\right )}{9 e} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.96 \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=\left [\frac {2 \, {\left (3 \, b d^{\frac {3}{2}} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (8 \, b d n - 3 \, a d + {\left (2 \, b e n - 3 \, a e\right )} x - 3 \, {\left (b e x + b d\right )} \log \left (c\right ) - 3 \, {\left (b e n x + b d n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{9 \, e}, -\frac {2 \, {\left (6 \, b \sqrt {-d} d n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (8 \, b d n - 3 \, a d + {\left (2 \, b e n - 3 \, a e\right )} x - 3 \, {\left (b e x + b d\right )} \log \left (c\right ) - 3 \, {\left (b e n x + b d n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{9 \, e}\right ] \]

[In]

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

[Out]

[2/9*(3*b*d^(3/2)*n*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - (8*b*d*n - 3*a*d + (2*b*e*n - 3*a*e)*x - 3*
(b*e*x + b*d)*log(c) - 3*(b*e*n*x + b*d*n)*log(x))*sqrt(e*x + d))/e, -2/9*(6*b*sqrt(-d)*d*n*arctan(sqrt(e*x +
d)*sqrt(-d)/d) + (8*b*d*n - 3*a*d + (2*b*e*n - 3*a*e)*x - 3*(b*e*x + b*d)*log(c) - 3*(b*e*n*x + b*d*n)*log(x))
*sqrt(e*x + d))/e]

Sympy [A] (verification not implemented)

Time = 29.54 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.53 \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=a \left (\begin {cases} \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {for}\: e \neq 0 \\\sqrt {d} x & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {16 d^{\frac {3}{2}} \sqrt {1 + \frac {e x}{d}}}{9 e} + \frac {2 d^{\frac {3}{2}} \log {\left (\frac {e x}{d} \right )}}{3 e} - \frac {4 d^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {e x}{d}} + 1 \right )}}{3 e} + \frac {4 \sqrt {d} x \sqrt {1 + \frac {e x}{d}}}{9} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\sqrt {d} x & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {for}\: e \neq 0 \\\sqrt {d} x & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]

[In]

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

[Out]

a*Piecewise((2*(d + e*x)**(3/2)/(3*e), Ne(e, 0)), (sqrt(d)*x, True)) - b*n*Piecewise((16*d**(3/2)*sqrt(1 + e*x
/d)/(9*e) + 2*d**(3/2)*log(e*x/d)/(3*e) - 4*d**(3/2)*log(sqrt(1 + e*x/d) + 1)/(3*e) + 4*sqrt(d)*x*sqrt(1 + e*x
/d)/9, (e > -oo) & (e < oo) & Ne(e, 0)), (sqrt(d)*x, True)) + b*Piecewise((2*(d + e*x)**(3/2)/(3*e), Ne(e, 0))
, (sqrt(d)*x, True))*log(c*x**n)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} b \log \left (c x^{n}\right )}{3 \, e} - \frac {2 \, {\left (3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right ) + 2 \, {\left (e x + d\right )}^{\frac {3}{2}} + 6 \, \sqrt {e x + d} d\right )} b n}{9 \, e} + \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} a}{3 \, e} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x} \,d x \]

[In]

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

[Out]

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