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*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)
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Rubi [A] time = 0.0450409, antiderivative size = 73, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 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]
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
]
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*}
Mathematica [A] time = 0.0202206, 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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Maple [C] time = 0.139, size = 716, normalized size = 9.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^(5/2)*(a+b*ln(c*x^n))^2,x)
[Out]
2/7*d^3*b^2*x^4/(d*x)^(1/2)*ln(x^n)^2+2/49*d^3*b*x^4*(7*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-7*I*b*Pi*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)-7*I*b*Pi*csgn(I*c*x^n)^3+7*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+14*b*ln(c)-4*b*n+14*a)/(
d*x)^(1/2)*ln(x^n)+1/686*d^3*(-196*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3+56*I*Pi*b^2*n*csgn(I*c*x^n)^3-196*I*Pi*a*b*c
sgn(I*c*x^n)^3+196*ln(c)^2*b^2-49*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-112*a*b*n+32*b^2*n^2+196*a^2+98*Pi^2*b^
2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+98*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-49*Pi^2*b^2*csgn
(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-196*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+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*x^n)*csgn(I*c*x^n)^2-49*Pi^2*b^2*csgn(I*c*x^n)^6+392*ln(c)*a*b-112*ln(c)*b^2*n-196*I*Pi*a*b*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+98*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+98*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^
5+196*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+196*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+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)-49*Pi^2*b^2*csgn(I*x^n)^2*c
sgn(I*c*x^n)^4)*x^4/(d*x)^(1/2)
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Maxima [A] time = 1.15542, size = 138, normalized size = 1.89 \begin{align*} \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} \end{align*}
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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Fricas [B] time = 0.796186, size = 321, normalized size = 4.4 \begin{align*} \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} \end{align*}
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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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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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Giac [C] time = 1.99799, size = 574, normalized size = 7.86 \begin{align*} \text{result too large to display} \end{align*}
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)