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*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

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Rubi [A]  time = 0.0409136, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 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]

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
]

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*}

Mathematica [A]  time = 0.0132299, 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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Maple [A]  time = 0.054, size = 107, normalized size = 1.6 \begin{align*} 2\,{\frac{{b}^{2}\sqrt{dx} \left ( \ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) \right ) ^{2}}{d}}-8\,{\frac{{b}^{2}n\sqrt{dx}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{d}}+16\,{\frac{{b}^{2}{n}^{2}\sqrt{dx}}{d}}+4\,{\frac{\sqrt{dx}ab\ln \left ( c{x}^{n} \right ) }{d}}-8\,{\frac{\sqrt{dx}abn}{d}}+2\,{\frac{\sqrt{dx}{a}^{2}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^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 = 1.18133, size = 138, normalized size = 2.06 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 0.956475, size = 203, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} - 4 \, a b n + a^{2} - 2 \,{\left (2 \, b^{2} n - a b\right )} \log \left (c\right ) - 2 \,{\left (2 \, b^{2} n^{2} - b^{2} n \log \left (c\right ) - a b n\right )} \log \left (x\right )\right )} \sqrt{d x}}{d} \end{align*}

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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Sympy [B]  time = 4.27413, size = 199, normalized size = 2.97 \begin{align*} \frac{2 a^{2} \sqrt{x}}{\sqrt{d}} + \frac{4 a b n \sqrt{x} \log{\left (x \right )}}{\sqrt{d}} - \frac{8 a b n \sqrt{x}}{\sqrt{d}} + \frac{4 a b \sqrt{x} \log{\left (c \right )}}{\sqrt{d}} + \frac{2 b^{2} n^{2} \sqrt{x} \log{\left (x \right )}^{2}}{\sqrt{d}} - \frac{8 b^{2} n^{2} \sqrt{x} \log{\left (x \right )}}{\sqrt{d}} + \frac{16 b^{2} n^{2} \sqrt{x}}{\sqrt{d}} + \frac{4 b^{2} n \sqrt{x} \log{\left (c \right )} \log{\left (x \right )}}{\sqrt{d}} - \frac{8 b^{2} n \sqrt{x} \log{\left (c \right )}}{\sqrt{d}} + \frac{2 b^{2} \sqrt{x} \log{\left (c \right )}^{2}}{\sqrt{d}} \end{align*}

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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Giac [A]  time = 1.30331, size = 159, normalized size = 2.37 \begin{align*} \frac{2 \,{\left ({\left (\sqrt{d x} \log \left (x\right )^{2} - 4 \, \sqrt{d x} \log \left (x\right ) + 8 \, \sqrt{d x}\right )} b^{2} n^{2} + 2 \,{\left (\sqrt{d x} \log \left (x\right ) - 2 \, \sqrt{d x}\right )} b^{2} n \log \left (c\right ) + \sqrt{d x} b^{2} \log \left (c\right )^{2} + 2 \,{\left (\sqrt{d x} \log \left (x\right ) - 2 \, \sqrt{d x}\right )} a b n + 2 \, \sqrt{d x} a b \log \left (c\right ) + \sqrt{d x} a^{2}\right )}}{d} \end{align*}

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