Optimal. Leaf size=83 \[ -\frac {a^3}{3 x^3}-\frac {3 a^2 b}{x}+3 a \left (b^2+a c\right ) x+\frac {1}{3} b \left (b^2+6 a c\right ) x^3+\frac {3}{5} c \left (b^2+a c\right ) x^5+\frac {3}{7} b c^2 x^7+\frac {c^3 x^9}{9} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, 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}{3 x^3}-\frac {3 a^2 b}{x}+\frac {3}{5} c x^5 \left (a c+b^2\right )+\frac {1}{3} b x^3 \left (6 a c+b^2\right )+3 a x \left (a c+b^2\right )+\frac {3}{7} b c^2 x^7+\frac {c^3 x^9}{9} \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^4} \, dx &=\int \left (3 a \left (b^2+a c\right )+\frac {a^3}{x^4}+\frac {3 a^2 b}{x^2}+b \left (b^2+6 a c\right ) x^2+3 c \left (b^2+a c\right ) x^4+3 b c^2 x^6+c^3 x^8\right ) \, dx\\ &=-\frac {a^3}{3 x^3}-\frac {3 a^2 b}{x}+3 a \left (b^2+a c\right ) x+\frac {1}{3} b \left (b^2+6 a c\right ) x^3+\frac {3}{5} c \left (b^2+a c\right ) x^5+\frac {3}{7} b c^2 x^7+\frac {c^3 x^9}{9}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 83, normalized size = 1.00 \begin {gather*} -\frac {a^3}{3 x^3}-\frac {3 a^2 b}{x}+3 a \left (b^2+a c\right ) x+\frac {1}{3} b \left (b^2+6 a c\right ) x^3+\frac {3}{5} c \left (b^2+a c\right ) x^5+\frac {3}{7} b c^2 x^7+\frac {c^3 x^9}{9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 84, normalized size = 1.01
method | result | size |
default | \(\frac {c^{3} x^{9}}{9}+\frac {3 b \,c^{2} x^{7}}{7}+\frac {3 a \,c^{2} x^{5}}{5}+\frac {3 b^{2} c \,x^{5}}{5}+2 a b c \,x^{3}+\frac {b^{3} x^{3}}{3}+3 a^{2} c x +3 a \,b^{2} x -\frac {3 a^{2} b}{x}-\frac {a^{3}}{3 x^{3}}\) | \(84\) |
norman | \(\frac {\frac {c^{3} x^{12}}{9}+\frac {3 b \,c^{2} x^{10}}{7}+\left (\frac {3}{5} c^{2} a +\frac {3}{5} b^{2} c \right ) x^{8}+\left (2 a b c +\frac {1}{3} b^{3}\right ) x^{6}+\left (3 a^{2} c +3 a \,b^{2}\right ) x^{4}-3 a^{2} b \,x^{2}-\frac {a^{3}}{3}}{x^{3}}\) | \(86\) |
risch | \(\frac {c^{3} x^{9}}{9}+\frac {3 b \,c^{2} x^{7}}{7}+\frac {3 a \,c^{2} x^{5}}{5}+\frac {3 b^{2} c \,x^{5}}{5}+2 a b c \,x^{3}+\frac {b^{3} x^{3}}{3}+3 a^{2} c x +3 a \,b^{2} x +\frac {-3 a^{2} b \,x^{2}-\frac {1}{3} a^{3}}{x^{3}}\) | \(86\) |
gosper | \(-\frac {-35 c^{3} x^{12}-135 b \,c^{2} x^{10}-189 a \,c^{2} x^{8}-189 b^{2} c \,x^{8}-630 a b c \,x^{6}-105 b^{3} x^{6}-945 a^{2} c \,x^{4}-945 a \,b^{2} x^{4}+945 a^{2} b \,x^{2}+105 a^{3}}{315 x^{3}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 80, normalized size = 0.96 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} + \frac {3}{7} \, b c^{2} x^{7} + \frac {3}{5} \, {\left (b^{2} c + a c^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x - \frac {9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \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 = 1.00 \begin {gather*} \frac {35 \, c^{3} x^{12} + 135 \, b c^{2} x^{10} + 189 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 105 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} - 945 \, a^{2} b x^{2} + 945 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 105 \, a^{3}}{315 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 90, normalized size = 1.08 \begin {gather*} \frac {3 b c^{2} x^{7}}{7} + \frac {c^{3} x^{9}}{9} + x^{5} \cdot \left (\frac {3 a c^{2}}{5} + \frac {3 b^{2} c}{5}\right ) + x^{3} \cdot \left (2 a b c + \frac {b^{3}}{3}\right ) + x \left (3 a^{2} c + 3 a b^{2}\right ) + \frac {- a^{3} - 9 a^{2} 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 = 4.17, size = 84, normalized size = 1.01 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} + \frac {3}{7} \, b c^{2} x^{7} + \frac {3}{5} \, b^{2} c x^{5} + \frac {3}{5} \, a c^{2} x^{5} + \frac {1}{3} \, b^{3} x^{3} + 2 \, a b c x^{3} + 3 \, a b^{2} x + 3 \, a^{2} c x - \frac {9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 77, normalized size = 0.93 \begin {gather*} x^3\,\left (\frac {b^3}{3}+2\,a\,c\,b\right )-\frac {\frac {a^3}{3}+3\,b\,a^2\,x^2}{x^3}+\frac {c^3\,x^9}{9}+\frac {3\,b\,c^2\,x^7}{7}+3\,a\,x\,\left (b^2+a\,c\right )+\frac {3\,c\,x^5\,\left (b^2+a\,c\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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