Integrand size = 22, antiderivative size = 123 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=\frac {1}{2} a c^2 (2 b c+3 a d) x^2+\frac {1}{4} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+\frac {1}{6} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+\frac {1}{8} b d^2 (3 b c+2 a d) x^8+\frac {1}{10} b^2 d^3 x^{10}+a^2 c^3 \log (x) \]
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Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=\frac {1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac {1}{2} a c^2 x^2 (3 a d+2 b c)+\frac {1}{8} b d^2 x^8 (2 a d+3 b c)+\frac {1}{10} b^2 d^3 x^{10} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^3}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (a c^2 (2 b c+3 a d)+\frac {a^2 c^3}{x}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^2+b d^2 (3 b c+2 a d) x^3+b^2 d^3 x^4\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} a c^2 (2 b c+3 a d) x^2+\frac {1}{4} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+\frac {1}{6} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+\frac {1}{8} b d^2 (3 b c+2 a d) x^8+\frac {1}{10} b^2 d^3 x^{10}+a^2 c^3 \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=\frac {1}{2} a c^2 (2 b c+3 a d) x^2+\frac {1}{4} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+\frac {1}{6} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+\frac {1}{8} b d^2 (3 b c+2 a d) x^8+\frac {1}{10} b^2 d^3 x^{10}+a^2 c^3 \log (x) \]
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Time = 2.70 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\left (\frac {1}{4} a b \,d^{3}+\frac {3}{8} b^{2} c \,d^{2}\right ) x^{8}+\left (\frac {3}{2} a^{2} c^{2} d +a b \,c^{3}\right ) x^{2}+\left (\frac {1}{6} a^{2} d^{3}+a b c \,d^{2}+\frac {1}{2} b^{2} c^{2} d \right ) x^{6}+\left (\frac {3}{4} c \,a^{2} d^{2}+\frac {3}{2} a b \,c^{2} d +\frac {1}{4} b^{2} c^{3}\right ) x^{4}+\frac {b^{2} d^{3} x^{10}}{10}+a^{2} c^{3} \ln \left (x \right )\) | \(122\) |
default | \(\frac {b^{2} d^{3} x^{10}}{10}+\frac {a b \,d^{3} x^{8}}{4}+\frac {3 b^{2} c \,d^{2} x^{8}}{8}+\frac {a^{2} d^{3} x^{6}}{6}+x^{6} d^{2} a b c +\frac {b^{2} c^{2} d \,x^{6}}{2}+\frac {3 a^{2} c \,d^{2} x^{4}}{4}+\frac {3 a b \,c^{2} d \,x^{4}}{2}+\frac {b^{2} c^{3} x^{4}}{4}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}+a b \,c^{3} x^{2}+a^{2} c^{3} \ln \left (x \right )\) | \(132\) |
risch | \(\frac {b^{2} d^{3} x^{10}}{10}+\frac {a b \,d^{3} x^{8}}{4}+\frac {3 b^{2} c \,d^{2} x^{8}}{8}+\frac {a^{2} d^{3} x^{6}}{6}+x^{6} d^{2} a b c +\frac {b^{2} c^{2} d \,x^{6}}{2}+\frac {3 a^{2} c \,d^{2} x^{4}}{4}+\frac {3 a b \,c^{2} d \,x^{4}}{2}+\frac {b^{2} c^{3} x^{4}}{4}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}+a b \,c^{3} x^{2}+a^{2} c^{3} \ln \left (x \right )\) | \(132\) |
parallelrisch | \(\frac {b^{2} d^{3} x^{10}}{10}+\frac {a b \,d^{3} x^{8}}{4}+\frac {3 b^{2} c \,d^{2} x^{8}}{8}+\frac {a^{2} d^{3} x^{6}}{6}+x^{6} d^{2} a b c +\frac {b^{2} c^{2} d \,x^{6}}{2}+\frac {3 a^{2} c \,d^{2} x^{4}}{4}+\frac {3 a b \,c^{2} d \,x^{4}}{2}+\frac {b^{2} c^{3} x^{4}}{4}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}+a b \,c^{3} x^{2}+a^{2} c^{3} \ln \left (x \right )\) | \(132\) |
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Time = 0.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=\frac {1}{10} \, b^{2} d^{3} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + \frac {1}{6} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + a^{2} c^{3} \log \left (x\right ) + \frac {1}{4} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]
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Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=a^{2} c^{3} \log {\left (x \right )} + \frac {b^{2} d^{3} x^{10}}{10} + x^{8} \left (\frac {a b d^{3}}{4} + \frac {3 b^{2} c d^{2}}{8}\right ) + x^{6} \left (\frac {a^{2} d^{3}}{6} + a b c d^{2} + \frac {b^{2} c^{2} d}{2}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c d^{2}}{4} + \frac {3 a b c^{2} d}{2} + \frac {b^{2} c^{3}}{4}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=\frac {1}{10} \, b^{2} d^{3} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + \frac {1}{6} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + \frac {1}{2} \, a^{2} c^{3} \log \left (x^{2}\right ) + \frac {1}{4} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=\frac {1}{10} \, b^{2} d^{3} x^{10} + \frac {3}{8} \, b^{2} c d^{2} x^{8} + \frac {1}{4} \, a b d^{3} x^{8} + \frac {1}{2} \, b^{2} c^{2} d x^{6} + a b c d^{2} x^{6} + \frac {1}{6} \, a^{2} d^{3} x^{6} + \frac {1}{4} \, b^{2} c^{3} x^{4} + \frac {3}{2} \, a b c^{2} d x^{4} + \frac {3}{4} \, a^{2} c d^{2} x^{4} + a b c^{3} x^{2} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {1}{2} \, a^{2} c^{3} \log \left (x^{2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx=x^4\,\left (\frac {3\,a^2\,c\,d^2}{4}+\frac {3\,a\,b\,c^2\,d}{2}+\frac {b^2\,c^3}{4}\right )+x^6\,\left (\frac {a^2\,d^3}{6}+a\,b\,c\,d^2+\frac {b^2\,c^2\,d}{2}\right )+\frac {b^2\,d^3\,x^{10}}{10}+a^2\,c^3\,\ln \left (x\right )+\frac {a\,c^2\,x^2\,\left (3\,a\,d+2\,b\,c\right )}{2}+\frac {b\,d^2\,x^8\,\left (2\,a\,d+3\,b\,c\right )}{8} \]
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