Integrand size = 22, antiderivative size = 120 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=-\frac {a^2 c^3}{3 x^3}-\frac {a c^2 (2 b c+3 a d)}{x}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x+\frac {1}{3} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^3+\frac {1}{5} b d^2 (3 b c+2 a d) x^5+\frac {1}{7} b^2 d^3 x^7 \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=\frac {1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{3 x^3}-\frac {a c^2 (3 a d+2 b c)}{x}+\frac {1}{5} b d^2 x^5 (2 a d+3 b c)+\frac {1}{7} b^2 d^3 x^7 \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )+\frac {a^2 c^3}{x^4}+\frac {a c^2 (2 b c+3 a d)}{x^2}+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^4+b^2 d^3 x^6\right ) \, dx \\ & = -\frac {a^2 c^3}{3 x^3}-\frac {a c^2 (2 b c+3 a d)}{x}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x+\frac {1}{3} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^3+\frac {1}{5} b d^2 (3 b c+2 a d) x^5+\frac {1}{7} b^2 d^3 x^7 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=-\frac {a^2 c^3}{3 x^3}-\frac {a c^2 (2 b c+3 a d)}{x}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x+\frac {1}{3} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^3+\frac {1}{5} b d^2 (3 b c+2 a d) x^5+\frac {1}{7} b^2 d^3 x^7 \]
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Time = 2.82 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {b^{2} d^{3} x^{7}}{7}+\frac {2 a b \,d^{3} x^{5}}{5}+\frac {3 b^{2} c \,d^{2} x^{5}}{5}+\frac {a^{2} d^{3} x^{3}}{3}+2 x^{3} b \,d^{2} a c +b^{2} c^{2} d \,x^{3}+3 c \,a^{2} d^{2} x +6 a b \,c^{2} d x +b^{2} x \,c^{3}-\frac {a^{2} c^{3}}{3 x^{3}}-\frac {a \,c^{2} \left (3 a d +2 b c \right )}{x}\) | \(124\) |
norman | \(\frac {\frac {b^{2} d^{3} x^{10}}{7}+\left (\frac {2}{5} a b \,d^{3}+\frac {3}{5} b^{2} c \,d^{2}\right ) x^{8}+\left (\frac {1}{3} a^{2} d^{3}+2 a b c \,d^{2}+b^{2} c^{2} d \right ) x^{6}+\left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{4}+\left (-3 a^{2} c^{2} d -2 a b \,c^{3}\right ) x^{2}-\frac {a^{2} c^{3}}{3}}{x^{3}}\) | \(125\) |
risch | \(\frac {b^{2} d^{3} x^{7}}{7}+\frac {2 a b \,d^{3} x^{5}}{5}+\frac {3 b^{2} c \,d^{2} x^{5}}{5}+\frac {a^{2} d^{3} x^{3}}{3}+2 x^{3} b \,d^{2} a c +b^{2} c^{2} d \,x^{3}+3 c \,a^{2} d^{2} x +6 a b \,c^{2} d x +b^{2} x \,c^{3}+\frac {\left (-3 a^{2} c^{2} d -2 a b \,c^{3}\right ) x^{2}-\frac {a^{2} c^{3}}{3}}{x^{3}}\) | \(129\) |
gosper | \(-\frac {-15 b^{2} d^{3} x^{10}-42 a b \,d^{3} x^{8}-63 b^{2} c \,d^{2} x^{8}-35 a^{2} d^{3} x^{6}-210 x^{6} d^{2} a b c -105 b^{2} c^{2} d \,x^{6}-315 a^{2} c \,d^{2} x^{4}-630 a b \,c^{2} d \,x^{4}-105 b^{2} c^{3} x^{4}+315 a^{2} c^{2} d \,x^{2}+210 a b \,c^{3} x^{2}+35 a^{2} c^{3}}{105 x^{3}}\) | \(138\) |
parallelrisch | \(\frac {15 b^{2} d^{3} x^{10}+42 a b \,d^{3} x^{8}+63 b^{2} c \,d^{2} x^{8}+35 a^{2} d^{3} x^{6}+210 x^{6} d^{2} a b c +105 b^{2} c^{2} d \,x^{6}+315 a^{2} c \,d^{2} x^{4}+630 a b \,c^{2} d \,x^{4}+105 b^{2} c^{3} x^{4}-315 a^{2} c^{2} d \,x^{2}-210 a b \,c^{3} x^{2}-35 a^{2} c^{3}}{105 x^{3}}\) | \(138\) |
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Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=\frac {15 \, b^{2} d^{3} x^{10} + 21 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 35 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 35 \, a^{2} c^{3} + 105 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 105 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{105 \, x^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=\frac {b^{2} d^{3} x^{7}}{7} + x^{5} \cdot \left (\frac {2 a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{3} + 2 a b c d^{2} + b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right ) + \frac {- a^{2} c^{3} + x^{2} \left (- 9 a^{2} c^{2} d - 6 a b c^{3}\right )}{3 x^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=\frac {1}{7} \, b^{2} d^{3} x^{7} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac {1}{3} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x - \frac {a^{2} c^{3} + 3 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{3 \, x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=\frac {1}{7} \, b^{2} d^{3} x^{7} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {2}{5} \, a b d^{3} x^{5} + b^{2} c^{2} d x^{3} + 2 \, a b c d^{2} x^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + b^{2} c^{3} x + 6 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x - \frac {6 \, a b c^{3} x^{2} + 9 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}}{3 \, x^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx=x^3\,\left (\frac {a^2\,d^3}{3}+2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )-\frac {x^2\,\left (3\,d\,a^2\,c^2+2\,b\,a\,c^3\right )+\frac {a^2\,c^3}{3}}{x^3}+x\,\left (3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d+b^2\,c^3\right )+\frac {b^2\,d^3\,x^7}{7}+\frac {b\,d^2\,x^5\,\left (2\,a\,d+3\,b\,c\right )}{5} \]
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