3.95 \(\int (d x)^{5/2} (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=73 \[ \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {8 b n (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{49 d}+\frac {16 b^2 n^2 (d x)^{7/2}}{343 d} \]

[Out]

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

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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} \[ \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {8 b n (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{49 d}+\frac {16 b^2 n^2 (d x)^{7/2}}{343 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*(a + b*Log[c*x^n])^2,x]

[Out]

(16*b^2*n^2*(d*x)^(7/2))/(343*d) - (8*b*n*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c
*x^n])^2)/(7*d)

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 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {1}{7} (4 b n) \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, 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}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.84 \[ \frac {2}{343} x (d x)^{5/2} \left (49 a^2+14 b (7 a-2 b n) \log \left (c x^n\right )-28 a b n+49 b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*(a + b*Log[c*x^n])^2,x]

[Out]

(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

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fricas [B]  time = 0.46, size = 141, normalized size = 1.93 \[ \frac {2}{343} \, {\left (49 \, b^{2} d^{2} n^{2} x^{3} \log \relax (x)^{2} + 49 \, b^{2} d^{2} x^{3} \log \relax (c)^{2} - 14 \, {\left (2 \, b^{2} d^{2} n - 7 \, a b d^{2}\right )} x^{3} \log \relax (c) + {\left (8 \, b^{2} d^{2} n^{2} - 28 \, a b d^{2} n + 49 \, a^{2} d^{2}\right )} x^{3} + 14 \, {\left (7 \, b^{2} d^{2} n x^{3} \log \relax (c) - {\left (2 \, b^{2} d^{2} n^{2} - 7 \, a b d^{2} n\right )} x^{3}\right )} \log \relax (x)\right )} \sqrt {d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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)*l
og(x))*sqrt(d*x)

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giac [C]  time = 1.31, size = 425, normalized size = 5.82 \[ \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}\relax (d)\right ) \log \relax (x)^{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 \relax (x)^{2} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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}\relax (d)\right ) \log \relax (x) + \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}\relax (d)\right ) \log \relax (c) \log \relax (x) + \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 \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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 \relax (c) \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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}\relax (d)\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}\relax (d)\right ) \log \relax (c) + \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}\relax (d)\right ) \log \relax (x) - \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}\relax (d)\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 \relax (c) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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 \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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}\relax (d)\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}\relax (d)\right ) + \frac {2}{7} \, b^{2} d^{\frac {5}{2}} x^{\frac {7}{2}} \log \relax (c)^{2} + \frac {4}{7} \, a b d^{\frac {5}{2}} x^{\frac {7}{2}} \log \relax (c) + \frac {2}{7} \, a^{2} d^{\frac {5}{2}} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

(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(a
bs(d))*cos(1/4*pi*sgn(d))*log(x) + (2/7*I + 2/7)*sqrt(2)*b^2*d^2*n*x^(7/2)*sqrt(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
)*sqrt(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*p
i*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(abs(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)

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maple [C]  time = 0.17, size = 716, normalized size = 9.81 \[ \frac {2 b^{2} d^{3} x^{4} \ln \left (x^{n}\right )^{2}}{7 \sqrt {d x}}+\frac {2 \left (-7 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+7 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+7 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-7 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b n +14 b \ln \relax (c )+14 a \right ) b \,d^{3} x^{4} \ln \left (x^{n}\right )}{49 \sqrt {d x}}+\frac {\left (-49 \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}+98 \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}+98 \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}-196 \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}+56 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-196 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-196 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+196 a^{2}-56 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-56 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+196 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+196 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+196 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+196 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}-49 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+98 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-49 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+98 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+392 a b \ln \relax (c )-112 b^{2} n \ln \relax (c )+196 b^{2} \ln \relax (c )^{2}-112 a b n -49 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-196 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 )-196 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 )+56 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 )\right ) d^{3} x^{4}}{686 \sqrt {d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+7*I*Pi*b*c
sgn(I*c)*csgn(I*c*x^n)^2+7*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-7*I*Pi*b*csgn(I*c*x^n)^3-4*b*n+14*b*ln(c)+14*a)/
(d*x)^(1/2)*ln(x^n)+1/686*d^3*(-49*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-196*Pi^2*b^2*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)^4+98*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+98*Pi^2*b^2*csgn(I*c)*csgn(I*x^
n)^2*csgn(I*c*x^n)^3+196*a^2+32*b^2*n^2-196*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+392*a*b*ln(c)-112*b^2
*n*ln(c)+196*b^2*ln(c)^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)*csgn(I*c*x^n)^5-112
*a*b*n-49*Pi^2*b^2*csgn(I*c*x^n)^6+98*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5-49*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)
^4+56*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-196*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+56
*I*Pi*b^2*n*csgn(I*c*x^n)^3-196*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*ln(c)*Pi*b^2
*csgn(I*x^n)*csgn(I*c*x^n)^2+196*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-56*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-
56*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+196*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+196*I*Pi*a*b*csgn(I*x^n)*
csgn(I*c*x^n)^2)*x^4/(d*x)^(1/2)

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maxima [A]  time = 0.59, size = 102, normalized size = 1.40 \[ \frac {2 \, \left (d x\right )^{\frac {7}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{7 \, d} - \frac {8 \, \left (d x\right )^{\frac {7}{2}} a b n}{49 \, d} + \frac {4 \, \left (d x\right )^{\frac {7}{2}} a b \log \left (c x^{n}\right )}{7 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} a^{2}}{7 \, d} + \frac {8}{343} \, {\left (\frac {2 \, \left (d x\right )^{\frac {7}{2}} n^{2}}{d} - \frac {7 \, \left (d x\right )^{\frac {7}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*(a+b*ln(c*x**n))**2,x)

[Out]

Timed out

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