Integrand size = 20, antiderivative size = 73 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {16 b^2 n^2 (d x)^{7/2}}{343 d}-\frac {8 b n (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d} \]
16/343*b^2*n^2*(d*x)^(7/2)/d-8/49*b*n*(d*x)^(7/2)*(a+b*ln(c*x^n))/d+2/7*(d *x)^(7/2)*(a+b*ln(c*x^n))^2/d
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{343} x (d x)^{5/2} \left (49 a^2-28 a b n+8 b^2 n^2+14 b (7 a-2 b n) \log \left (c x^n\right )+49 b^2 \log ^2\left (c x^n\right )\right ) \]
(2*x*(d*x)^(5/2)*(49*a^2 - 28*a*b*n + 8*b^2*n^2 + 14*b*(7*a - 2*b*n)*Log[c *x^n] + 49*b^2*Log[c*x^n]^2))/343
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {4}{7} b n \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )dx\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {4}{7} b n \left (\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac {4 b n (d x)^{7/2}}{49 d}\right )\) |
(2*(d*x)^(7/2)*(a + b*Log[c*x^n])^2)/(7*d) - (4*b*n*((-4*b*n*(d*x)^(7/2))/ (49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(7*d)))/7
3.1.95.3.1 Defintions of rubi rules used
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 = 0.21 (sec) , antiderivative size = 716, normalized size of antiderivative = 9.81
| method | result | size |
| risch | \(\frac {2 d^{3} x^{4} b^{2} \ln \left (x^{n}\right )^{2}}{7 \sqrt {d x}}+\frac {2 d^{3} b \,x^{4} \left (-7 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+7 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+7 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-7 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+14 b \ln \left (c \right )-4 b n +14 a \right ) \ln \left (x^{n}\right )}{49 \sqrt {d x}}+\frac {d^{3} \left (196 a^{2}+56 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+98 \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}-196 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-196 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-49 \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}+56 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 )-196 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 )-196 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 )+98 \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}-196 \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}+98 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-49 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+98 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-49 \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}+392 \ln \left (c \right ) a b +196 \ln \left (c \right )^{2} b^{2}-112 b^{2} \ln \left (c \right ) n -112 a b n -49 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+196 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+196 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-56 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+196 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-56 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+196 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^{4}}{686 \sqrt {d x}}\) | \(716\) |
2/7*d^3*x^4*b^2/(d*x)^(1/2)*ln(x^n)^2+2/49*d^3*b*x^4*(-7*I*b*Pi*csgn(I*c)* csgn(I*x^n)*csgn(I*c*x^n)+7*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+7*I*b*Pi*csgn (I*x^n)*csgn(I*c*x^n)^2-7*I*b*Pi*csgn(I*c*x^n)^3+14*b*ln(c)-4*b*n+14*a)/(d *x)^(1/2)*ln(x^n)+1/686*d^3*(196*a^2+56*I*Pi*b^2*n*csgn(I*c*x^n)^3+98*Pi^2 *b^2*csgn(I*c)*csgn(I*c*x^n)^5+32*b^2*n^2-196*I*ln(c)*Pi*b^2*csgn(I*c)*csg n(I*x^n)*csgn(I*c*x^n)-196*I*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+56 *I*Pi*b^2*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+392*ln(c)*a*b+196*ln(c)^2* b^2-112*b^2*ln(c)*n-112*a*b*n-56*I*Pi*b^2*n*csgn(I*c)*csgn(I*c*x^n)^2-56*I *Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+196*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^ n)^2-49*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+98*Pi^2*b^2*csgn(I*x^n)*csg n(I*c*x^n)^5+98*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-196*Pi^2* b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+196*I*ln(c)*Pi*b^2*csgn(I*x^n)*c sgn(I*c*x^n)^2-196*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-196*I*Pi*a*b*csgn(I*c*x^ n)^3+196*I*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2+98*Pi^2*b^2*csgn(I*c)^2*csgn(I *x^n)*csgn(I*c*x^n)^3-49*Pi^2*b^2*csgn(I*c*x^n)^6+196*I*ln(c)*Pi*b^2*csgn( I*c)*csgn(I*c*x^n)^2-49*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-49*Pi^2*b^2*c sgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2)*x^4/(d*x)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{343} \, {\left (49 \, b^{2} d^{2} n^{2} x^{3} \log \left (x\right )^{2} + 49 \, b^{2} d^{2} x^{3} \log \left (c\right )^{2} - 14 \, {\left (2 \, b^{2} d^{2} n - 7 \, a b d^{2}\right )} x^{3} \log \left (c\right ) + {\left (8 \, b^{2} d^{2} n^{2} - 28 \, a b d^{2} n + 49 \, a^{2} d^{2}\right )} x^{3} + 14 \, {\left (7 \, b^{2} d^{2} n x^{3} \log \left (c\right ) - {\left (2 \, b^{2} d^{2} n^{2} - 7 \, a b d^{2} n\right )} x^{3}\right )} \log \left (x\right )\right )} \sqrt {d x} \]
2/343*(49*b^2*d^2*n^2*x^3*log(x)^2 + 49*b^2*d^2*x^3*log(c)^2 - 14*(2*b^2*d ^2*n - 7*a*b*d^2)*x^3*log(c) + (8*b^2*d^2*n^2 - 28*a*b*d^2*n + 49*a^2*d^2) *x^3 + 14*(7*b^2*d^2*n*x^3*log(c) - (2*b^2*d^2*n^2 - 7*a*b*d^2*n)*x^3)*log (x))*sqrt(d*x)
Time = 22.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 a^{2} x \left (d x\right )^{\frac {5}{2}}}{7} - \frac {8 a b n x \left (d x\right )^{\frac {5}{2}}}{49} + \frac {4 a b x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{7} + \frac {16 b^{2} n^{2} x \left (d x\right )^{\frac {5}{2}}}{343} - \frac {8 b^{2} n x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{49} + \frac {2 b^{2} x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}^{2}}{7} \]
2*a**2*x*(d*x)**(5/2)/7 - 8*a*b*n*x*(d*x)**(5/2)/49 + 4*a*b*x*(d*x)**(5/2) *log(c*x**n)/7 + 16*b**2*n**2*x*(d*x)**(5/2)/343 - 8*b**2*n*x*(d*x)**(5/2) *log(c*x**n)/49 + 2*b**2*x*(d*x)**(5/2)*log(c*x**n)**2/7
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 \, \left (d x\right )^{\frac {7}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{7 \, d} - \frac {8 \, \left (d x\right )^{\frac {7}{2}} a b n}{49 \, d} + \frac {4 \, \left (d x\right )^{\frac {7}{2}} a b \log \left (c x^{n}\right )}{7 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} a^{2}}{7 \, d} + \frac {8}{343} \, {\left (\frac {2 \, \left (d x\right )^{\frac {7}{2}} n^{2}}{d} - \frac {7 \, \left (d x\right )^{\frac {7}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} \]
2/7*(d*x)^(7/2)*b^2*log(c*x^n)^2/d - 8/49*(d*x)^(7/2)*a*b*n/d + 4/7*(d*x)^ (7/2)*a*b*log(c*x^n)/d + 2/7*(d*x)^(7/2)*a^2/d + 8/343*(2*(d*x)^(7/2)*n^2/ d - 7*(d*x)^(7/2)*n*log(c*x^n)/d)*b^2
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 425, normalized size of antiderivative = 5.82 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\left (\frac {1}{7} i + \frac {1}{7}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{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}{7} i - \frac {1}{7}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{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}{49} i + \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) + \left (\frac {2}{7} i + \frac {2}{7}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{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}{49} i - \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{7} i - \frac {2}{7}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{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}{343} i + \frac {8}{343}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{49} i + \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (c\right ) + \left (\frac {2}{7} i + \frac {2}{7}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) - \left (\frac {8}{343} i - \frac {8}{343}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {4}{49} i - \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (c\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{7} i - \frac {2}{7}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{49} i + \frac {4}{49}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {4}{49} i - \frac {4}{49}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \frac {2}{7} \, b^{2} d^{\frac {5}{2}} x^{\frac {7}{2}} \log \left (c\right )^{2} + \frac {4}{7} \, a b d^{\frac {5}{2}} x^{\frac {7}{2}} \log \left (c\right ) + \frac {2}{7} \, a^{2} d^{\frac {5}{2}} x^{\frac {7}{2}} \]
(1/7*I + 1/7)*sqrt(2)*b^2*d^2*n^2*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))* log(x)^2 - (1/7*I - 1/7)*sqrt(2)*b^2*d^2*n^2*x^(7/2)*sqrt(abs(d))*log(x)^2 *sin(1/4*pi*sgn(d)) - (4/49*I + 4/49)*sqrt(2)*b^2*d^2*n^2*x^(7/2)*sqrt(abs (d))*cos(1/4*pi*sgn(d))*log(x) + (2/7*I + 2/7)*sqrt(2)*b^2*d^2*n*x^(7/2)*s qrt(abs(d))*cos(1/4*pi*sgn(d))*log(c)*log(x) + (4/49*I - 4/49)*sqrt(2)*b^2 *d^2*n^2*x^(7/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/7*I - 2/7)*sq rt(2)*b^2*d^2*n*x^(7/2)*sqrt(abs(d))*log(c)*log(x)*sin(1/4*pi*sgn(d)) + (8 /343*I + 8/343)*sqrt(2)*b^2*d^2*n^2*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d) ) - (4/49*I + 4/49)*sqrt(2)*b^2*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn( d))*log(c) + (2/7*I + 2/7)*sqrt(2)*a*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4* pi*sgn(d))*log(x) - (8/343*I - 8/343)*sqrt(2)*b^2*d^2*n^2*x^(7/2)*sqrt(abs (d))*sin(1/4*pi*sgn(d)) + (4/49*I - 4/49)*sqrt(2)*b^2*d^2*n*x^(7/2)*sqrt(a bs(d))*log(c)*sin(1/4*pi*sgn(d)) - (2/7*I - 2/7)*sqrt(2)*a*b*d^2*n*x^(7/2) *sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (4/49*I + 4/49)*sqrt(2)*a*b*d^2* n*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) + (4/49*I - 4/49)*sqrt(2)*a*b*d^ 2*n*x^(7/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/7*b^2*d^(5/2)*x^(7/2)*log( c)^2 + 4/7*a*b*d^(5/2)*x^(7/2)*log(c) + 2/7*a^2*d^(5/2)*x^(7/2)
Timed out. \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int {\left (d\,x\right )}^{5/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]