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

Optimal. Leaf size=41 \[ \frac{2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d}-\frac{4 b n (d x)^{3/2}}{9 d} \]

[Out]

(-4*b*n*(d*x)^(3/2))/(9*d) + (2*(d*x)^(3/2)*(a + b*Log[c*x^n]))/(3*d)

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Rubi [A]  time = 0.013953, antiderivative size = 41, 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} \[ \frac{2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d}-\frac{4 b n (d x)^{3/2}}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*(a + b*Log[c*x^n]),x]

[Out]

(-4*b*n*(d*x)^(3/2))/(9*d) + (2*(d*x)^(3/2)*(a + b*Log[c*x^n]))/(3*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 \sqrt{d x} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{4 b n (d x)^{3/2}}{9 d}+\frac{2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0070536, size = 29, normalized size = 0.71 \[ \frac{2}{9} x \sqrt{d x} \left (3 a+3 b \log \left (c x^n\right )-2 b n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[d*x]*(3*a - 2*b*n + 3*b*Log[c*x^n]))/9

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Maple [C]  time = 0.08, size = 124, normalized size = 3. \begin{align*}{\frac{2\,bd{x}^{2}\ln \left ({x}^{n} \right ) }{3}{\frac{1}{\sqrt{dx}}}}+{\frac{d \left ( 3\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -3\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+3\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +6\,b\ln \left ( c \right ) -4\,bn+6\,a \right ){x}^{2}}{9}{\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d*b*x^2/(d*x)^(1/2)*ln(x^n)+1/9*d*(3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-3*I*b*Pi*csgn(I*c*x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)-4*b*n+6*a)*x^2/(d*x)^(1/2)

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Maxima [A]  time = 1.13549, size = 55, normalized size = 1.34 \begin{align*} -\frac{4 \, \left (d x\right )^{\frac{3}{2}} b n}{9 \, d} + \frac{2 \, \left (d x\right )^{\frac{3}{2}} b \log \left (c x^{n}\right )}{3 \, d} + \frac{2 \, \left (d x\right )^{\frac{3}{2}} a}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*(d*x)^(3/2)*b*n/d + 2/3*(d*x)^(3/2)*b*log(c*x^n)/d + 2/3*(d*x)^(3/2)*a/d

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Fricas [A]  time = 0.915775, size = 88, normalized size = 2.15 \begin{align*} \frac{2}{9} \,{\left (3 \, b n x \log \left (x\right ) + 3 \, b x \log \left (c\right ) -{\left (2 \, b n - 3 \, a\right )} x\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*b*n*x*log(x) + 3*b*x*log(c) - (2*b*n - 3*a)*x)*sqrt(d*x)

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Sympy [A]  time = 2.5384, size = 70, normalized size = 1.71 \begin{align*} \frac{2 a \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{2 b \sqrt{d} n x^{\frac{3}{2}} \log{\left (x \right )}}{3} - \frac{4 b \sqrt{d} n x^{\frac{3}{2}}}{9} + \frac{2 b \sqrt{d} x^{\frac{3}{2}} \log{\left (c \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(d)*x**(3/2)/3 + 2*b*sqrt(d)*n*x**(3/2)*log(x)/3 - 4*b*sqrt(d)*n*x**(3/2)/9 + 2*b*sqrt(d)*x**(3/2)*log
(c)/3

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Giac [C]  time = 1.55264, size = 142, normalized size = 3.46 \begin{align*} \left (\frac{1}{3} i + \frac{1}{3}\right ) \, \sqrt{2} b n x^{\frac{3}{2}} \sqrt{{\left | d \right |}} \cos \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) \log \left (x\right ) - \left (\frac{1}{3} i - \frac{1}{3}\right ) \, \sqrt{2} b n x^{\frac{3}{2}} \sqrt{{\left | d \right |}} \log \left (x\right ) \sin \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) - \left (\frac{2}{9} i + \frac{2}{9}\right ) \, \sqrt{2} b n x^{\frac{3}{2}} \sqrt{{\left | d \right |}} \cos \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) + \left (\frac{2}{9} i - \frac{2}{9}\right ) \, \sqrt{2} b n x^{\frac{3}{2}} \sqrt{{\left | d \right |}} \sin \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) + \frac{2}{3} \, b \sqrt{d} x^{\frac{3}{2}} \log \left (c\right ) + \frac{2}{3} \, a \sqrt{d} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/3*I + 1/3)*sqrt(2)*b*n*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) - (1/3*I - 1/3)*sqrt(2)*b*n*x^(3/2)*s
qrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/9*I + 2/9)*sqrt(2)*b*n*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) + (2
/9*I - 2/9)*sqrt(2)*b*n*x^(3/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/3*b*sqrt(d)*x^(3/2)*log(c) + 2/3*a*sqrt(d)
*x^(3/2)