Integrand size = 20, antiderivative size = 67 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=-\frac {16 b^2 n^2}{d \sqrt {d x}}-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}} \] Output:
-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)
Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=-\frac {2 x \left (a^2+4 a b n+8 b^2 n^2+2 b (a+2 b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right )}{(d x)^{3/2}} \] Input:
Integrate[(a + b*Log[c*x^n])^2/(d*x)^(3/2),x]
Output:
(-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)
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2742, 2741}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle 4 b n \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}}dx-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle 4 b n \left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {4 b n}{d \sqrt {d x}}\right )-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}}\) |
Input:
Int[(a + b*Log[c*x^n])^2/(d*x)^(3/2),x]
Output:
(-2*(a + b*Log[c*x^n])^2)/(d*Sqrt[d*x]) + 4*b*n*((-4*b*n)/(d*Sqrt[d*x]) - (2*(a + b*Log[c*x^n]))/(d*Sqrt[d*x]))
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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.31 (sec) , antiderivative size = 707, normalized size of antiderivative = 10.55
method | result | size |
risch | \(-\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{d \sqrt {d x}}-\frac {2 b \left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+4 n b +2 a \right ) \ln \left (x^{n}\right )}{d \sqrt {d x}}-\frac {-4 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-4 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-8 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+4 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+16 b^{2} \ln \left (c \right ) n +8 a b \ln \left (c \right )+4 a^{2}+4 b^{2} \ln \left (c \right )^{2}+32 b^{2} n^{2}-\pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}+8 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+8 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+16 a n b -4 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4} \operatorname {csgn}\left (i c \right )-4 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{5} \operatorname {csgn}\left (i c \right )-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4} \operatorname {csgn}\left (i c \right )^{2}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3} \operatorname {csgn}\left (i c \right )^{2}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} \operatorname {csgn}\left (i c \right )-8 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )^{2}+4 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 d \sqrt {d x}}\) | \(707\) |
Input:
int((a+b*ln(c*x^n))^2/(d*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/d*b^2/(d*x)^(1/2)*ln(x^n)^2-2/d*b*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I *Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*cs gn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+4*n*b+2*a)/(d*x)^(1/2)*ln(x^n)-1/2/d*(8* I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-8*I*Pi*b^2*n*csgn(I*c*x^n)^3+8*I*Pi*b ^2*n*csgn(I*x^n)*csgn(I*c*x^n)^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*cs gn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2 +4*I*Pi*ln(c)*b^2*csgn(I*c*x^n)^2*csgn(I*c)+16*b^2*ln(c)*n+8*a*b*ln(c)+4*a ^2+4*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)+4*b^2*ln(c)^2+32*b^2*n^2-4*I*Pi*ln (c)*b^2*csgn(I*c*x^n)^3+2*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)-Pi^2*b^2*csgn (I*c*x^n)^4*csgn(I*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)^5-8*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I* c)-4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2*Pi^2*b^2*csgn(I*x^n)*c sgn(I*c*x^n)^3*csgn(I*c)^2+4*I*Pi*ln(c)*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-4* I*Pi*ln(c)*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b*csgn(I*c*x^n )^3+16*a*n*b-Pi^2*b^2*csgn(I*c*x^n)^6)/(d*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} + 4 \, a b n + a^{2} + 2 \, {\left (2 \, b^{2} n + a b\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \sqrt {d x}}{d^{2} x} \] Input:
integrate((a+b*log(c*x^n))^2/(d*x)^(3/2),x, algorithm="fricas")
Output:
-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)
Time = 0.55 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=- \frac {2 a^{2} x}{\left (d x\right )^{\frac {3}{2}}} - \frac {8 a b n x}{\left (d x\right )^{\frac {3}{2}}} - \frac {4 a b x \log {\left (c x^{n} \right )}}{\left (d x\right )^{\frac {3}{2}}} - \frac {16 b^{2} n^{2} x}{\left (d x\right )^{\frac {3}{2}}} - \frac {8 b^{2} n x \log {\left (c x^{n} \right )}}{\left (d x\right )^{\frac {3}{2}}} - \frac {2 b^{2} x \log {\left (c x^{n} \right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \] Input:
integrate((a+b*ln(c*x**n))**2/(d*x)**(3/2),x)
Output:
-2*a**2*x/(d*x)**(3/2) - 8*a*b*n*x/(d*x)**(3/2) - 4*a*b*x*log(c*x**n)/(d*x )**(3/2) - 16*b**2*n**2*x/(d*x)**(3/2) - 8*b**2*n*x*log(c*x**n)/(d*x)**(3/ 2) - 2*b**2*x*log(c*x**n)**2/(d*x)**(3/2)
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=-8 \, b^{2} {\left (\frac {2 \, n^{2}}{\sqrt {d x} d} + \frac {n \log \left (c x^{n}\right )}{\sqrt {d x} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{\sqrt {d x} d} - \frac {8 \, a b n}{\sqrt {d x} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{\sqrt {d x} d} - \frac {2 \, a^{2}}{\sqrt {d x} d} \] Input:
integrate((a+b*log(c*x^n))^2/(d*x)^(3/2),x, algorithm="maxima")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (61) = 122\).
Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {b^{2} n^{2} \log \left (d x\right )^{2}}{\sqrt {d x}} - \frac {2 \, {\left (b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n^{2} - b^{2} n \log \left (c\right ) - a b n\right )} \log \left (d x\right )}{\sqrt {d x}} + \frac {b^{2} n^{2} \log \left (d\right )^{2} - 4 \, b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} n^{2} + 4 \, b^{2} n \log \left (c\right ) + b^{2} \log \left (c\right )^{2} - 2 \, a b n \log \left (d\right ) + 4 \, a b n + 2 \, a b \log \left (c\right ) + a^{2}}{\sqrt {d x}}\right )}}{d} \] Input:
integrate((a+b*log(c*x^n))^2/(d*x)^(3/2),x, algorithm="giac")
Output:
-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*l og(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
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*log(c*x^n))^2/(d*x)^(3/2),x)
Output:
int((a + b*log(c*x^n))^2/(d*x)^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx=\frac {2 \sqrt {d}\, \left (-\mathrm {log}\left (x^{n} c \right )^{2} b^{2}-2 \,\mathrm {log}\left (x^{n} c \right ) a b -4 \,\mathrm {log}\left (x^{n} c \right ) b^{2} n -a^{2}-4 a b n -8 b^{2} n^{2}\right )}{\sqrt {x}\, d^{2}} \] Input:
int((a+b*log(c*x^n))^2/(d*x)^(3/2),x)
Output:
(2*sqrt(d)*( - log(x**n*c)**2*b**2 - 2*log(x**n*c)*a*b - 4*log(x**n*c)*b** 2*n - a**2 - 4*a*b*n - 8*b**2*n**2))/(sqrt(x)*d**2)