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

Optimal. Leaf size=73 \[ \frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac{8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac{16 b^2 n^2 (d x)^{5/2}}{125 d} \]

[Out]

(16*b^2*n^2*(d*x)^(5/2))/(125*d) - (8*b*n*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c
*x^n])^2)/(5*d)

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Rubi [A]  time = 0.0464597, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2305, 2304} \[ \frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac{8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac{16 b^2 n^2 (d x)^{5/2}}{125 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b^2*n^2*(d*x)^(5/2))/(125*d) - (8*b*n*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c
*x^n])^2)/(5*d)

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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 )^2 \, dx &=\frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac{1}{5} (4 b n) \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{16 b^2 n^2 (d x)^{5/2}}{125 d}-\frac{8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac{2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0177044, size = 61, normalized size = 0.84 \[ \frac{2}{125} x (d x)^{3/2} \left (25 a^2+10 b (5 a-2 b n) \log \left (c x^n\right )-20 a b n+25 b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(3/2)*(25*a^2 - 20*a*b*n + 8*b^2*n^2 + 10*b*(5*a - 2*b*n)*Log[c*x^n] + 25*b^2*Log[c*x^n]^2))/125

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Maple [C]  time = 0.134, size = 716, normalized size = 9.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*d^2*b^2*x^3/(d*x)^(1/2)*ln(x^n)^2+2/25*d^2*b*x^3*(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)/(
d*x)^(1/2)*ln(x^n)+1/250*d^2*(-100*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3+40*I*Pi*b^2*n*csgn(I*c*x^n)^3-100*I*Pi*a*b*c
sgn(I*c*x^n)^3+100*ln(c)^2*b^2-25*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-80*a*b*n+32*b^2*n^2+100*a^2+50*Pi^2*b^2
*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+50*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-25*Pi^2*b^2*csgn(
I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-100*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+100*I*Pi*a*b*csgn(I*c*
x^n)^2*csgn(I*c)-40*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-40*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+100*I*ln(c)
*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-25*Pi^2*b^2*csgn(I*c*x^n)^6+200*ln(c)*a*b-80*ln(c)*b^2*n+50*Pi^2*b^2*csgn(
I*c*x^n)^5*csgn(I*c)+50*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-100*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+
100*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+100*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+40*I*Pi*b^2*n*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)-100*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-25*Pi^2*b^2*csgn(I*x^n)^2*csg
n(I*c*x^n)^4)*x^3/(d*x)^(1/2)

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Maxima [A]  time = 1.10065, size = 138, normalized size = 1.89 \begin{align*} \frac{2 \, \left (d x\right )^{\frac{5}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{5 \, d} - \frac{8 \, \left (d x\right )^{\frac{5}{2}} a b n}{25 \, d} + \frac{4 \, \left (d x\right )^{\frac{5}{2}} a b \log \left (c x^{n}\right )}{5 \, d} + \frac{2 \, \left (d x\right )^{\frac{5}{2}} a^{2}}{5 \, d} + \frac{8}{125} \,{\left (\frac{2 \, \left (d x\right )^{\frac{5}{2}} n^{2}}{d} - \frac{5 \, \left (d x\right )^{\frac{5}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(d*x)^(5/2)*b^2*log(c*x^n)^2/d - 8/25*(d*x)^(5/2)*a*b*n/d + 4/5*(d*x)^(5/2)*a*b*log(c*x^n)/d + 2/5*(d*x)^(
5/2)*a^2/d + 8/125*(2*(d*x)^(5/2)*n^2/d - 5*(d*x)^(5/2)*n*log(c*x^n)/d)*b^2

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Fricas [A]  time = 0.898863, size = 294, normalized size = 4.03 \begin{align*} \frac{2}{125} \,{\left (25 \, b^{2} d n^{2} x^{2} \log \left (x\right )^{2} + 25 \, b^{2} d x^{2} \log \left (c\right )^{2} - 10 \,{\left (2 \, b^{2} d n - 5 \, a b d\right )} x^{2} \log \left (c\right ) +{\left (8 \, b^{2} d n^{2} - 20 \, a b d n + 25 \, a^{2} d\right )} x^{2} + 10 \,{\left (5 \, b^{2} d n x^{2} \log \left (c\right ) -{\left (2 \, b^{2} d n^{2} - 5 \, a b d n\right )} x^{2}\right )} \log \left (x\right )\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))^2,x, algorithm="fricas")

[Out]

2/125*(25*b^2*d*n^2*x^2*log(x)^2 + 25*b^2*d*x^2*log(c)^2 - 10*(2*b^2*d*n - 5*a*b*d)*x^2*log(c) + (8*b^2*d*n^2
- 20*a*b*d*n + 25*a^2*d)*x^2 + 10*(5*b^2*d*n*x^2*log(c) - (2*b^2*d*n^2 - 5*a*b*d*n)*x^2)*log(x))*sqrt(d*x)

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Sympy [B]  time = 124.624, size = 216, normalized size = 2.96 \begin{align*} \frac{2 a^{2} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a b d^{\frac{3}{2}} n x^{\frac{5}{2}} \log{\left (x \right )}}{5} - \frac{8 a b d^{\frac{3}{2}} n x^{\frac{5}{2}}}{25} + \frac{4 a b d^{\frac{3}{2}} x^{\frac{5}{2}} \log{\left (c \right )}}{5} + \frac{2 b^{2} d^{\frac{3}{2}} n^{2} x^{\frac{5}{2}} \log{\left (x \right )}^{2}}{5} - \frac{8 b^{2} d^{\frac{3}{2}} n^{2} x^{\frac{5}{2}} \log{\left (x \right )}}{25} + \frac{16 b^{2} d^{\frac{3}{2}} n^{2} x^{\frac{5}{2}}}{125} + \frac{4 b^{2} d^{\frac{3}{2}} n x^{\frac{5}{2}} \log{\left (c \right )} \log{\left (x \right )}}{5} - \frac{8 b^{2} d^{\frac{3}{2}} n x^{\frac{5}{2}} \log{\left (c \right )}}{25} + \frac{2 b^{2} d^{\frac{3}{2}} x^{\frac{5}{2}} \log{\left (c \right )}^{2}}{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))**2,x)

[Out]

2*a**2*d**(3/2)*x**(5/2)/5 + 4*a*b*d**(3/2)*n*x**(5/2)*log(x)/5 - 8*a*b*d**(3/2)*n*x**(5/2)/25 + 4*a*b*d**(3/2
)*x**(5/2)*log(c)/5 + 2*b**2*d**(3/2)*n**2*x**(5/2)*log(x)**2/5 - 8*b**2*d**(3/2)*n**2*x**(5/2)*log(x)/25 + 16
*b**2*d**(3/2)*n**2*x**(5/2)/125 + 4*b**2*d**(3/2)*n*x**(5/2)*log(c)*log(x)/5 - 8*b**2*d**(3/2)*n*x**(5/2)*log
(c)/25 + 2*b**2*d**(3/2)*x**(5/2)*log(c)**2/5

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Giac [C]  time = 1.94437, size = 521, normalized size = 7.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/125*(-(25*I + 25)*sqrt(2)*b^2*n^2*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x)^2 + (25*I - 25)*sqrt(2)*b^
2*n^2*x^(5/2)*sqrt(abs(d))*log(x)^2*sin(1/4*pi*sgn(d)) + (20*I + 20)*sqrt(2)*b^2*n^2*x^(5/2)*sqrt(abs(d))*cos(
1/4*pi*sgn(d))*log(x) - (50*I + 50)*sqrt(2)*b^2*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(c)*log(x) - (20*
I - 20)*sqrt(2)*b^2*n^2*x^(5/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) + (50*I - 50)*sqrt(2)*b^2*n*x^(5/2)*sqr
t(abs(d))*log(c)*log(x)*sin(1/4*pi*sgn(d)) - (8*I + 8)*sqrt(2)*b^2*n^2*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))
+ (20*I + 20)*sqrt(2)*b^2*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(c) - (50*I + 50)*sqrt(2)*a*b*n*x^(5/2
)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) + (8*I - 8)*sqrt(2)*b^2*n^2*x^(5/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) -
(20*I - 20)*sqrt(2)*b^2*n*x^(5/2)*sqrt(abs(d))*log(c)*sin(1/4*pi*sgn(d)) + (50*I - 50)*sqrt(2)*a*b*n*x^(5/2)*
sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) + (20*I + 20)*sqrt(2)*a*b*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) - (
20*I - 20)*sqrt(2)*a*b*n*x^(5/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) - 50*b^2*sqrt(d)*x^(5/2)*log(c)^2 - 100*a*b*s
qrt(d)*x^(5/2)*log(c) - 50*a^2*sqrt(d)*x^(5/2))*d