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/(d*x)^(1/2)-8*b*n*(a+b*ln(c*x^n))/d/(d*x)^(1/2)-2*(a+b*ln(c*x^n))^2/d/(d*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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
]

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]

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 = 0.01, 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)

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 87, normalized size = 1.30 \[ -\frac {2 \, {\left (b^{2} n^{2} \log \relax (x)^{2} + 8 \, b^{2} n^{2} + b^{2} \log \relax (c)^{2} + 4 \, a b n + a^{2} + 2 \, {\left (2 \, b^{2} n + a b\right )} \log \relax (c) + 2 \, {\left (2 \, b^{2} n^{2} + b^{2} n \log \relax (c) + a b n\right )} \log \relax (x)\right )} \sqrt {d x}}{d^{2} x} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.41, size = 149, normalized size = 2.22 \[ -\frac {2 \, {\left (\frac {b^{2} n^{2} \log \left (d x\right )^{2}}{\sqrt {d x}} - \frac {2 \, {\left (b^{2} n^{2} \log \relax (d) - 2 \, b^{2} n^{2} - b^{2} n \log \relax (c) - a b n\right )} \log \left (d x\right )}{\sqrt {d x}} + \frac {b^{2} n^{2} \log \relax (d)^{2} - 4 \, b^{2} n^{2} \log \relax (d) - 2 \, b^{2} n \log \relax (c) \log \relax (d) + 8 \, b^{2} n^{2} + 4 \, b^{2} n \log \relax (c) + b^{2} \log \relax (c)^{2} - 2 \, a b n \log \relax (d) + 4 \, a b n + 2 \, a b \log \relax (c) + a^{2}}{\sqrt {d x}}\right )}}{d} \]

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

________________________________________________________________________________________

maple [C]  time = 0.17, size = 707, normalized size = 10.55 \[ -\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{\sqrt {d x}\, d}-\frac {2 \left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +2 b \ln \relax (c )+2 a \right ) b \ln \left (x^{n}\right )}{\sqrt {d x}\, d}-\frac {-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}-8 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-4 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 a^{2}+8 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+32 b^{2} n^{2}-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+8 a b \ln \relax (c )+16 b^{2} n \ln \relax (c )+4 b^{2} \ln \relax (c )^{2}+16 a b n -\pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )-4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-8 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 \sqrt {d x}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.63, size = 101, normalized size = 1.51 \[ -8 \, b^{2} {\left (\frac {2 \, n^{2}}{\sqrt {d x} d} + \frac {n \log \left (c x^{n}\right )}{\sqrt {d x} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{\sqrt {d x} d} - \frac {8 \, a b n}{\sqrt {d x} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{\sqrt {d x} d} - \frac {2 \, a^{2}}{\sqrt {d x} d} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 2.90, size = 201, normalized size = 3.00 \[ - \frac {2 a^{2}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 a b n \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {8 a b n}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 a b \log {\relax (c )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b^{2} n^{2} \log {\relax (x )}^{2}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {8 b^{2} n^{2} \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {16 b^{2} n^{2}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 b^{2} n \log {\relax (c )} \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {8 b^{2} n \log {\relax (c )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b^{2} \log {\relax (c )}^{2}}{d^{\frac {3}{2}} \sqrt {x}} \]

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))

________________________________________________________________________________________