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

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

[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

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Rubi [A]  time = 0.03, 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 = {2319, 50, 63, 208} \[ \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {4 b n \sqrt {d+e x}}{e}+\frac {4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e} \]

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 50

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 63

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 208

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

Rule 2319

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) \operatorname {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*}

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Mathematica [A]  time = 0.04, size = 55, normalized size = 0.80 \[ \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )-2 b n\right )+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e} \]

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

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fricas [A]  time = 0.50, size = 116, normalized size = 1.68 \[ \left [\frac {2 \, {\left (b \sqrt {d} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (b n \log \relax (x) - 2 \, b n + b \log \relax (c) + a\right )} \sqrt {e x + d}\right )}}{e}, -\frac {2 \, {\left (2 \, b \sqrt {-d} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) - {\left (b n \log \relax (x) - 2 \, b n + b \log \relax (c) + a\right )} \sqrt {e x + d}\right )}}{e}\right ] \]

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

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giac [A]  time = 0.34, size = 78, normalized size = 1.13 \[ -2 \, {\left ({\left (\frac {2 \, d \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} - \sqrt {x e + d} \log \relax (x) + 2 \, \sqrt {x e + d}\right )} b n - \sqrt {x e + d} b \log \relax (c) - \sqrt {x e + d} a\right )} e^{\left (-1\right )} \]

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)

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maple [A]  time = 0.04, size = 70, normalized size = 1.01 \[ \frac {4 b \sqrt {d}\, n \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e}-\frac {4 \sqrt {e x +d}\, b n}{e}+\frac {2 \sqrt {e x +d}\, b \ln \left (c \,x^{n}\right )}{e}+\frac {2 \sqrt {e x +d}\, a}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 82, normalized size = 1.19 \[ -\frac {2 \, {\left (\sqrt {d} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right ) + 2 \, \sqrt {e x + d}\right )} b n}{e} + \frac {2 \, \sqrt {e x + d} b \log \left (c x^{n}\right )}{e} + \frac {2 \, \sqrt {e x + d} a}{e} \]

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

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

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)

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sympy [A]  time = 27.88, size = 252, normalized size = 3.65 \[ \begin {cases} \frac {- \frac {2 a d}{\sqrt {d + e x}} - 2 a \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 b d \left (\frac {\log {\left (c x^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - 2 b \left (- d \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a x + b \left (- n x + x \log {\left (c x^{n} \right )}\right )}{\sqrt {d}} & \text {otherwise} \end {cases} \]

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))

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