\(\int (d x)^{3/2} (a+b \log (c x^n)) \, dx\) [90]

Optimal result

Integrand size = 18, antiderivative size = 41 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4 b n (d x)^{5/2}}{25 d}+\frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d} \]

[Out]

-4/25*b*n*(d*x)^(5/2)/d+2/5*(d*x)^(5/2)*(a+b*ln(c*x^n))/d

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2341} \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d}-\frac {4 b n (d x)^{5/2}}{25 d} \]

[In]

Int[(d*x)^(3/2)*(a + b*Log[c*x^n]),x]

[Out]

(-4*b*n*(d*x)^(5/2))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(5*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
]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 b n (d x)^{5/2}}{25 d}+\frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2}{25} x (d x)^{3/2} \left (5 a-2 b n+5 b \log \left (c x^n\right )\right ) \]

[In]

Integrate[(d*x)^(3/2)*(a + b*Log[c*x^n]),x]

[Out]

(2*x*(d*x)^(3/2)*(5*a - 2*b*n + 5*b*Log[c*x^n]))/25

Maple [C] (warning: unable to verify)

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

Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.12

method	result	size
risch	\(\frac {2 d^{2} b \,x^{3} \ln \left (x^{n}\right )}{5 \sqrt {d x}}+\frac {d^{2} \left (-5 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+5 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+5 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-5 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+10 b \ln \left (c \right )-4 b n +10 a \right ) x^{3}}{25 \sqrt {d x}}\)	\(128\)

[In]

int((d*x)^(3/2)*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

2/5*d^2*b*x^3/(d*x)^(1/2)*ln(x^n)+1/25*d^2*(-5*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+5*I*b*Pi*csgn(I*c)*c
sgn(I*c*x^n)^2+5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-5*I*b*Pi*csgn(I*c*x^n)^3+10*b*ln(c)-4*b*n+10*a)*x^3/(d*x)^
(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2}{25} \, {\left (5 \, b d n x^{2} \log \left (x\right ) + 5 \, b d x^{2} \log \left (c\right ) - {\left (2 \, b d n - 5 \, a d\right )} x^{2}\right )} \sqrt {d x} \]

[In]

integrate((d*x)^(3/2)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

2/25*(5*b*d*n*x^2*log(x) + 5*b*d*x^2*log(c) - (2*b*d*n - 5*a*d)*x^2)*sqrt(d*x)

Sympy [A] (verification not implemented)

Time = 1.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 a x \left (d x\right )^{\frac {3}{2}}}{5} - \frac {4 b n x \left (d x\right )^{\frac {3}{2}}}{25} + \frac {2 b x \left (d x\right )^{\frac {3}{2}} \log {\left (c x^{n} \right )}}{5} \]

[In]

integrate((d*x)**(3/2)*(a+b*ln(c*x**n)),x)

[Out]

2*a*x*(d*x)**(3/2)/5 - 4*b*n*x*(d*x)**(3/2)/25 + 2*b*x*(d*x)**(3/2)*log(c*x**n)/5

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4 \, \left (d x\right )^{\frac {5}{2}} b n}{25 \, d} + \frac {2 \, \left (d x\right )^{\frac {5}{2}} b \log \left (c x^{n}\right )}{5 \, d} + \frac {2 \, \left (d x\right )^{\frac {5}{2}} a}{5 \, d} \]

[In]

integrate((d*x)^(3/2)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-4/25*(d*x)^(5/2)*b*n/d + 2/5*(d*x)^(5/2)*b*log(c*x^n)/d + 2/5*(d*x)^(5/2)*a/d

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.63 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, {\left (-\left (5 i + 5\right ) \, \sqrt {2} b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) + \left (5 i - 5\right ) \, \sqrt {2} b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (2 i + 2\right ) \, \sqrt {2} b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (2 i - 2\right ) \, \sqrt {2} b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - 10 \, b \sqrt {d} x^{\frac {5}{2}} \log \left (c\right ) - 10 \, a \sqrt {d} x^{\frac {5}{2}}\right )} d \]

[In]

integrate((d*x)^(3/2)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

-1/25*(-(5*I + 5)*sqrt(2)*b*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) + (5*I - 5)*sqrt(2)*b*n*x^(5/2)*s
qrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) + (2*I + 2)*sqrt(2)*b*n*x^(5/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) - (2*I -
2)*sqrt(2)*b*n*x^(5/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) - 10*b*sqrt(d)*x^(5/2)*log(c) - 10*a*sqrt(d)*x^(5/2))*
d

Mupad [F(-1)]

Timed out. \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (d\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

int((d*x)^(3/2)*(a + b*log(c*x^n)),x)

[Out]

int((d*x)^(3/2)*(a + b*log(c*x^n)), x)