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

Optimal. Leaf size=67 \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt{d x}}-\frac{16 b^2 n^2}{d \sqrt{d x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0468623, antiderivative size = 67, 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} \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt{d x}}-\frac{16 b^2 n^2}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0127942, size = 54, normalized size = 0.81 \[ -\frac{2 x \left (a^2+2 b (a+2 b n) \log \left (c x^n\right )+4 a b n+b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*(a^2 + 4*a*b*n + 8*b^2*n^2 + 2*b*(a + 2*b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2))/(d*x)^(3/2)

________________________________________________________________________________________

Maple [C]  time = 0.131, size = 707, normalized size = 10.6 \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)^(3/2),x)

[Out]

-2/d*b^2/(d*x)^(1/2)*ln(x^n)^2-2/d*b*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)-I*b*Pi*csgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+4*b*n+2*a)/(d*x)^(1/2)*ln(x^n)-1/2/d*(
4*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+4*ln(c)^2*b^2-Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2+16*a*b*n+32*b^
2*n^2+4*a^2+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+2*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I
*c)-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-4*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+4*I*ln
(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*
c)-4*I*Pi*a*b*csgn(I*c*x^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3+8*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+8*I*Pi
*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-4*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)-Pi^2*b^2*csgn(I*c*x^n)^6+8*ln(c)*a*b+16*ln(c)*b^2*n-8*I*Pi*b^2*n*csgn(I*c*x^n)^3+2*Pi^2
*b^2*csgn(I*c*x^n)^5*csgn(I*c)+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-8*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4)/(d*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.08424, size = 136, normalized size = 2.03 \begin{align*} -8 \, b^{2}{\left (\frac{2 \, n^{2}}{\sqrt{d x} d} + \frac{n \log \left (c x^{n}\right )}{\sqrt{d x} d}\right )} - \frac{2 \, b^{2} \log \left (c x^{n}\right )^{2}}{\sqrt{d x} d} - \frac{8 \, a b n}{\sqrt{d x} d} - \frac{4 \, a b \log \left (c x^{n}\right )}{\sqrt{d x} d} - \frac{2 \, a^{2}}{\sqrt{d x} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.938875, size = 212, normalized size = 3.16 \begin{align*} -\frac{2 \,{\left (b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} + 4 \, a b n + a^{2} + 2 \,{\left (2 \, b^{2} n + a b\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \sqrt{d x}}{d^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 5.55021, size = 201, normalized size = 3. \begin{align*} - \frac{2 a^{2}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 a b n \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{8 a b n}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 a b \log{\left (c \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b^{2} n^{2} \log{\left (x \right )}^{2}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{8 b^{2} n^{2} \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{16 b^{2} n^{2}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{8 b^{2} n \log{\left (c \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b^{2} \log{\left (c \right )}^{2}}{d^{\frac{3}{2}} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(d**(3/2)*sqrt(x)) - 4*a*b*n*log(x)/(d**(3/2)*sqrt(x)) - 8*a*b*n/(d**(3/2)*sqrt(x)) - 4*a*b*log(c)/(d*
*(3/2)*sqrt(x)) - 2*b**2*n**2*log(x)**2/(d**(3/2)*sqrt(x)) - 8*b**2*n**2*log(x)/(d**(3/2)*sqrt(x)) - 16*b**2*n
**2/(d**(3/2)*sqrt(x)) - 4*b**2*n*log(c)*log(x)/(d**(3/2)*sqrt(x)) - 8*b**2*n*log(c)/(d**(3/2)*sqrt(x)) - 2*b*
*2*log(c)**2/(d**(3/2)*sqrt(x))

________________________________________________________________________________________

Giac [B]  time = 1.26972, size = 201, normalized size = 3. \begin{align*} -\frac{2 \,{\left (\frac{b^{2} n^{2} \log \left (d x\right )^{2}}{\sqrt{d x}} - \frac{2 \,{\left (b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n^{2} - b^{2} n \log \left (c\right ) - a b n\right )} \log \left (d x\right )}{\sqrt{d x}} + \frac{b^{2} n^{2} \log \left (d\right )^{2} - 4 \, b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} n^{2} + 4 \, b^{2} n \log \left (c\right ) + b^{2} \log \left (c\right )^{2} - 2 \, a b n \log \left (d\right ) + 4 \, a b n + 2 \, a b \log \left (c\right ) + a^{2}}{\sqrt{d x}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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