Optimal. Leaf size=50 \[ -\frac {a^4 c^3}{4 x^4}+\frac {2 a^3 b c^3}{3 x^3}-\frac {2 a b^3 c^3}{x}-b^4 c^3 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 50, 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 = {76}
\begin {gather*} -\frac {a^4 c^3}{4 x^4}+\frac {2 a^3 b c^3}{3 x^3}-\frac {2 a b^3 c^3}{x}-b^4 c^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^3}{x^5} \, dx &=\int \left (\frac {a^4 c^3}{x^5}-\frac {2 a^3 b c^3}{x^4}+\frac {2 a b^3 c^3}{x^2}-\frac {b^4 c^3}{x}\right ) \, dx\\ &=-\frac {a^4 c^3}{4 x^4}+\frac {2 a^3 b c^3}{3 x^3}-\frac {2 a b^3 c^3}{x}-b^4 c^3 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 42, normalized size = 0.84 \begin {gather*} c^3 \left (-\frac {a^4}{4 x^4}+\frac {2 a^3 b}{3 x^3}-\frac {2 a b^3}{x}-b^4 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 39, normalized size = 0.78
method | result | size |
default | \(c^{3} \left (-\frac {2 a \,b^{3}}{x}-\frac {a^{4}}{4 x^{4}}-b^{4} \ln \left (x \right )+\frac {2 a^{3} b}{3 x^{3}}\right )\) | \(39\) |
norman | \(\frac {-\frac {1}{4} a^{4} c^{3}-2 a \,b^{3} c^{3} x^{3}+\frac {2}{3} a^{3} b \,c^{3} x}{x^{4}}-b^{4} c^{3} \ln \left (x \right )\) | \(47\) |
risch | \(\frac {-\frac {1}{4} a^{4} c^{3}-2 a \,b^{3} c^{3} x^{3}+\frac {2}{3} a^{3} b \,c^{3} x}{x^{4}}-b^{4} c^{3} \ln \left (x \right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 47, normalized size = 0.94 \begin {gather*} -b^{4} c^{3} \log \left (x\right ) - \frac {24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.91, size = 49, normalized size = 0.98 \begin {gather*} -\frac {12 \, b^{4} c^{3} x^{4} \log \left (x\right ) + 24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.11, size = 49, normalized size = 0.98 \begin {gather*} - b^{4} c^{3} \log {\left (x \right )} - \frac {3 a^{4} c^{3} - 8 a^{3} b c^{3} x + 24 a b^{3} c^{3} x^{3}}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.56, size = 48, normalized size = 0.96 \begin {gather*} -b^{4} c^{3} \log \left ({\left | x \right |}\right ) - \frac {24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 40, normalized size = 0.80 \begin {gather*} -\frac {c^3\,\left (3\,a^4+24\,a\,b^3\,x^3+12\,b^4\,x^4\,\ln \left (x\right )-8\,a^3\,b\,x\right )}{12\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________