3.5.5 \(\int \frac {a+b \log (c (d+e \sqrt {x})^n)}{x^2} \, dx\) [405]

Optimal. Leaf size=70 \[ -\frac {b e n}{d \sqrt {x}}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}-\frac {b e^2 n \log (x)}{2 d^2} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 46} \begin {gather*} -\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {b e^2 n \log (x)}{2 d^2}-\frac {b e n}{d \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^2,x]

[Out]

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

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 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 \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+(b e n) \text {Subst}\left (\int \frac {1}{x^2 (d+e x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+(b e n) \text {Subst}\left (\int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {b e n}{d \sqrt {x}}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}-\frac {b e^2 n \log (x)}{2 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.96 \begin {gather*} -\frac {a}{x}-\frac {b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+b e n \left (-\frac {1}{d \sqrt {x}}+\frac {e \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {e \log (x)}{2 d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^2,x]

[Out]

-(a/x) - (b*Log[c*(d + e*Sqrt[x])^n])/x + b*e*n*(-(1/(d*Sqrt[x])) + (e*Log[d + e*Sqrt[x]])/d^2 - (e*Log[x])/(2
*d^2))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 66, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, b n {\left (\frac {2 \, e \log \left (\sqrt {x} e + d\right )}{d^{2}} - \frac {e \log \left (x\right )}{d^{2}} - \frac {2}{d \sqrt {x}}\right )} e - \frac {b \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )}{x} - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 64, normalized size = 0.91 \begin {gather*} -\frac {b n x e^{2} \log \left (\sqrt {x}\right ) + b d n \sqrt {x} e + b d^{2} \log \left (c\right ) + a d^{2} + {\left (b d^{2} n - b n x e^{2}\right )} \log \left (\sqrt {x} e + d\right )}{d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (66) = 132\).
time = 14.69, size = 410, normalized size = 5.86 \begin {gather*} \begin {cases} - \frac {2 a d^{3} \sqrt {x}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 a d^{2} e x}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{3} \sqrt {x} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e n x}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e x \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {b d e^{2} n x^{\frac {3}{2}} \log {\left (x \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d e^{2} n x^{\frac {3}{2}}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b d e^{2} x^{\frac {3}{2}} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {b e^{3} n x^{2} \log {\left (x \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b e^{3} x^{2} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} & \text {for}\: d \neq 0 \\- \frac {a}{x} - \frac {b n}{2 x} - \frac {b \log {\left (c \left (e \sqrt {x}\right )^{n} \right )}}{x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a*d**3*sqrt(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*a*d**2*e*x/(2*d**3*x**(3/2) + 2*d**2*e*x**2
) - 2*b*d**3*sqrt(x)*log(c*(d + e*sqrt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*n*x/(2*d**3*x**(
3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*x*log(c*(d + e*sqrt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - b*d*e**2*n*
x**(3/2)*log(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d*e**2*n*x**(3/2)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) +
2*b*d*e**2*x**(3/2)*log(c*(d + e*sqrt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - b*e**3*n*x**2*log(x)/(2*d**3
*x**(3/2) + 2*d**2*e*x**2) + 2*b*e**3*x**2*log(c*(d + e*sqrt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x**2), Ne(d,
0)), (-a/x - b*n/(2*x) - b*log(c*(e*sqrt(x))**n)/x, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (63) = 126\).
time = 3.94, size = 187, normalized size = 2.67 \begin {gather*} \frac {{\left ({\left (\sqrt {x} e + d\right )}^{2} b n e^{3} \log \left (\sqrt {x} e + d\right ) - 2 \, {\left (\sqrt {x} e + d\right )} b d n e^{3} \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} b n e^{3} \log \left (\sqrt {x} e\right ) + 2 \, {\left (\sqrt {x} e + d\right )} b d n e^{3} \log \left (\sqrt {x} e\right ) - b d^{2} n e^{3} \log \left (\sqrt {x} e\right ) - {\left (\sqrt {x} e + d\right )} b d n e^{3} + b d^{2} n e^{3} - b d^{2} e^{3} \log \left (c\right ) - a d^{2} e^{3}\right )} e^{\left (-1\right )}}{{\left (\sqrt {x} e + d\right )}^{2} d^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d^{3} + d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((sqrt(x)*e + d)^2*b*n*e^3*log(sqrt(x)*e + d) - 2*(sqrt(x)*e + d)*b*d*n*e^3*log(sqrt(x)*e + d) - (sqrt(x)*e +
d)^2*b*n*e^3*log(sqrt(x)*e) + 2*(sqrt(x)*e + d)*b*d*n*e^3*log(sqrt(x)*e) - b*d^2*n*e^3*log(sqrt(x)*e) - (sqrt(
x)*e + d)*b*d*n*e^3 + b*d^2*n*e^3 - b*d^2*e^3*log(c) - a*d^2*e^3)*e^(-1)/((sqrt(x)*e + d)^2*d^2 - 2*(sqrt(x)*e
+ d)*d^3 + d^4)

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Mupad [B]
time = 0.80, size = 58, normalized size = 0.83 \begin {gather*} \frac {2\,b\,e^2\,n\,\mathrm {atanh}\left (\frac {2\,e\,\sqrt {x}}{d}+1\right )}{d^2}-\frac {b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x}-\frac {b\,e\,n}{d\,\sqrt {x}}-\frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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