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

Optimal. Leaf size=73 \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac{16 b^2 n^2}{27 d (d x)^{3/2}} \]

[Out]

(-16*b^2*n^2)/(27*d*(d*x)^(3/2)) - (8*b*n*(a + b*Log[c*x^n]))/(9*d*(d*x)^(3/2)) - (2*(a + b*Log[c*x^n])^2)/(3*
d*(d*x)^(3/2))

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Rubi [A]  time = 0.0464623, 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} \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac{16 b^2 n^2}{27 d (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*b^2*n^2)/(27*d*(d*x)^(3/2)) - (8*b*n*(a + b*Log[c*x^n]))/(9*d*(d*x)^(3/2)) - (2*(a + b*Log[c*x^n])^2)/(3*
d*(d*x)^(3/2))

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 \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx &=-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}+\frac{1}{3} (4 b n) \int \frac{a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx\\ &=-\frac{16 b^2 n^2}{27 d (d x)^{3/2}}-\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0146199, size = 61, normalized size = 0.84 \[ -\frac{2 x \left (9 a^2+6 b (3 a+2 b n) \log \left (c x^n\right )+12 a b n+9 b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{27 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*(9*a^2 + 12*a*b*n + 8*b^2*n^2 + 6*b*(3*a + 2*b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2))/(27*(d*x)^(5/2))

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Maple [C]  time = 0.137, 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((a+b*ln(c*x^n))^2/(d*x)^(5/2),x)

[Out]

-2/3/d^2*b^2/x/(d*x)^(1/2)*ln(x^n)^2-2/9/d^2*b*(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/(d*x)^(
1/2)*ln(x^n)-1/54/d^2*(36*ln(c)^2*b^2-9*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2+48*a*b*n+32*b^2*n^2+36*a^2+36*I*P
i*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)+18*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^
3*csgn(I*c)^2+18*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-9*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csg
n(I*c)^2-36*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-24*I*Pi*b^2*n*csgn(I*c*x^n)^3-36*I*ln(c)*Pi*b^2*csg
n(I*c*x^n)^3-36*I*Pi*a*b*csgn(I*c*x^n)^3+36*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*ln(c)*Pi*b^2*csgn(
I*c*x^n)^2*csgn(I*c)-9*Pi^2*b^2*csgn(I*c*x^n)^6+72*ln(c)*a*b+48*ln(c)*b^2*n-36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)+18*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+18*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-24*I*Pi*b^2*n*csgn
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+24*I*Pi*b^2*n*csgn(I*x^n
)*csgn(I*c*x^n)^2+24*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-9*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4)/x/(d*x)^(1
/2)

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Maxima [A]  time = 1.13115, size = 138, normalized size = 1.89 \begin{align*} -\frac{8}{27} \, b^{2}{\left (\frac{2 \, n^{2}}{\left (d x\right )^{\frac{3}{2}} d} + \frac{3 \, n \log \left (c x^{n}\right )}{\left (d x\right )^{\frac{3}{2}} d}\right )} - \frac{2 \, b^{2} \log \left (c x^{n}\right )^{2}}{3 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{8 \, a b n}{9 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{4 \, a b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{2 \, a^{2}}{3 \, \left (d x\right )^{\frac{3}{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/27*b^2*(2*n^2/((d*x)^(3/2)*d) + 3*n*log(c*x^n)/((d*x)^(3/2)*d)) - 2/3*b^2*log(c*x^n)^2/((d*x)^(3/2)*d) - 8/
9*a*b*n/((d*x)^(3/2)*d) - 4/3*a*b*log(c*x^n)/((d*x)^(3/2)*d) - 2/3*a^2/((d*x)^(3/2)*d)

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Fricas [A]  time = 0.848769, size = 236, normalized size = 3.23 \begin{align*} -\frac{2 \,{\left (9 \, b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + 9 \, b^{2} \log \left (c\right )^{2} + 12 \, a b n + 9 \, a^{2} + 6 \,{\left (2 \, b^{2} n + 3 \, a b\right )} \log \left (c\right ) + 6 \,{\left (2 \, b^{2} n^{2} + 3 \, b^{2} n \log \left (c\right ) + 3 \, a b n\right )} \log \left (x\right )\right )} \sqrt{d x}}{27 \, d^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*(9*b^2*n^2*log(x)^2 + 8*b^2*n^2 + 9*b^2*log(c)^2 + 12*a*b*n + 9*a^2 + 6*(2*b^2*n + 3*a*b)*log(c) + 6*(2*
b^2*n^2 + 3*b^2*n*log(c) + 3*a*b*n)*log(x))*sqrt(d*x)/(d^3*x^2)

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Sympy [B]  time = 42.5333, size = 218, normalized size = 2.99 \begin{align*} - \frac{2 a^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{4 a b n \log{\left (x \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{8 a b n}{9 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{4 a b \log{\left (c \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b^{2} n^{2} \log{\left (x \right )}^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{8 b^{2} n^{2} \log{\left (x \right )}}{9 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{16 b^{2} n^{2}}{27 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{4 b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{8 b^{2} n \log{\left (c \right )}}{9 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b^{2} \log{\left (c \right )}^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(3*d**(5/2)*x**(3/2)) - 4*a*b*n*log(x)/(3*d**(5/2)*x**(3/2)) - 8*a*b*n/(9*d**(5/2)*x**(3/2)) - 4*a*b*l
og(c)/(3*d**(5/2)*x**(3/2)) - 2*b**2*n**2*log(x)**2/(3*d**(5/2)*x**(3/2)) - 8*b**2*n**2*log(x)/(9*d**(5/2)*x**
(3/2)) - 16*b**2*n**2/(27*d**(5/2)*x**(3/2)) - 4*b**2*n*log(c)*log(x)/(3*d**(5/2)*x**(3/2)) - 8*b**2*n*log(c)/
(9*d**(5/2)*x**(3/2)) - 2*b**2*log(c)**2/(3*d**(5/2)*x**(3/2))

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Giac [B]  time = 1.30257, size = 288, normalized size = 3.95 \begin{align*} -\frac{2 \,{\left (\frac{9 \, b^{2} d n^{2} \log \left (d x\right )^{2}}{\sqrt{d x} x} - \frac{6 \,{\left (3 \, b^{2} d^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} d^{2} n^{2} - 3 \, b^{2} d^{2} n \log \left (c\right ) - 3 \, a b d^{2} n\right )} \log \left (d x\right )}{\sqrt{d x} d x} + \frac{9 \, b^{2} d^{2} n^{2} \log \left (d\right )^{2} - 12 \, b^{2} d^{2} n^{2} \log \left (d\right ) - 18 \, b^{2} d^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} d^{2} n^{2} + 12 \, b^{2} d^{2} n \log \left (c\right ) + 9 \, b^{2} d^{2} \log \left (c\right )^{2} - 18 \, a b d^{2} n \log \left (d\right ) + 12 \, a b d^{2} n + 18 \, a b d^{2} \log \left (c\right ) + 9 \, a^{2} d^{2}}{\sqrt{d x} d x}\right )}}{27 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*(9*b^2*d*n^2*log(d*x)^2/(sqrt(d*x)*x) - 6*(3*b^2*d^2*n^2*log(d) - 2*b^2*d^2*n^2 - 3*b^2*d^2*n*log(c) - 3
*a*b*d^2*n)*log(d*x)/(sqrt(d*x)*d*x) + (9*b^2*d^2*n^2*log(d)^2 - 12*b^2*d^2*n^2*log(d) - 18*b^2*d^2*n*log(c)*l
og(d) + 8*b^2*d^2*n^2 + 12*b^2*d^2*n*log(c) + 9*b^2*d^2*log(c)^2 - 18*a*b*d^2*n*log(d) + 12*a*b*d^2*n + 18*a*b
*d^2*log(c) + 9*a^2*d^2)/(sqrt(d*x)*d*x))/d^3