\(\int \sqrt {d x} (a+b \log (c x^n))^2 \, dx\) [97]

Optimal result

Integrand size = 20, antiderivative size = 73 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {16 b^2 n^2 (d x)^{3/2}}{27 d}-\frac {8 b n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d}+\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2342, 2341} \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {8 b n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d}+\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{3 d}+\frac {16 b^2 n^2 (d x)^{3/2}}{27 d} \]

[In]

Int[Sqrt[d*x]*(a + b*Log[c*x^n])^2,x]

[Out]

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

Rule 2341

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 2342

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{27} x \sqrt {d x} \left (9 a^2-12 a b n+8 b^2 n^2+6 b (3 a-2 b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )\right ) \]

[In]

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

[Out]

(2*x*Sqrt[d*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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 710, normalized size of antiderivative = 9.73

method	result	size
risch	\(\frac {2 d \,b^{2} x^{2} \ln \left (x^{n}\right )^{2}}{3 \sqrt {d x}}+\frac {2 d b \,x^{2} \left (-3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+6 b \ln \left (c \right )-4 b n +6 a \right ) \ln \left (x^{n}\right )}{9 \sqrt {d x}}+\frac {d \left (36 a^{2}+24 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+32 b^{2} n^{2}+72 \ln \left (c \right ) a b +36 \ln \left (c \right )^{2} b^{2}-48 b^{2} \ln \left (c \right ) n -48 a b n -9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+36 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}\right ) x^{2}}{54 \sqrt {d x}}\)	\(710\)

[In]

int((d*x)^(1/2)*(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{27} \, {\left (9 \, b^{2} n^{2} x \log \left (x\right )^{2} + 9 \, b^{2} x \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{2} n - 3 \, a b\right )} x \log \left (c\right ) + {\left (8 \, b^{2} n^{2} - 12 \, a b n + 9 \, a^{2}\right )} x + 6 \, {\left (3 \, b^{2} n x \log \left (c\right ) - {\left (2 \, b^{2} n^{2} - 3 \, a b n\right )} x\right )} \log \left (x\right )\right )} \sqrt {d x} \]

[In]

integrate((d*x)^(1/2)*(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 a^{2} x \sqrt {d x}}{3} - \frac {8 a b n x \sqrt {d x}}{9} + \frac {4 a b x \sqrt {d x} \log {\left (c x^{n} \right )}}{3} + \frac {16 b^{2} n^{2} x \sqrt {d x}}{27} - \frac {8 b^{2} n x \sqrt {d x} \log {\left (c x^{n} \right )}}{9} + \frac {2 b^{2} x \sqrt {d x} \log {\left (c x^{n} \right )}^{2}}{3} \]

[In]

integrate((d*x)**(1/2)*(a+b*ln(c*x**n))**2,x)

[Out]

2*a**2*x*sqrt(d*x)/3 - 8*a*b*n*x*sqrt(d*x)/9 + 4*a*b*x*sqrt(d*x)*log(c*x**n)/3 + 16*b**2*n**2*x*sqrt(d*x)/27 -
8*b**2*n*x*sqrt(d*x)*log(c*x**n)/9 + 2*b**2*x*sqrt(d*x)*log(c*x**n)**2/3

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 \, \left (d x\right )^{\frac {3}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{3 \, d} - \frac {8 \, \left (d x\right )^{\frac {3}{2}} a b n}{9 \, d} + \frac {4 \, \left (d x\right )^{\frac {3}{2}} a b \log \left (c x^{n}\right )}{3 \, d} + \frac {8}{27} \, {\left (\frac {2 \, \left (d x\right )^{\frac {3}{2}} n^{2}}{d} - \frac {3 \, \left (d x\right )^{\frac {3}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} + \frac {2 \, \left (d x\right )^{\frac {3}{2}} a^{2}}{3 \, d} \]

[In]

integrate((d*x)^(1/2)*(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 383, normalized size of antiderivative = 5.25 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\left (\frac {1}{3} i + \frac {1}{3}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right )^{2} - \left (\frac {1}{3} i - \frac {1}{3}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \left (x\right )^{2} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{9} i + \frac {4}{9}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) + \left (\frac {2}{3} i + \frac {2}{3}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (c\right ) \log \left (x\right ) + \left (\frac {4}{9} i - \frac {4}{9}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{3} i - \frac {2}{3}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \left (c\right ) \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {8}{27} i + \frac {8}{27}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{9} i + \frac {4}{9}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (c\right ) + \left (\frac {2}{3} i + \frac {2}{3}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) - \left (\frac {8}{27} i - \frac {8}{27}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {4}{9} i - \frac {4}{9}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \left (c\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{3} i - \frac {2}{3}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{9} i + \frac {4}{9}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {4}{9} i - \frac {4}{9}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \frac {2}{3} \, b^{2} \sqrt {d} x^{\frac {3}{2}} \log \left (c\right )^{2} + \frac {4}{3} \, a b \sqrt {d} x^{\frac {3}{2}} \log \left (c\right ) + \frac {2}{3} \, a^{2} \sqrt {d} x^{\frac {3}{2}} \]

[In]

integrate((d*x)^(1/2)*(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

(1/3*I + 1/3)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x)^2 - (1/3*I - 1/3)*sqrt(2)*b^2*n^2
*x^(3/2)*sqrt(abs(d))*log(x)^2*sin(1/4*pi*sgn(d)) - (4/9*I + 4/9)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*cos(1/4
*pi*sgn(d))*log(x) + (2/3*I + 2/3)*sqrt(2)*b^2*n*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(c)*log(x) + (4/9*
I - 4/9)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/3*I - 2/3)*sqrt(2)*b^2*n*x^(3/2)*
sqrt(abs(d))*log(c)*log(x)*sin(1/4*pi*sgn(d)) + (8/27*I + 8/27)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*cos(1/4*p
i*sgn(d)) - (4/9*I + 4/9)*sqrt(2)*b^2*n*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(c) + (2/3*I + 2/3)*sqrt(2)
*a*b*n*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) - (8/27*I - 8/27)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*s
in(1/4*pi*sgn(d)) + (4/9*I - 4/9)*sqrt(2)*b^2*n*x^(3/2)*sqrt(abs(d))*log(c)*sin(1/4*pi*sgn(d)) - (2/3*I - 2/3)
*sqrt(2)*a*b*n*x^(3/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (4/9*I + 4/9)*sqrt(2)*a*b*n*x^(3/2)*sqrt(abs(d
))*cos(1/4*pi*sgn(d)) + (4/9*I - 4/9)*sqrt(2)*a*b*n*x^(3/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/3*b^2*sqrt(d)*
x^(3/2)*log(c)^2 + 4/3*a*b*sqrt(d)*x^(3/2)*log(c) + 2/3*a^2*sqrt(d)*x^(3/2)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int \sqrt {d\,x}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]

[In]

int((d*x)^(1/2)*(a + b*log(c*x^n))^2,x)

[Out]

int((d*x)^(1/2)*(a + b*log(c*x^n))^2, x)