3.1.52 \(\int \frac {\log (c (a+b \sqrt {x})^p)}{x^3} \, dx\) [52]

Optimal. Leaf size=100 \[ -\frac {b p}{6 a x^{3/2}}+\frac {b^2 p}{4 a^2 x}-\frac {b^3 p}{2 a^3 \sqrt {x}}+\frac {b^4 p \log \left (a+b \sqrt {x}\right )}{2 a^4}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}-\frac {b^4 p \log (x)}{4 a^4} \]

[Out]

-1/6*b*p/a/x^(3/2)+1/4*b^2*p/a^2/x-1/4*b^4*p*ln(x)/a^4+1/2*b^4*p*ln(a+b*x^(1/2))/a^4-1/2*ln(c*(a+b*x^(1/2))^p)
/x^2-1/2*b^3*p/a^3/x^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 46} \begin {gather*} \frac {b^4 p \log \left (a+b \sqrt {x}\right )}{2 a^4}-\frac {b^4 p \log (x)}{4 a^4}-\frac {b^3 p}{2 a^3 \sqrt {x}}+\frac {b^2 p}{4 a^2 x}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}-\frac {b p}{6 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*Sqrt[x])^p]/x^3,x]

[Out]

-1/6*(b*p)/(a*x^(3/2)) + (b^2*p)/(4*a^2*x) - (b^3*p)/(2*a^3*Sqrt[x]) + (b^4*p*Log[a + b*Sqrt[x]])/(2*a^4) - Lo
g[c*(a + b*Sqrt[x])^p]/(2*x^2) - (b^4*p*Log[x])/(4*a^4)

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 {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx &=2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^5} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}+\frac {1}{2} (b p) \text {Subst}\left (\int \frac {1}{x^4 (a+b x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}+\frac {1}{2} (b p) \text {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {b p}{6 a x^{3/2}}+\frac {b^2 p}{4 a^2 x}-\frac {b^3 p}{2 a^3 \sqrt {x}}+\frac {b^4 p \log \left (a+b \sqrt {x}\right )}{2 a^4}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}-\frac {b^4 p \log (x)}{4 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 90, normalized size = 0.90 \begin {gather*} \frac {a b p \sqrt {x} \left (-2 a^2+3 a b \sqrt {x}-6 b^2 x\right )+6 b^4 p x^2 \log \left (a+b \sqrt {x}\right )-6 a^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right )-3 b^4 p x^2 \log (x)}{12 a^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*Sqrt[x])^p]/x^3,x]

[Out]

(a*b*p*Sqrt[x]*(-2*a^2 + 3*a*b*Sqrt[x] - 6*b^2*x) + 6*b^4*p*x^2*Log[a + b*Sqrt[x]] - 6*a^4*Log[c*(a + b*Sqrt[x
])^p] - 3*b^4*p*x^2*Log[x])/(12*a^4*x^2)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a+b*x^(1/2))^p)/x^3,x)

[Out]

int(ln(c*(a+b*x^(1/2))^p)/x^3,x)

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Maxima [A]
time = 0.35, size = 76, normalized size = 0.76 \begin {gather*} \frac {1}{12} \, b p {\left (\frac {6 \, b^{3} \log \left (b \sqrt {x} + a\right )}{a^{4}} - \frac {3 \, b^{3} \log \left (x\right )}{a^{4}} - \frac {6 \, b^{2} x - 3 \, a b \sqrt {x} + 2 \, a^{2}}{a^{3} x^{\frac {3}{2}}}\right )} - \frac {\log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b*x^(1/2))^p)/x^3,x, algorithm="maxima")

[Out]

1/12*b*p*(6*b^3*log(b*sqrt(x) + a)/a^4 - 3*b^3*log(x)/a^4 - (6*b^2*x - 3*a*b*sqrt(x) + 2*a^2)/(a^3*x^(3/2))) -
1/2*log((b*sqrt(x) + a)^p*c)/x^2

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Fricas [A]
time = 0.40, size = 84, normalized size = 0.84 \begin {gather*} -\frac {6 \, b^{4} p x^{2} \log \left (\sqrt {x}\right ) - 3 \, a^{2} b^{2} p x + 6 \, a^{4} \log \left (c\right ) - 6 \, {\left (b^{4} p x^{2} - a^{4} p\right )} \log \left (b \sqrt {x} + a\right ) + 2 \, {\left (3 \, a b^{3} p x + a^{3} b p\right )} \sqrt {x}}{12 \, a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b*x^(1/2))^p)/x^3,x, algorithm="fricas")

[Out]

-1/12*(6*b^4*p*x^2*log(sqrt(x)) - 3*a^2*b^2*p*x + 6*a^4*log(c) - 6*(b^4*p*x^2 - a^4*p)*log(b*sqrt(x) + a) + 2*
(3*a*b^3*p*x + a^3*b*p)*sqrt(x))/(a^4*x^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (90) = 180\).
time = 64.49, size = 403, normalized size = 4.03 \begin {gather*} \begin {cases} - \frac {6 a^{5} \sqrt {x} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {2 a^{4} b p x}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {6 a^{4} b x \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} + \frac {a^{3} b^{2} p x^{\frac {3}{2}}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {3 a^{2} b^{3} p x^{2}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {3 a b^{4} p x^{\frac {5}{2}} \log {\left (x \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {6 a b^{4} p x^{\frac {5}{2}}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} + \frac {6 a b^{4} x^{\frac {5}{2}} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {3 b^{5} p x^{3} \log {\left (x \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} + \frac {6 b^{5} x^{3} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} & \text {for}\: a \neq 0 \\- \frac {p}{8 x^{2}} - \frac {\log {\left (c \left (b \sqrt {x}\right )^{p} \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a+b*x**(1/2))**p)/x**3,x)

[Out]

Piecewise((-6*a**5*sqrt(x)*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 2*a**4*b*p*x/(12*a*
*5*x**(5/2) + 12*a**4*b*x**3) - 6*a**4*b*x*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) + a**
3*b**2*p*x**(3/2)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 3*a**2*b**3*p*x**2/(12*a**5*x**(5/2) + 12*a**4*b*x**3)
- 3*a*b**4*p*x**(5/2)*log(x)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 6*a*b**4*p*x**(5/2)/(12*a**5*x**(5/2) + 12
*a**4*b*x**3) + 6*a*b**4*x**(5/2)*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 3*b**5*p*x**
3*log(x)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) + 6*b**5*x**3*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a*
*4*b*x**3), Ne(a, 0)), (-p/(8*x**2) - log(c*(b*sqrt(x))**p)/(2*x**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (80) = 160\).
time = 3.99, size = 232, normalized size = 2.32 \begin {gather*} -\frac {\frac {6 \, b^{5} p \log \left (b \sqrt {x} + a\right )}{{\left (b \sqrt {x} + a\right )}^{4} - 4 \, {\left (b \sqrt {x} + a\right )}^{3} a + 6 \, {\left (b \sqrt {x} + a\right )}^{2} a^{2} - 4 \, {\left (b \sqrt {x} + a\right )} a^{3} + a^{4}} - \frac {6 \, b^{5} p \log \left (b \sqrt {x} + a\right )}{a^{4}} + \frac {6 \, b^{5} p \log \left (b \sqrt {x}\right )}{a^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{3} b^{5} p - 21 \, {\left (b \sqrt {x} + a\right )}^{2} a b^{5} p + 26 \, {\left (b \sqrt {x} + a\right )} a^{2} b^{5} p - 11 \, a^{3} b^{5} p + 6 \, a^{3} b^{5} \log \left (c\right )}{{\left (b \sqrt {x} + a\right )}^{4} a^{3} - 4 \, {\left (b \sqrt {x} + a\right )}^{3} a^{4} + 6 \, {\left (b \sqrt {x} + a\right )}^{2} a^{5} - 4 \, {\left (b \sqrt {x} + a\right )} a^{6} + a^{7}}}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b*x^(1/2))^p)/x^3,x, algorithm="giac")

[Out]

-1/12*(6*b^5*p*log(b*sqrt(x) + a)/((b*sqrt(x) + a)^4 - 4*(b*sqrt(x) + a)^3*a + 6*(b*sqrt(x) + a)^2*a^2 - 4*(b*
sqrt(x) + a)*a^3 + a^4) - 6*b^5*p*log(b*sqrt(x) + a)/a^4 + 6*b^5*p*log(b*sqrt(x))/a^4 + (6*(b*sqrt(x) + a)^3*b
^5*p - 21*(b*sqrt(x) + a)^2*a*b^5*p + 26*(b*sqrt(x) + a)*a^2*b^5*p - 11*a^3*b^5*p + 6*a^3*b^5*log(c))/((b*sqrt
(x) + a)^4*a^3 - 4*(b*sqrt(x) + a)^3*a^4 + 6*(b*sqrt(x) + a)^2*a^5 - 4*(b*sqrt(x) + a)*a^6 + a^7))/b

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Mupad [B]
time = 0.46, size = 72, normalized size = 0.72 \begin {gather*} \frac {b^4\,p\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^4}-\frac {\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{2\,x^2}-\frac {\frac {b\,p}{3\,a}-\frac {b^2\,p\,\sqrt {x}}{2\,a^2}+\frac {b^3\,p\,x}{a^3}}{2\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x^(1/2))^p)/x^3,x)

[Out]

(b^4*p*atanh((2*b*x^(1/2))/a + 1))/a^4 - log(c*(a + b*x^(1/2))^p)/(2*x^2) - ((b*p)/(3*a) - (b^2*p*x^(1/2))/(2*
a^2) + (b^3*p*x)/a^3)/(2*x^(3/2))

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