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

Optimal. Leaf size=37 \[ \frac{2 \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{4 b n \sqrt{d x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.0145035, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2304} \[ \frac{2 \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{4 b n \sqrt{d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/Sqrt[d*x],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.005951, size = 24, normalized size = 0.65 \[ \frac{2 x \left (a+b \log \left (c x^n\right )-2 b n\right )}{\sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 42, normalized size = 1.1 \begin{align*} 2\,{\frac{\sqrt{dx}b\ln \left ( c{x}^{n} \right ) }{d}}-4\,{\frac{bn\sqrt{dx}}{d}}+2\,{\frac{\sqrt{dx}a}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17155, size = 55, normalized size = 1.49 \begin{align*} -\frac{4 \, \sqrt{d x} b n}{d} + \frac{2 \, \sqrt{d x} b \log \left (c x^{n}\right )}{d} + \frac{2 \, \sqrt{d x} a}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.869924, size = 69, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\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))/(d*x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.6799, size = 63, normalized size = 1.7 \begin{align*} \frac{2 a \sqrt{x}}{\sqrt{d}} + \frac{2 b n \sqrt{x} \log{\left (x \right )}}{\sqrt{d}} - \frac{4 b n \sqrt{x}}{\sqrt{d}} + \frac{2 b \sqrt{x} \log{\left (c \right )}}{\sqrt{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.37406, size = 55, normalized size = 1.49 \begin{align*} \frac{2 \,{\left ({\left (\sqrt{d x} \log \left (x\right ) - 2 \, \sqrt{d x}\right )} b n + \sqrt{d x} b \log \left (c\right ) + \sqrt{d x} a\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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