3.51 \(\int \frac{\log (c (a+b \sqrt{x})^p)}{x^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{b^2 p \log \left (a+b \sqrt{x}\right )}{a^2}-\frac{b^2 p \log (x)}{2 a^2}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x}-\frac{b p}{a \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0452999, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2454, 2395, 44} \[ \frac{b^2 p \log \left (a+b \sqrt{x}\right )}{a^2}-\frac{b^2 p \log (x)}{2 a^2}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x}-\frac{b p}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*Sqrt[x])^p]/x^2,x]

[Out]

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

Rule 2454

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])

Rule 2395

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 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x}+(b p) \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x}+(b p) \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{b p}{a \sqrt{x}}+\frac{b^2 p \log \left (a+b \sqrt{x}\right )}{a^2}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x}-\frac{b^2 p \log (x)}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.039548, size = 55, normalized size = 0.87 \[ -\frac{b p \left (-2 b \log \left (a+b \sqrt{x}\right )+\frac{2 a}{\sqrt{x}}+b \log (x)\right )}{2 a^2}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\ln \left ( c \left ( a+b\sqrt{x} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a+b*x^(1/2))^p)/x^2,x)

[Out]

int(ln(c*(a+b*x^(1/2))^p)/x^2,x)

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Maxima [A]  time = 1.03496, size = 72, normalized size = 1.14 \begin{align*} \frac{1}{2} \, b p{\left (\frac{2 \, b \log \left (b \sqrt{x} + a\right )}{a^{2}} - \frac{b \log \left (x\right )}{a^{2}} - \frac{2}{a \sqrt{x}}\right )} - \frac{\log \left ({\left (b \sqrt{x} + a\right )}^{p} c\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*p*(2*b*log(b*sqrt(x) + a)/a^2 - b*log(x)/a^2 - 2/(a*sqrt(x))) - log((b*sqrt(x) + a)^p*c)/x

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Fricas [A]  time = 2.2627, size = 136, normalized size = 2.16 \begin{align*} -\frac{b^{2} p x \log \left (\sqrt{x}\right ) + a b p \sqrt{x} + a^{2} \log \left (c\right ) -{\left (b^{2} p x - a^{2} p\right )} \log \left (b \sqrt{x} + a\right )}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*p*x*log(sqrt(x)) + a*b*p*sqrt(x) + a^2*log(c) - (b^2*p*x - a^2*p)*log(b*sqrt(x) + a))/(a^2*x)

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Sympy [A]  time = 91.8096, size = 406, normalized size = 6.44 \begin{align*} \begin{cases} - \frac{a^{3} p \sqrt{x} \log{\left (a + b \sqrt{x} \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} - \frac{a^{3} \sqrt{x} \log{\left (c \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} - \frac{a^{2} b p x \log{\left (a + b \sqrt{x} \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} - \frac{a^{2} b p x}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} - \frac{a^{2} b x \log{\left (c \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} - \frac{a b^{2} p x^{\frac{3}{2}} \log{\left (x \right )}}{2 \left (a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}\right )} + \frac{a b^{2} p x^{\frac{3}{2}} \log{\left (a + b \sqrt{x} \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} + \frac{a b^{2} x^{\frac{3}{2}} \log{\left (c \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} - \frac{b^{3} p x^{2} \log{\left (x \right )}}{2 \left (a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}\right )} + \frac{b^{3} p x^{2} \log{\left (a + b \sqrt{x} \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} + \frac{b^{3} p x^{2}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} + \frac{b^{3} x^{2} \log{\left (c \right )}}{a^{3} x^{\frac{3}{2}} + a^{2} b x^{2}} & \text{for}\: a \neq 0 \\- \frac{p \log{\left (b \right )}}{x} - \frac{p \log{\left (x \right )}}{2 x} - \frac{p}{2 x} - \frac{\log{\left (c \right )}}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a+b*x**(1/2))**p)/x**2,x)

[Out]

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

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Giac [B]  time = 1.20244, size = 178, normalized size = 2.83 \begin{align*} -\frac{\frac{b^{3} p \log \left (b \sqrt{x} + a\right )}{{\left (b \sqrt{x} + a\right )}^{2} - 2 \,{\left (b \sqrt{x} + a\right )} a + a^{2}} - \frac{b^{3} p \log \left (b \sqrt{x} + a\right )}{a^{2}} + \frac{b^{3} p \log \left (b \sqrt{x}\right )}{a^{2}} + \frac{{\left (b \sqrt{x} + a\right )} b^{3} p - a b^{3} p + a b^{3} \log \left (c\right )}{{\left (b \sqrt{x} + a\right )}^{2} a - 2 \,{\left (b \sqrt{x} + a\right )} a^{2} + a^{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^3*p*log(b*sqrt(x) + a)/((b*sqrt(x) + a)^2 - 2*(b*sqrt(x) + a)*a + a^2) - b^3*p*log(b*sqrt(x) + a)/a^2 + b^
3*p*log(b*sqrt(x))/a^2 + ((b*sqrt(x) + a)*b^3*p - a*b^3*p + a*b^3*log(c))/((b*sqrt(x) + a)^2*a - 2*(b*sqrt(x)
+ a)*a^2 + a^3))/b