3.90 \(\int (d x)^{3/2} (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=41 \[ \frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d}-\frac{4 b n (d x)^{5/2}}{25 d} \]

[Out]

(-4*b*n*(d*x)^(5/2))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(5*d)

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Rubi [A]  time = 0.0159575, 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} \[ \frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d}-\frac{4 b n (d x)^{5/2}}{25 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(3/2)*(a + b*Log[c*x^n]),x]

[Out]

(-4*b*n*(d*x)^(5/2))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(5*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 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{4 b n (d x)^{5/2}}{25 d}+\frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0100356, size = 29, normalized size = 0.71 \[ \frac{2}{25} x (d x)^{3/2} \left (5 a+5 b \log \left (c x^n\right )-2 b n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(3/2)*(a + b*Log[c*x^n]),x]

[Out]

(2*x*(d*x)^(3/2)*(5*a - 2*b*n + 5*b*Log[c*x^n]))/25

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Maple [C]  time = 0.082, size = 128, normalized size = 3.1 \begin{align*}{\frac{2\,b{d}^{2}{x}^{3}\ln \left ({x}^{n} \right ) }{5}{\frac{1}{\sqrt{dx}}}}+{\frac{{d}^{2} \left ( 5\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-5\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -5\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+5\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +10\,b\ln \left ( c \right ) -4\,bn+10\,a \right ){x}^{3}}{25}{\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(3/2)*(a+b*ln(c*x^n)),x)

[Out]

2/5*d^2*b*x^3/(d*x)^(1/2)*ln(x^n)+1/25*d^2*(5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-5*I*b*Pi*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)-5*I*b*Pi*csgn(I*c*x^n)^3+5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+10*b*ln(c)-4*b*n+10*a)*x^3/(d*x)^(
1/2)

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Maxima [A]  time = 1.01124, size = 55, normalized size = 1.34 \begin{align*} -\frac{4 \, \left (d x\right )^{\frac{5}{2}} b n}{25 \, d} + \frac{2 \, \left (d x\right )^{\frac{5}{2}} b \log \left (c x^{n}\right )}{5 \, d} + \frac{2 \, \left (d x\right )^{\frac{5}{2}} a}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-4/25*(d*x)^(5/2)*b*n/d + 2/5*(d*x)^(5/2)*b*log(c*x^n)/d + 2/5*(d*x)^(5/2)*a/d

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Fricas [A]  time = 0.934176, size = 108, normalized size = 2.63 \begin{align*} \frac{2}{25} \,{\left (5 \, b d n x^{2} \log \left (x\right ) + 5 \, b d x^{2} \log \left (c\right ) -{\left (2 \, b d n - 5 \, a d\right )} x^{2}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

2/25*(5*b*d*n*x^2*log(x) + 5*b*d*x^2*log(c) - (2*b*d*n - 5*a*d)*x^2)*sqrt(d*x)

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Sympy [A]  time = 46.0505, size = 70, normalized size = 1.71 \begin{align*} \frac{2 a d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{2 b d^{\frac{3}{2}} n x^{\frac{5}{2}} \log{\left (x \right )}}{5} - \frac{4 b d^{\frac{3}{2}} n x^{\frac{5}{2}}}{25} + \frac{2 b d^{\frac{3}{2}} x^{\frac{5}{2}} \log{\left (c \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(3/2)*(a+b*ln(c*x**n)),x)

[Out]

2*a*d**(3/2)*x**(5/2)/5 + 2*b*d**(3/2)*n*x**(5/2)*log(x)/5 - 4*b*d**(3/2)*n*x**(5/2)/25 + 2*b*d**(3/2)*x**(5/2
)*log(c)/5

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Giac [C]  time = 1.52479, size = 146, normalized size = 3.56 \begin{align*} -\frac{1}{25} \,{\left (-\left (5 i + 5\right ) \, \sqrt{2} b n x^{\frac{5}{2}} \sqrt{{\left | d \right |}} \cos \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) \log \left (x\right ) + \left (5 i - 5\right ) \, \sqrt{2} b n x^{\frac{5}{2}} \sqrt{{\left | d \right |}} \log \left (x\right ) \sin \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) + \left (2 i + 2\right ) \, \sqrt{2} b n x^{\frac{5}{2}} \sqrt{{\left | d \right |}} \cos \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) - \left (2 i - 2\right ) \, \sqrt{2} b n x^{\frac{5}{2}} \sqrt{{\left | d \right |}} \sin \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) - 10 \, b \sqrt{d} x^{\frac{5}{2}} \log \left (c\right ) - 10 \, a \sqrt{d} x^{\frac{5}{2}}\right )} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

-1/25*(-(5*I + 5)*sqrt(2)*b*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) + (5*I - 5)*sqrt(2)*b*n*x^(5/2)*s
qrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) + (2*I + 2)*sqrt(2)*b*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) - (2*I -
2)*sqrt(2)*b*n*x^(5/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) - 10*b*sqrt(d)*x^(5/2)*log(c) - 10*a*sqrt(d)*x^(5/2))*
d