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]

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

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Rubi [A]  time = 0.05, antiderivative size = 73, 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} \[ -\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac {16 b^2 n^2}{27 d (d x)^{3/2}} \]

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 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)^{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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.46, size = 94, normalized size = 1.29 \[ -\frac {2 \, {\left (9 \, b^{2} n^{2} \log \relax (x)^{2} + 8 \, b^{2} n^{2} + 9 \, b^{2} \log \relax (c)^{2} + 12 \, a b n + 9 \, a^{2} + 6 \, {\left (2 \, b^{2} n + 3 \, a b\right )} \log \relax (c) + 6 \, {\left (2 \, b^{2} n^{2} + 3 \, b^{2} n \log \relax (c) + 3 \, a b n\right )} \log \relax (x)\right )} \sqrt {d x}}{27 \, d^{3} x^{2}} \]

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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giac [B]  time = 0.41, size = 213, normalized size = 2.92 \[ -\frac {2 \, {\left (\frac {9 \, b^{2} d n^{2} \log \left (d x\right )^{2}}{\sqrt {d x} x} - \frac {6 \, {\left (3 \, b^{2} d^{2} n^{2} \log \relax (d) - 2 \, b^{2} d^{2} n^{2} - 3 \, b^{2} d^{2} n \log \relax (c) - 3 \, a b d^{2} n\right )} \log \left (d x\right )}{\sqrt {d x} d x} + \frac {9 \, b^{2} d^{2} n^{2} \log \relax (d)^{2} - 12 \, b^{2} d^{2} n^{2} \log \relax (d) - 18 \, b^{2} d^{2} n \log \relax (c) \log \relax (d) + 8 \, b^{2} d^{2} n^{2} + 12 \, b^{2} d^{2} n \log \relax (c) + 9 \, b^{2} d^{2} \log \relax (c)^{2} - 18 \, a b d^{2} n \log \relax (d) + 12 \, a b d^{2} n + 18 \, a b d^{2} \log \relax (c) + 9 \, a^{2} d^{2}}{\sqrt {d x} d x}\right )}}{27 \, d^{3}} \]

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

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maple [C]  time = 0.18, size = 716, normalized size = 9.81 \[ -\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{3 \sqrt {d x}\, d^{2} x}-\frac {2 \left (-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +6 b \ln \relax (c )+6 a \right ) b \ln \left (x^{n}\right )}{9 \sqrt {d x}\, d^{2} x}-\frac {-9 \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}+18 \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}+18 \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}-36 \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}-24 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-36 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+36 a^{2}+24 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+36 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+36 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 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}-9 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+72 a b \ln \relax (c )+48 b^{2} n \ln \relax (c )+36 b^{2} \ln \relax (c )^{2}+48 a b n -9 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-36 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 )-36 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 )-24 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 )}{54 \sqrt {d x}\, d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.66, size = 102, normalized size = 1.40 \[ -\frac {8}{27} \, b^{2} {\left (\frac {2 \, n^{2}}{\left (d x\right )^{\frac {3}{2}} d} + \frac {3 \, n \log \left (c x^{n}\right )}{\left (d x\right )^{\frac {3}{2}} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {8 \, a b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 26.71, size = 218, normalized size = 2.99 \[ - \frac {2 a^{2}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 a b n \log {\relax (x )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {8 a b n}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 a b \log {\relax (c )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b^{2} n^{2} \log {\relax (x )}^{2}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {8 b^{2} n^{2} \log {\relax (x )}}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {16 b^{2} n^{2}}{27 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 b^{2} n \log {\relax (c )} \log {\relax (x )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {8 b^{2} n \log {\relax (c )}}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b^{2} \log {\relax (c )}^{2}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} \]

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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