Optimal. Leaf size=98 \[ -\frac {a^8}{3 x^3}-\frac {8 a^7 b}{x}+28 a^6 b^2 x+\frac {56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac {28}{9} a^2 b^6 x^9+\frac {8}{11} a b^7 x^{11}+\frac {b^8 x^{13}}{13} \]
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Rubi [A]
time = 0.03, antiderivative size = 98, 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 = {276}
\begin {gather*} -\frac {a^8}{3 x^3}-\frac {8 a^7 b}{x}+28 a^6 b^2 x+\frac {56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac {28}{9} a^2 b^6 x^9+\frac {8}{11} a b^7 x^{11}+\frac {b^8 x^{13}}{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^4} \, dx &=\int \left (28 a^6 b^2+\frac {a^8}{x^4}+\frac {8 a^7 b}{x^2}+56 a^5 b^3 x^2+70 a^4 b^4 x^4+56 a^3 b^5 x^6+28 a^2 b^6 x^8+8 a b^7 x^{10}+b^8 x^{12}\right ) \, dx\\ &=-\frac {a^8}{3 x^3}-\frac {8 a^7 b}{x}+28 a^6 b^2 x+\frac {56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac {28}{9} a^2 b^6 x^9+\frac {8}{11} a b^7 x^{11}+\frac {b^8 x^{13}}{13}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 98, normalized size = 1.00 \begin {gather*} -\frac {a^8}{3 x^3}-\frac {8 a^7 b}{x}+28 a^6 b^2 x+\frac {56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac {28}{9} a^2 b^6 x^9+\frac {8}{11} a b^7 x^{11}+\frac {b^8 x^{13}}{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 89, normalized size = 0.91
method | result | size |
default | \(-\frac {a^{8}}{3 x^{3}}-\frac {8 a^{7} b}{x}+28 a^{6} b^{2} x +\frac {56 a^{5} b^{3} x^{3}}{3}+14 a^{4} b^{4} x^{5}+8 a^{3} b^{5} x^{7}+\frac {28 a^{2} b^{6} x^{9}}{9}+\frac {8 a \,b^{7} x^{11}}{11}+\frac {b^{8} x^{13}}{13}\) | \(89\) |
risch | \(\frac {b^{8} x^{13}}{13}+\frac {8 a \,b^{7} x^{11}}{11}+\frac {28 a^{2} b^{6} x^{9}}{9}+8 a^{3} b^{5} x^{7}+14 a^{4} b^{4} x^{5}+\frac {56 a^{5} b^{3} x^{3}}{3}+28 a^{6} b^{2} x +\frac {-8 a^{7} b \,x^{2}-\frac {1}{3} a^{8}}{x^{3}}\) | \(91\) |
norman | \(\frac {-\frac {1}{3} a^{8}-8 a^{7} b \,x^{2}+28 a^{6} b^{2} x^{4}+\frac {56}{3} a^{5} b^{3} x^{6}+14 a^{4} b^{4} x^{8}+8 a^{3} b^{5} x^{10}+\frac {28}{9} a^{2} b^{6} x^{12}+\frac {8}{11} a \,b^{7} x^{14}+\frac {1}{13} b^{8} x^{16}}{x^{3}}\) | \(92\) |
gosper | \(-\frac {-99 b^{8} x^{16}-936 a \,b^{7} x^{14}-4004 a^{2} b^{6} x^{12}-10296 a^{3} b^{5} x^{10}-18018 a^{4} b^{4} x^{8}-24024 a^{5} b^{3} x^{6}-36036 a^{6} b^{2} x^{4}+10296 a^{7} b \,x^{2}+429 a^{8}}{1287 x^{3}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 89, normalized size = 0.91 \begin {gather*} \frac {1}{13} \, b^{8} x^{13} + \frac {8}{11} \, a b^{7} x^{11} + \frac {28}{9} \, a^{2} b^{6} x^{9} + 8 \, a^{3} b^{5} x^{7} + 14 \, a^{4} b^{4} x^{5} + \frac {56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} x - \frac {24 \, a^{7} b x^{2} + a^{8}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 92, normalized size = 0.94 \begin {gather*} \frac {99 \, b^{8} x^{16} + 936 \, a b^{7} x^{14} + 4004 \, a^{2} b^{6} x^{12} + 10296 \, a^{3} b^{5} x^{10} + 18018 \, a^{4} b^{4} x^{8} + 24024 \, a^{5} b^{3} x^{6} + 36036 \, a^{6} b^{2} x^{4} - 10296 \, a^{7} b x^{2} - 429 \, a^{8}}{1287 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 100, normalized size = 1.02 \begin {gather*} 28 a^{6} b^{2} x + \frac {56 a^{5} b^{3} x^{3}}{3} + 14 a^{4} b^{4} x^{5} + 8 a^{3} b^{5} x^{7} + \frac {28 a^{2} b^{6} x^{9}}{9} + \frac {8 a b^{7} x^{11}}{11} + \frac {b^{8} x^{13}}{13} + \frac {- a^{8} - 24 a^{7} b x^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 89, normalized size = 0.91 \begin {gather*} \frac {1}{13} \, b^{8} x^{13} + \frac {8}{11} \, a b^{7} x^{11} + \frac {28}{9} \, a^{2} b^{6} x^{9} + 8 \, a^{3} b^{5} x^{7} + 14 \, a^{4} b^{4} x^{5} + \frac {56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} x - \frac {24 \, a^{7} b x^{2} + a^{8}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 91, normalized size = 0.93 \begin {gather*} \frac {b^8\,x^{13}}{13}-\frac {\frac {a^8}{3}+8\,b\,a^7\,x^2}{x^3}+28\,a^6\,b^2\,x+\frac {8\,a\,b^7\,x^{11}}{11}+\frac {56\,a^5\,b^3\,x^3}{3}+14\,a^4\,b^4\,x^5+8\,a^3\,b^5\,x^7+\frac {28\,a^2\,b^6\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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