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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = 0.01, 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]

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 25, normalized size = 0.68 \[ \frac {2 \, {\left (b n \log \relax (x) - 2 \, b n + b \log \relax (c) + a\right )} \sqrt {d x}}{d} \]

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

________________________________________________________________________________________

giac [A]  time = 0.29, size = 41, normalized size = 1.11 \[ \frac {2 \, {\left ({\left (\sqrt {d x} \log \relax (x) - 2 \, \sqrt {d x}\right )} b n + \sqrt {d x} b \log \relax (c) + \sqrt {d x} a\right )}}{d} \]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 42, normalized size = 1.14 \[ -\frac {4 \sqrt {d x}\, b n}{d}+\frac {2 \sqrt {d x}\, b \ln \left (c \,x^{n}\right )}{d}+\frac {2 \sqrt {d x}\, a}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)/(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

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 41, normalized size = 1.11 \[ -\frac {4 \, \sqrt {d x} b n}{d} + \frac {2 \, \sqrt {d x} b \log \left (c x^{n}\right )}{d} + \frac {2 \, \sqrt {d x} a}{d} \]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d*x)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.77, size = 63, normalized size = 1.70 \[ \frac {2 a \sqrt {x}}{\sqrt {d}} + \frac {2 b n \sqrt {x} \log {\relax (x )}}{\sqrt {d}} - \frac {4 b n \sqrt {x}}{\sqrt {d}} + \frac {2 b \sqrt {x} \log {\relax (c )}}{\sqrt {d}} \]

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)

________________________________________________________________________________________