Integrand size = 20, antiderivative size = 99 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=-\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a \left (b^2+a c\right ) x^{3/2}+\frac {2}{7} b \left (b^2+6 a c\right ) x^{7/2}+\frac {6}{11} c \left (b^2+a c\right ) x^{11/2}+\frac {2}{5} b c^2 x^{15/2}+\frac {2}{19} c^3 x^{19/2} \]
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Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1122} \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=-\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+\frac {6}{11} c x^{11/2} \left (a c+b^2\right )+\frac {2}{7} b x^{7/2} \left (6 a c+b^2\right )+2 a x^{3/2} \left (a c+b^2\right )+\frac {2}{5} b c^2 x^{15/2}+\frac {2}{19} c^3 x^{19/2} \]
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Rule 1122
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3}{x^{7/2}}+\frac {3 a^2 b}{x^{3/2}}+3 a \left (b^2+a c\right ) \sqrt {x}+b \left (b^2+6 a c\right ) x^{5/2}+3 c \left (b^2+a c\right ) x^{9/2}+3 b c^2 x^{13/2}+c^3 x^{17/2}\right ) \, dx \\ & = -\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a \left (b^2+a c\right ) x^{3/2}+\frac {2}{7} b \left (b^2+6 a c\right ) x^{7/2}+\frac {6}{11} c \left (b^2+a c\right ) x^{11/2}+\frac {2}{5} b c^2 x^{15/2}+\frac {2}{19} c^3 x^{19/2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=-\frac {2 \left (1463 a^3+21945 a^2 b x^2-7315 a b^2 x^4-7315 a^2 c x^4-1045 b^3 x^6-6270 a b c x^6-1995 b^2 c x^8-1995 a c^2 x^8-1463 b c^2 x^{10}-385 c^3 x^{12}\right )}{7315 x^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 c^{3} x^{\frac {19}{2}}}{19}+\frac {2 b \,c^{2} x^{\frac {15}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {11}{2}}}{11}+\frac {6 b^{2} c \,x^{\frac {11}{2}}}{11}+\frac {12 a b c \,x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {7}{2}}}{7}+2 a^{2} c \,x^{\frac {3}{2}}+2 a \,b^{2} x^{\frac {3}{2}}-\frac {6 a^{2} b}{\sqrt {x}}-\frac {2 a^{3}}{5 x^{\frac {5}{2}}}\) | \(88\) |
default | \(\frac {2 c^{3} x^{\frac {19}{2}}}{19}+\frac {2 b \,c^{2} x^{\frac {15}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {11}{2}}}{11}+\frac {6 b^{2} c \,x^{\frac {11}{2}}}{11}+\frac {12 a b c \,x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {7}{2}}}{7}+2 a^{2} c \,x^{\frac {3}{2}}+2 a \,b^{2} x^{\frac {3}{2}}-\frac {6 a^{2} b}{\sqrt {x}}-\frac {2 a^{3}}{5 x^{\frac {5}{2}}}\) | \(88\) |
gosper | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
trager | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
risch | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
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Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2 \, {\left (385 \, c^{3} x^{12} + 1463 \, b c^{2} x^{10} + 1995 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 1045 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} - 21945 \, a^{2} b x^{2} + 7315 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 1463 \, a^{3}\right )}}{7315 \, x^{\frac {5}{2}}} \]
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Time = 1.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=- \frac {2 a^{3}}{5 x^{\frac {5}{2}}} - \frac {6 a^{2} b}{\sqrt {x}} + 2 a^{2} c x^{\frac {3}{2}} + 2 a b^{2} x^{\frac {3}{2}} + \frac {12 a b c x^{\frac {7}{2}}}{7} + \frac {6 a c^{2} x^{\frac {11}{2}}}{11} + \frac {2 b^{3} x^{\frac {7}{2}}}{7} + \frac {6 b^{2} c x^{\frac {11}{2}}}{11} + \frac {2 b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 c^{3} x^{\frac {19}{2}}}{19} \]
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Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b c^{2} x^{\frac {15}{2}} + \frac {6}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {7}{2}} + 2 \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b c^{2} x^{\frac {15}{2}} + \frac {6}{11} \, b^{2} c x^{\frac {11}{2}} + \frac {6}{11} \, a c^{2} x^{\frac {11}{2}} + \frac {2}{7} \, b^{3} x^{\frac {7}{2}} + \frac {12}{7} \, a b c x^{\frac {7}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} + 2 \, a^{2} c x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=x^{7/2}\,\left (\frac {2\,b^3}{7}+\frac {12\,a\,c\,b}{7}\right )-\frac {\frac {2\,a^3}{5}+6\,b\,a^2\,x^2}{x^{5/2}}+\frac {2\,c^3\,x^{19/2}}{19}+\frac {2\,b\,c^2\,x^{15/2}}{5}+2\,a\,x^{3/2}\,\left (b^2+a\,c\right )+\frac {6\,c\,x^{11/2}\,\left (b^2+a\,c\right )}{11} \]
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