\(\int (d x)^{5/2} (a+b \log (c x^n)) \, dx\) [89]

Optimal result

Integrand size = 18, antiderivative size = 41 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4 b n (d x)^{7/2}}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d} \]

[Out]

-4/49*b*n*(d*x)^(7/2)/d+2/7*(d*x)^(7/2)*(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} \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac {4 b n (d x)^{7/2}}{49 d} \]

[In]

Int[(d*x)^(5/2)*(a + b*Log[c*x^n]),x]

[Out]

(-4*b*n*(d*x)^(7/2))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(7*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)^{7/2}}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2}{49} x (d x)^{5/2} \left (7 a-2 b n+7 b \log \left (c x^n\right )\right ) \]

[In]

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

[Out]

(2*x*(d*x)^(5/2)*(7*a - 2*b*n + 7*b*Log[c*x^n]))/49

Maple [C] (warning: unable to verify)

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

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.12

method	result	size
risch	\(\frac {2 d^{3} b \,x^{4} \ln \left (x^{n}\right )}{7 \sqrt {d x}}+\frac {d^{3} \left (-7 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+7 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+7 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-7 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+14 b \ln \left (c \right )-4 b n +14 a \right ) x^{4}}{49 \sqrt {d x}}\)	\(128\)

[In]

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

[Out]

2/7*d^3*b*x^4/(d*x)^(1/2)*ln(x^n)+1/49*d^3*(-7*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+7*I*b*Pi*csgn(I*c)*c
sgn(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)*x^4/(d*x)^
(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.22 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2}{49} \, {\left (7 \, b d^{2} n x^{3} \log \left (x\right ) + 7 \, b d^{2} x^{3} \log \left (c\right ) - {\left (2 \, b d^{2} n - 7 \, a d^{2}\right )} x^{3}\right )} \sqrt {d x} \]

[In]

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

[Out]

2/49*(7*b*d^2*n*x^3*log(x) + 7*b*d^2*x^3*log(c) - (2*b*d^2*n - 7*a*d^2)*x^3)*sqrt(d*x)

Sympy [A] (verification not implemented)

Time = 13.70 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 a x \left (d x\right )^{\frac {5}{2}}}{7} - \frac {4 b n x \left (d x\right )^{\frac {5}{2}}}{49} + \frac {2 b x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{7} \]

[In]

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

[Out]

2*a*x*(d*x)**(5/2)/7 - 4*b*n*x*(d*x)**(5/2)/49 + 2*b*x*(d*x)**(5/2)*log(c*x**n)/7

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4 \, \left (d x\right )^{\frac {7}{2}} b n}{49 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} b \log \left (c x^{n}\right )}{7 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} a}{7 \, d} \]

[In]

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

[Out]

-4/49*(d*x)^(7/2)*b*n/d + 2/7*(d*x)^(7/2)*b*log(c*x^n)/d + 2/7*(d*x)^(7/2)*a/d

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.85 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\left (\frac {1}{7} i + \frac {1}{7}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) - \left (\frac {1}{7} i - \frac {1}{7}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{49} i + \frac {2}{49}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {2}{49} i - \frac {2}{49}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \frac {2}{7} \, b d^{\frac {5}{2}} x^{\frac {7}{2}} \log \left (c\right ) + \frac {2}{7} \, a d^{\frac {5}{2}} x^{\frac {7}{2}} \]

[In]

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

[Out]

(1/7*I + 1/7)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) - (1/7*I - 1/7)*sqrt(2)*b*d^2*n*x
^(7/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/49*I + 2/49)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*p
i*sgn(d)) + (2/49*I - 2/49)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/7*b*d^(5/2)*x^(7/2)*lo
g(c) + 2/7*a*d^(5/2)*x^(7/2)

Mupad [F(-1)]

Timed out. \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (d\,x\right )}^{5/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

int((d*x)^(5/2)*(a + b*log(c*x^n)),x)

[Out]

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