Optimal. Leaf size=113 \[ -\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 113, 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} \[ -\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a^6 b c^6}{6 x^6}-\frac {a^7 c^6}{7 x^7}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx &=\int \left (\frac {a^7 c^6}{x^8}-\frac {5 a^6 b c^6}{x^7}+\frac {9 a^5 b^2 c^6}{x^6}-\frac {5 a^4 b^3 c^6}{x^5}-\frac {5 a^3 b^4 c^6}{x^4}+\frac {9 a^2 b^5 c^6}{x^3}-\frac {5 a b^6 c^6}{x^2}+\frac {b^7 c^6}{x}\right ) \, dx\\ &=-\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 113, normalized size = 1.00 \[ -\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 105, normalized size = 0.93 \[ \frac {420 \, b^{7} c^{6} x^{7} \log \relax (x) + 2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.03, size = 103, normalized size = 0.91 \[ b^{7} c^{6} \log \left ({\left | x \right |}\right ) + \frac {2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 102, normalized size = 0.90 \[ b^{7} c^{6} \ln \relax (x )+\frac {5 a \,b^{6} c^{6}}{x}-\frac {9 a^{2} b^{5} c^{6}}{2 x^{2}}+\frac {5 a^{3} b^{4} c^{6}}{3 x^{3}}+\frac {5 a^{4} b^{3} c^{6}}{4 x^{4}}-\frac {9 a^{5} b^{2} c^{6}}{5 x^{5}}+\frac {5 a^{6} b \,c^{6}}{6 x^{6}}-\frac {a^{7} c^{6}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.10, size = 102, normalized size = 0.90 \[ b^{7} c^{6} \log \relax (x) + \frac {2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.31, size = 82, normalized size = 0.73 \[ \frac {c^6\,\left (5\,a\,b^6\,x^6-\frac {a^7}{7}-\frac {9\,a^5\,b^2\,x^2}{5}+\frac {5\,a^4\,b^3\,x^3}{4}+\frac {5\,a^3\,b^4\,x^4}{3}-\frac {9\,a^2\,b^5\,x^5}{2}+b^7\,x^7\,\ln \relax (x)+\frac {5\,a^6\,b\,x}{6}\right )}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.57, size = 109, normalized size = 0.96 \[ b^{7} c^{6} \log {\relax (x )} + \frac {- 60 a^{7} c^{6} + 350 a^{6} b c^{6} x - 756 a^{5} b^{2} c^{6} x^{2} + 525 a^{4} b^{3} c^{6} x^{3} + 700 a^{3} b^{4} c^{6} x^{4} - 1890 a^{2} b^{5} c^{6} x^{5} + 2100 a b^{6} c^{6} x^{6}}{420 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________