Optimal. Leaf size=80 \[ -\frac {a^3}{x}+3 a^2 b x+a \left (b^2+a c\right ) x^3+\frac {1}{5} b \left (b^2+6 a c\right ) x^5+\frac {3}{7} c \left (b^2+a c\right ) x^7+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1122}
\begin {gather*} -\frac {a^3}{x}+3 a^2 b x+\frac {3}{7} c x^7 \left (a c+b^2\right )+\frac {1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 1122
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^3}{x^2} \, dx &=\int \left (3 a^2 b+\frac {a^3}{x^2}+3 a \left (b^2+a c\right ) x^2+b \left (b^2+6 a c\right ) x^4+3 c \left (b^2+a c\right ) x^6+3 b c^2 x^8+c^3 x^{10}\right ) \, dx\\ &=-\frac {a^3}{x}+3 a^2 b x+a \left (b^2+a c\right ) x^3+\frac {1}{5} b \left (b^2+6 a c\right ) x^5+\frac {3}{7} c \left (b^2+a c\right ) x^7+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 80, normalized size = 1.00 \begin {gather*} -\frac {a^3}{x}+3 a^2 b x+a \left (b^2+a c\right ) x^3+\frac {1}{5} b \left (b^2+6 a c\right ) x^5+\frac {3}{7} c \left (b^2+a c\right ) x^7+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 84, normalized size = 1.05
method | result | size |
default | \(\frac {c^{3} x^{11}}{11}+\frac {b \,c^{2} x^{9}}{3}+\frac {3 a \,c^{2} x^{7}}{7}+\frac {3 b^{2} c \,x^{7}}{7}+\frac {6 a b c \,x^{5}}{5}+\frac {b^{3} x^{5}}{5}+a^{2} c \,x^{3}+a \,b^{2} x^{3}+3 a^{2} b x -\frac {a^{3}}{x}\) | \(84\) |
norman | \(\frac {\frac {c^{3} x^{12}}{11}+\frac {b \,c^{2} x^{10}}{3}+\left (\frac {3}{7} c^{2} a +\frac {3}{7} b^{2} c \right ) x^{8}+\left (\frac {6}{5} a b c +\frac {1}{5} b^{3}\right ) x^{6}+\left (a^{2} c +a \,b^{2}\right ) x^{4}+3 a^{2} b \,x^{2}-a^{3}}{x}\) | \(84\) |
risch | \(\frac {c^{3} x^{11}}{11}+\frac {b \,c^{2} x^{9}}{3}+\frac {3 a \,c^{2} x^{7}}{7}+\frac {3 b^{2} c \,x^{7}}{7}+\frac {6 a b c \,x^{5}}{5}+\frac {b^{3} x^{5}}{5}+a^{2} c \,x^{3}+a \,b^{2} x^{3}+3 a^{2} b x -\frac {a^{3}}{x}\) | \(84\) |
gosper | \(-\frac {-105 c^{3} x^{12}-385 b \,c^{2} x^{10}-495 a \,c^{2} x^{8}-495 b^{2} c \,x^{8}-1386 a b c \,x^{6}-231 b^{3} x^{6}-1155 a^{2} c \,x^{4}-1155 a \,b^{2} x^{4}-3465 a^{2} b \,x^{2}+1155 a^{3}}{1155 x}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 78, normalized size = 0.98 \begin {gather*} \frac {1}{11} \, c^{3} x^{11} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{7} \, {\left (b^{2} c + a c^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b^{3} + 6 \, a b c\right )} x^{5} + 3 \, a^{2} b x + {\left (a b^{2} + a^{2} c\right )} x^{3} - \frac {a^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 83, normalized size = 1.04 \begin {gather*} \frac {105 \, c^{3} x^{12} + 385 \, b c^{2} x^{10} + 495 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 231 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 3465 \, a^{2} b x^{2} + 1155 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 1155 \, a^{3}}{1155 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 82, normalized size = 1.02 \begin {gather*} - \frac {a^{3}}{x} + 3 a^{2} b x + \frac {b c^{2} x^{9}}{3} + \frac {c^{3} x^{11}}{11} + x^{7} \cdot \left (\frac {3 a c^{2}}{7} + \frac {3 b^{2} c}{7}\right ) + x^{5} \cdot \left (\frac {6 a b c}{5} + \frac {b^{3}}{5}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.87, size = 83, normalized size = 1.04 \begin {gather*} \frac {1}{11} \, c^{3} x^{11} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{7} \, b^{2} c x^{7} + \frac {3}{7} \, a c^{2} x^{7} + \frac {1}{5} \, b^{3} x^{5} + \frac {6}{5} \, a b c x^{5} + a b^{2} x^{3} + a^{2} c x^{3} + 3 \, a^{2} b x - \frac {a^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 73, normalized size = 0.91 \begin {gather*} x^5\,\left (\frac {b^3}{5}+\frac {6\,a\,c\,b}{5}\right )-\frac {a^3}{x}+\frac {c^3\,x^{11}}{11}+\frac {b\,c^2\,x^9}{3}+a\,x^3\,\left (b^2+a\,c\right )+\frac {3\,c\,x^7\,\left (b^2+a\,c\right )}{7}+3\,a^2\,b\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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