Optimal. Leaf size=102 \[ -\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {28}{5} a^2 b^6 x^5+\frac {56}{3} a^3 b^5 x^3-\frac {28 a^6 b^2}{3 x^3}+70 a^4 b^4 x-\frac {56 a^5 b^3}{x}-\frac {8 a^7 b}{5 x^5}-\frac {a^8}{7 x^7}+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^8} \, dx &=\int \left (70 a^4 b^4+\frac {a^8}{x^8}+\frac {8 a^7 b}{x^6}+\frac {28 a^6 b^2}{x^4}+\frac {56 a^5 b^3}{x^2}+56 a^3 b^5 x^2+28 a^2 b^6 x^4+8 a b^7 x^6+b^8 x^8\right ) \, dx\\ &=-\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 102, normalized size = 1.00 \[ -\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 92, normalized size = 0.90 \[ \frac {35 \, b^{8} x^{16} + 360 \, a b^{7} x^{14} + 1764 \, a^{2} b^{6} x^{12} + 5880 \, a^{3} b^{5} x^{10} + 22050 \, a^{4} b^{4} x^{8} - 17640 \, a^{5} b^{3} x^{6} - 2940 \, a^{6} b^{2} x^{4} - 504 \, a^{7} b x^{2} - 45 \, a^{8}}{315 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.93, size = 91, normalized size = 0.89 \[ \frac {1}{9} \, b^{8} x^{9} + \frac {8}{7} \, a b^{7} x^{7} + \frac {28}{5} \, a^{2} b^{6} x^{5} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} x - \frac {5880 \, a^{5} b^{3} x^{6} + 980 \, a^{6} b^{2} x^{4} + 168 \, a^{7} b x^{2} + 15 \, a^{8}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 89, normalized size = 0.87 \[ \frac {b^{8} x^{9}}{9}+\frac {8 a \,b^{7} x^{7}}{7}+\frac {28 a^{2} b^{6} x^{5}}{5}+\frac {56 a^{3} b^{5} x^{3}}{3}+70 a^{4} b^{4} x -\frac {56 a^{5} b^{3}}{x}-\frac {28 a^{6} b^{2}}{3 x^{3}}-\frac {8 a^{7} b}{5 x^{5}}-\frac {a^{8}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 91, normalized size = 0.89 \[ \frac {1}{9} \, b^{8} x^{9} + \frac {8}{7} \, a b^{7} x^{7} + \frac {28}{5} \, a^{2} b^{6} x^{5} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} x - \frac {5880 \, a^{5} b^{3} x^{6} + 980 \, a^{6} b^{2} x^{4} + 168 \, a^{7} b x^{2} + 15 \, a^{8}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.80, size = 91, normalized size = 0.89 \[ \frac {b^8\,x^9}{9}-\frac {\frac {a^8}{7}+\frac {8\,a^7\,b\,x^2}{5}+\frac {28\,a^6\,b^2\,x^4}{3}+56\,a^5\,b^3\,x^6}{x^7}+70\,a^4\,b^4\,x+\frac {8\,a\,b^7\,x^7}{7}+\frac {56\,a^3\,b^5\,x^3}{3}+\frac {28\,a^2\,b^6\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.33, size = 102, normalized size = 1.00 \[ 70 a^{4} b^{4} x + \frac {56 a^{3} b^{5} x^{3}}{3} + \frac {28 a^{2} b^{6} x^{5}}{5} + \frac {8 a b^{7} x^{7}}{7} + \frac {b^{8} x^{9}}{9} + \frac {- 15 a^{8} - 168 a^{7} b x^{2} - 980 a^{6} b^{2} x^{4} - 5880 a^{5} b^{3} x^{6}}{105 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________