Optimal. Leaf size=78 \[ -\frac {a^5 c^4}{2 x^2}+\frac {3 a^4 b c^4}{x}+2 a^3 b^2 c^4 \log (x)+2 a^2 b^3 c^4 x-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \[ 2 a^2 b^3 c^4 x+2 a^3 b^2 c^4 \log (x)+\frac {3 a^4 b c^4}{x}-\frac {a^5 c^4}{2 x^2}-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx &=\int \left (2 a^2 b^3 c^4+\frac {a^5 c^4}{x^3}-\frac {3 a^4 b c^4}{x^2}+\frac {2 a^3 b^2 c^4}{x}-3 a b^4 c^4 x+b^5 c^4 x^2\right ) \, dx\\ &=-\frac {a^5 c^4}{2 x^2}+\frac {3 a^4 b c^4}{x}+2 a^2 b^3 c^4 x-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3+2 a^3 b^2 c^4 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 78, normalized size = 1.00 \[ -\frac {a^5 c^4}{2 x^2}+\frac {3 a^4 b c^4}{x}+2 a^3 b^2 c^4 \log (x)+2 a^2 b^3 c^4 x-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 77, normalized size = 0.99 \[ \frac {2 \, b^{5} c^{4} x^{5} - 9 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} + 12 \, a^{3} b^{2} c^{4} x^{2} \log \relax (x) + 18 \, a^{4} b c^{4} x - 3 \, a^{5} c^{4}}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.98, size = 74, normalized size = 0.95 \[ \frac {1}{3} \, b^{5} c^{4} x^{3} - \frac {3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \left ({\left | x \right |}\right ) + \frac {6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 73, normalized size = 0.94 \[ \frac {b^{5} c^{4} x^{3}}{3}-\frac {3 a \,b^{4} c^{4} x^{2}}{2}+2 a^{3} b^{2} c^{4} \ln \relax (x )+2 a^{2} b^{3} c^{4} x +\frac {3 a^{4} b \,c^{4}}{x}-\frac {a^{5} c^{4}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 73, normalized size = 0.94 \[ \frac {1}{3} \, b^{5} c^{4} x^{3} - \frac {3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \relax (x) + \frac {6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.28, size = 73, normalized size = 0.94 \[ \frac {b^5\,c^4\,x^3}{3}-\frac {\frac {a^5\,c^4}{2}-3\,a^4\,b\,c^4\,x}{x^2}+2\,a^2\,b^3\,c^4\,x-\frac {3\,a\,b^4\,c^4\,x^2}{2}+2\,a^3\,b^2\,c^4\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.22, size = 78, normalized size = 1.00 \[ 2 a^{3} b^{2} c^{4} \log {\relax (x )} + 2 a^{2} b^{3} c^{4} x - \frac {3 a b^{4} c^{4} x^{2}}{2} + \frac {b^{5} c^{4} x^{3}}{3} + \frac {- a^{5} c^{4} + 6 a^{4} b c^{4} x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________