3.2.47 \(\int \frac {a+b \log (c x^n)}{\sqrt {d+e x}} \, dx\) [147]

Optimal. Leaf size=69 \[ -\frac {4 b n \sqrt {d+e x}}{e}+\frac {4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2356, 52, 65, 214} \begin {gather*} \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {4 b n \sqrt {d+e x}}{e}+\frac {4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*n*Sqrt[d + e*x])/e + (4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/e + (2*Sqrt[d + e*x]*(a + b*Log[c*x^
n]))/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*} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx &=\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(2 b n) \int \frac {\sqrt {d+e x}}{x} \, dx}{e}\\ &=-\frac {4 b n \sqrt {d+e x}}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(2 b d n) \int \frac {1}{x \sqrt {d+e x}} \, dx}{e}\\ &=-\frac {4 b n \sqrt {d+e x}}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(4 b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e^2}\\ &=-\frac {4 b n \sqrt {d+e x}}{e}+\frac {4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 55, normalized size = 0.80 \begin {gather*} \frac {4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 \sqrt {d+e x} \left (a-2 b n+b \log \left (c x^n\right )\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 61, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {2 \sqrt {e x +d}\, a +4 b n \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) \sqrt {d}+2 \ln \left (c \,x^{n}\right ) b \sqrt {e x +d}-4 b n \sqrt {e x +d}}{e}\)	\(61\)
default	\(\frac {2 \sqrt {e x +d}\, a +4 b n \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) \sqrt {d}+2 \ln \left (c \,x^{n}\right ) b \sqrt {e x +d}-4 b n \sqrt {e x +d}}{e}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 84, normalized size = 1.22 \begin {gather*} -2 \, {\left (\sqrt {d} \log \left (\frac {\sqrt {x e + d} - \sqrt {d}}{\sqrt {x e + d} + \sqrt {d}}\right ) + 2 \, \sqrt {x e + d}\right )} b n e^{\left (-1\right )} + 2 \, \sqrt {x e + d} b e^{\left (-1\right )} \log \left (c x^{n}\right ) + 2 \, \sqrt {x e + d} a e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 119, normalized size = 1.72 \begin {gather*} \left [2 \, {\left (b \sqrt {d} n \log \left (\frac {x e + 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}, -2 \, {\left (2 \, b \sqrt {-d} n \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) - {\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*(b*sqrt(d)*n*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + (b*n*log(x) - 2*b*n + b*log(c) + a)*sqrt(x*e +
d))*e^(-1), -2*(2*b*sqrt(-d)*n*arctan(sqrt(x*e + d)*sqrt(-d)/d) - (b*n*log(x) - 2*b*n + b*log(c) + a)*sqrt(x*e
+ d))*e^(-1)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (65) = 130\).
time = 10.99, size = 252, normalized size = 3.65 \begin {gather*} \begin {cases} \frac {- \frac {2 a d}{\sqrt {d + e x}} - 2 a \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 b d \left (\frac {\log {\left (c x^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - 2 b \left (- d \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a x + b \left (- n x + x \log {\left (c x^{n} \right )}\right )}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a*d/sqrt(d + e*x) - 2*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*b*d*(log(c*x**n)/sqrt(d + e*x) -
2*n*atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/(d*sqrt(-1/d))) - 2*b*(-d*(log(c*(-d/e + (d + e*x)/e)**n)/sqrt(d + e*x
) - 2*n*atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/(d*sqrt(-1/d))) - sqrt(d + e*x)*log(c*(-d/e + (d + e*x)/e)**n) - 2*
n*(-e*sqrt(d + e*x) - e*atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/sqrt(-1/d))/e))/e, Ne(e, 0)), ((a*x + b*(-n*x + x*l
og(c*x**n)))/sqrt(d), True))

________________________________________________________________________________________

Giac [A]
time = 3.21, size = 78, normalized size = 1.13 \begin {gather*} -2 \, {\left ({\left (\frac {2 \, d \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} - \sqrt {x e + d} \log \left (x\right ) + 2 \, \sqrt {x e + d}\right )} b n - \sqrt {x e + d} b \log \left (c\right ) - \sqrt {x e + d} a\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*d*arctan(sqrt(x*e + d)/sqrt(-d))/sqrt(-d) - sqrt(x*e + d)*log(x) + 2*sqrt(x*e + d))*b*n - sqrt(x*e + d)
*b*log(c) - sqrt(x*e + d)*a)*e^(-1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________