Optimal. Leaf size=67 \[ -\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}}-\frac {16 b^2 n^2}{d \sqrt {d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 67, 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} \[ -\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}}-\frac {16 b^2 n^2}{d \sqrt {d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}}+(4 b n) \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx\\ &=-\frac {16 b^2 n^2}{d \sqrt {d x}}-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt {d x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 54, normalized size = 0.81 \[ -\frac {2 x \left (a^2+2 b (a+2 b n) \log \left (c x^n\right )+4 a b n+b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 87, normalized size = 1.30 \[ -\frac {2 \, {\left (b^{2} n^{2} \log \relax (x)^{2} + 8 \, b^{2} n^{2} + b^{2} \log \relax (c)^{2} + 4 \, a b n + a^{2} + 2 \, {\left (2 \, b^{2} n + a b\right )} \log \relax (c) + 2 \, {\left (2 \, b^{2} n^{2} + b^{2} n \log \relax (c) + a b n\right )} \log \relax (x)\right )} \sqrt {d x}}{d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.41, size = 149, normalized size = 2.22 \[ -\frac {2 \, {\left (\frac {b^{2} n^{2} \log \left (d x\right )^{2}}{\sqrt {d x}} - \frac {2 \, {\left (b^{2} n^{2} \log \relax (d) - 2 \, b^{2} n^{2} - b^{2} n \log \relax (c) - a b n\right )} \log \left (d x\right )}{\sqrt {d x}} + \frac {b^{2} n^{2} \log \relax (d)^{2} - 4 \, b^{2} n^{2} \log \relax (d) - 2 \, b^{2} n \log \relax (c) \log \relax (d) + 8 \, b^{2} n^{2} + 4 \, b^{2} n \log \relax (c) + b^{2} \log \relax (c)^{2} - 2 \, a b n \log \relax (d) + 4 \, a b n + 2 \, a b \log \relax (c) + a^{2}}{\sqrt {d x}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.17, size = 707, normalized size = 10.55 \[ -\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{\sqrt {d x}\, d}-\frac {2 \left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +2 b \ln \relax (c )+2 a \right ) b \ln \left (x^{n}\right )}{\sqrt {d x}\, d}-\frac {-\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}+2 \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}+2 \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}-4 \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}-8 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-4 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 a^{2}+8 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 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}-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+8 a b \ln \relax (c )+16 b^{2} n \ln \relax (c )+4 b^{2} \ln \relax (c )^{2}+16 a b n -\pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 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 )-4 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 )-8 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 )}{2 \sqrt {d x}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.63, size = 101, normalized size = 1.51 \[ -8 \, b^{2} {\left (\frac {2 \, n^{2}}{\sqrt {d x} d} + \frac {n \log \left (c x^{n}\right )}{\sqrt {d x} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{\sqrt {d x} d} - \frac {8 \, a b n}{\sqrt {d x} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{\sqrt {d x} d} - \frac {2 \, a^{2}}{\sqrt {d x} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.90, size = 201, normalized size = 3.00 \[ - \frac {2 a^{2}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 a b n \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {8 a b n}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 a b \log {\relax (c )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b^{2} n^{2} \log {\relax (x )}^{2}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {8 b^{2} n^{2} \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {16 b^{2} n^{2}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 b^{2} n \log {\relax (c )} \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {8 b^{2} n \log {\relax (c )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b^{2} \log {\relax (c )}^{2}}{d^{\frac {3}{2}} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________