3.98 \(\int \frac {(a+b \log (c x^n))^2}{\sqrt {d x}} \, dx\)

Optimal. Leaf size=67 \[ -\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {16 b^2 n^2 \sqrt {d x}}{d} \]

[Out]

16*b^2*n^2*(d*x)^(1/2)/d-8*b*n*(a+b*ln(c*x^n))*(d*x)^(1/2)/d+2*(a+b*ln(c*x^n))^2*(d*x)^(1/2)/d

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2305, 2304} \[ -\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {16 b^2 n^2 \sqrt {d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^2/Sqrt[d*x],x]

[Out]

(16*b^2*n^2*Sqrt[d*x])/d - (8*b*n*Sqrt[d*x]*(a + b*Log[c*x^n]))/d + (2*Sqrt[d*x]*(a + b*Log[c*x^n])^2)/d

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}-(4 b n) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx\\ &=\frac {16 b^2 n^2 \sqrt {d x}}{d}-\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 54, normalized size = 0.81 \[ \frac {2 x \left (a^2+2 b (a-2 b n) \log \left (c x^n\right )-4 a b n+b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{\sqrt {d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(a^2 - 4*a*b*n + 8*b^2*n^2 + 2*b*(a - 2*b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2))/Sqrt[d*x]

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fricas [A]  time = 0.44, size = 87, normalized size = 1.30 \[ \frac {2 \, {\left (b^{2} n^{2} \log \relax (x)^{2} + 8 \, b^{2} n^{2} + b^{2} \log \relax (c)^{2} - 4 \, a b n + a^{2} - 2 \, {\left (2 \, b^{2} n - a b\right )} \log \relax (c) - 2 \, {\left (2 \, b^{2} n^{2} - b^{2} n \log \relax (c) - a b n\right )} \log \relax (x)\right )} \sqrt {d x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b^2*n^2*log(x)^2 + 8*b^2*n^2 + b^2*log(c)^2 - 4*a*b*n + a^2 - 2*(2*b^2*n - a*b)*log(c) - 2*(2*b^2*n^2 - b^2
*n*log(c) - a*b*n)*log(x))*sqrt(d*x)/d

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giac [A]  time = 0.43, size = 118, normalized size = 1.76 \[ \frac {2 \, {\left ({\left (\sqrt {d x} \log \relax (x)^{2} - 4 \, \sqrt {d x} \log \relax (x) + 8 \, \sqrt {d x}\right )} b^{2} n^{2} + 2 \, {\left (\sqrt {d x} \log \relax (x) - 2 \, \sqrt {d x}\right )} b^{2} n \log \relax (c) + \sqrt {d x} b^{2} \log \relax (c)^{2} + 2 \, {\left (\sqrt {d x} \log \relax (x) - 2 \, \sqrt {d x}\right )} a b n + 2 \, \sqrt {d x} a b \log \relax (c) + \sqrt {d x} a^{2}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((sqrt(d*x)*log(x)^2 - 4*sqrt(d*x)*log(x) + 8*sqrt(d*x))*b^2*n^2 + 2*(sqrt(d*x)*log(x) - 2*sqrt(d*x))*b^2*n*
log(c) + sqrt(d*x)*b^2*log(c)^2 + 2*(sqrt(d*x)*log(x) - 2*sqrt(d*x))*a*b*n + 2*sqrt(d*x)*a*b*log(c) + sqrt(d*x
)*a^2)/d

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maple [A]  time = 0.06, size = 107, normalized size = 1.60 \[ \frac {16 \sqrt {d x}\, b^{2} n^{2}}{d}-\frac {8 \sqrt {d x}\, b^{2} n \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{d}+\frac {2 \sqrt {d x}\, b^{2} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}}{d}-\frac {8 \sqrt {d x}\, a b n}{d}+\frac {4 \sqrt {d x}\, a b \ln \left (c \,x^{n}\right )}{d}+\frac {2 \sqrt {d x}\, a^{2}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*b^2*(d*x)^(1/2)*ln(c*exp(n*ln(x)))^2-8/d*b^2*n*(d*x)^(1/2)*ln(c*exp(n*ln(x)))+16*b^2*n^2*(d*x)^(1/2)/d+4/d
*(d*x)^(1/2)*a*b*ln(c*x^n)-8/d*(d*x)^(1/2)*a*b*n+2/d*(d*x)^(1/2)*a^2

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maxima [A]  time = 0.53, size = 102, normalized size = 1.52 \[ \frac {2 \, \sqrt {d x} b^{2} \log \left (c x^{n}\right )^{2}}{d} + 8 \, {\left (\frac {2 \, \sqrt {d x} n^{2}}{d} - \frac {\sqrt {d x} n \log \left (c x^{n}\right )}{d}\right )} b^{2} - \frac {8 \, \sqrt {d x} a b n}{d} + \frac {4 \, \sqrt {d x} a b \log \left (c x^{n}\right )}{d} + \frac {2 \, \sqrt {d x} a^{2}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(d*x)*b^2*log(c*x^n)^2/d + 8*(2*sqrt(d*x)*n^2/d - sqrt(d*x)*n*log(c*x^n)/d)*b^2 - 8*sqrt(d*x)*a*b*n/d +
4*sqrt(d*x)*a*b*log(c*x^n)/d + 2*sqrt(d*x)*a^2/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.46, size = 199, normalized size = 2.97 \[ \frac {2 a^{2} \sqrt {x}}{\sqrt {d}} + \frac {4 a b n \sqrt {x} \log {\relax (x )}}{\sqrt {d}} - \frac {8 a b n \sqrt {x}}{\sqrt {d}} + \frac {4 a b \sqrt {x} \log {\relax (c )}}{\sqrt {d}} + \frac {2 b^{2} n^{2} \sqrt {x} \log {\relax (x )}^{2}}{\sqrt {d}} - \frac {8 b^{2} n^{2} \sqrt {x} \log {\relax (x )}}{\sqrt {d}} + \frac {16 b^{2} n^{2} \sqrt {x}}{\sqrt {d}} + \frac {4 b^{2} n \sqrt {x} \log {\relax (c )} \log {\relax (x )}}{\sqrt {d}} - \frac {8 b^{2} n \sqrt {x} \log {\relax (c )}}{\sqrt {d}} + \frac {2 b^{2} \sqrt {x} \log {\relax (c )}^{2}}{\sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*sqrt(x)/sqrt(d) + 4*a*b*n*sqrt(x)*log(x)/sqrt(d) - 8*a*b*n*sqrt(x)/sqrt(d) + 4*a*b*sqrt(x)*log(c)/sqrt(
d) + 2*b**2*n**2*sqrt(x)*log(x)**2/sqrt(d) - 8*b**2*n**2*sqrt(x)*log(x)/sqrt(d) + 16*b**2*n**2*sqrt(x)/sqrt(d)
+ 4*b**2*n*sqrt(x)*log(c)*log(x)/sqrt(d) - 8*b**2*n*sqrt(x)*log(c)/sqrt(d) + 2*b**2*sqrt(x)*log(c)**2/sqrt(d)

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