3.89 \(\int (d x)^{5/2} (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=41 \[ \frac{2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac{4 b n (d x)^{7/2}}{49 d} \]

[Out]

(-4*b*n*(d*x)^(7/2))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(7*d)

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Rubi [A]  time = 0.0158954, 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} \[ \frac{2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac{4 b n (d x)^{7/2}}{49 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*(a + b*Log[c*x^n]),x]

[Out]

(-4*b*n*(d*x)^(7/2))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(7*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)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{4 b n (d x)^{7/2}}{49 d}+\frac{2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.0135287, size = 29, normalized size = 0.71 \[ \frac{2}{49} x (d x)^{5/2} \left (7 a+7 b \log \left (c x^n\right )-2 b n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*(a + b*Log[c*x^n]),x]

[Out]

(2*x*(d*x)^(5/2)*(7*a - 2*b*n + 7*b*Log[c*x^n]))/49

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Maple [C]  time = 0.108, size = 128, normalized size = 3.1 \begin{align*}{\frac{2\,{d}^{3}b{x}^{4}\ln \left ({x}^{n} \right ) }{7}{\frac{1}{\sqrt{dx}}}}+{\frac{{d}^{3} \left ( 7\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-7\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -7\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+7\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +14\,b\ln \left ( c \right ) -4\,bn+14\,a \right ){x}^{4}}{49}{\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(a+b*ln(c*x^n)),x)

[Out]

2/7*d^3*b*x^4/(d*x)^(1/2)*ln(x^n)+1/49*d^3*(7*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-7*I*b*Pi*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)-7*I*b*Pi*csgn(I*c*x^n)^3+7*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+14*b*ln(c)-4*b*n+14*a)*x^4/(d*x)^(
1/2)

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Maxima [A]  time = 1.19304, size = 55, normalized size = 1.34 \begin{align*} -\frac{4 \, \left (d x\right )^{\frac{7}{2}} b n}{49 \, d} + \frac{2 \, \left (d x\right )^{\frac{7}{2}} b \log \left (c x^{n}\right )}{7 \, d} + \frac{2 \, \left (d x\right )^{\frac{7}{2}} a}{7 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-4/49*(d*x)^(7/2)*b*n/d + 2/7*(d*x)^(7/2)*b*log(c*x^n)/d + 2/7*(d*x)^(7/2)*a/d

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Fricas [A]  time = 0.913255, size = 119, normalized size = 2.9 \begin{align*} \frac{2}{49} \,{\left (7 \, b d^{2} n x^{3} \log \left (x\right ) + 7 \, b d^{2} x^{3} \log \left (c\right ) -{\left (2 \, b d^{2} n - 7 \, a d^{2}\right )} x^{3}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

2/49*(7*b*d^2*n*x^3*log(x) + 7*b*d^2*x^3*log(c) - (2*b*d^2*n - 7*a*d^2)*x^3)*sqrt(d*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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Giac [C]  time = 1.43687, size = 158, normalized size = 3.85 \begin{align*} \left (\frac{1}{7} i + \frac{1}{7}\right ) \, \sqrt{2} b d^{2} n x^{\frac{7}{2}} \sqrt{{\left | d \right |}} \cos \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) \log \left (x\right ) - \left (\frac{1}{7} i - \frac{1}{7}\right ) \, \sqrt{2} b d^{2} n x^{\frac{7}{2}} \sqrt{{\left | d \right |}} \log \left (x\right ) \sin \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) - \left (\frac{2}{49} i + \frac{2}{49}\right ) \, \sqrt{2} b d^{2} n x^{\frac{7}{2}} \sqrt{{\left | d \right |}} \cos \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) + \left (\frac{2}{49} i - \frac{2}{49}\right ) \, \sqrt{2} b d^{2} n x^{\frac{7}{2}} \sqrt{{\left | d \right |}} \sin \left (\frac{1}{4} \, \pi \mathrm{sgn}\left (d\right )\right ) + \frac{2}{7} \, b d^{\frac{5}{2}} x^{\frac{7}{2}} \log \left (c\right ) + \frac{2}{7} \, a d^{\frac{5}{2}} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

(1/7*I + 1/7)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) - (1/7*I - 1/7)*sqrt(2)*b*d^2*n*x
^(7/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/49*I + 2/49)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*p
i*sgn(d)) + (2/49*I - 2/49)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/7*b*d^(5/2)*x^(7/2)*lo
g(c) + 2/7*a*d^(5/2)*x^(7/2)