Optimal. Leaf size=120 \[ \frac {1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac {1}{7} b d^2 x^7 (2 a d+3 b c)+\frac {1}{9} b^2 d^3 x^9 \]
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Rubi [A] time = 0.06, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ \frac {1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac {1}{7} b d^2 x^7 (2 a d+3 b c)+\frac {1}{9} b^2 d^3 x^9 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^2} \, dx &=\int \left (a c^2 (2 b c+3 a d)+\frac {a^2 c^3}{x^2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^2+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^4+b d^2 (3 b c+2 a d) x^6+b^2 d^3 x^8\right ) \, dx\\ &=-\frac {a^2 c^3}{x}+a c^2 (2 b c+3 a d) x+\frac {1}{3} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^3+\frac {1}{5} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^5+\frac {1}{7} b d^2 (3 b c+2 a d) x^7+\frac {1}{9} b^2 d^3 x^9\\ \end {align*}
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Mathematica [A] time = 0.04, size = 120, normalized size = 1.00 \[ \frac {1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac {1}{7} b d^2 x^7 (2 a d+3 b c)+\frac {1}{9} b^2 d^3 x^9 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 129, normalized size = 1.08 \[ \frac {35 \, b^{2} d^{3} x^{10} + 45 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 63 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 315 \, a^{2} c^{3} + 105 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 315 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{315 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 130, normalized size = 1.08 \[ \frac {1}{9} \, b^{2} d^{3} x^{9} + \frac {3}{7} \, b^{2} c d^{2} x^{7} + \frac {2}{7} \, a b d^{3} x^{7} + \frac {3}{5} \, b^{2} c^{2} d x^{5} + \frac {6}{5} \, a b c d^{2} x^{5} + \frac {1}{5} \, a^{2} d^{3} x^{5} + \frac {1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + 2 \, a b c^{3} x + 3 \, a^{2} c^{2} d x - \frac {a^{2} c^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 131, normalized size = 1.09 \[ \frac {b^{2} d^{3} x^{9}}{9}+\frac {2 a b \,d^{3} x^{7}}{7}+\frac {3 b^{2} c \,d^{2} x^{7}}{7}+\frac {a^{2} d^{3} x^{5}}{5}+\frac {6 a b c \,d^{2} x^{5}}{5}+\frac {3 b^{2} c^{2} d \,x^{5}}{5}+a^{2} c \,d^{2} x^{3}+2 a b \,c^{2} d \,x^{3}+\frac {b^{2} c^{3} x^{3}}{3}+3 a^{2} c^{2} d x +2 a b \,c^{3} x -\frac {a^{2} c^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 124, normalized size = 1.03 \[ \frac {1}{9} \, b^{2} d^{3} x^{9} + \frac {1}{7} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{5} - \frac {a^{2} c^{3}}{x} + \frac {1}{3} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 115, normalized size = 0.96 \[ x^3\,\left (a^2\,c\,d^2+2\,a\,b\,c^2\,d+\frac {b^2\,c^3}{3}\right )+x^5\,\left (\frac {a^2\,d^3}{5}+\frac {6\,a\,b\,c\,d^2}{5}+\frac {3\,b^2\,c^2\,d}{5}\right )-\frac {a^2\,c^3}{x}+\frac {b^2\,d^3\,x^9}{9}+\frac {b\,d^2\,x^7\,\left (2\,a\,d+3\,b\,c\right )}{7}+a\,c^2\,x\,\left (3\,a\,d+2\,b\,c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 131, normalized size = 1.09 \[ - \frac {a^{2} c^{3}}{x} + \frac {b^{2} d^{3} x^{9}}{9} + x^{7} \left (\frac {2 a b d^{3}}{7} + \frac {3 b^{2} c d^{2}}{7}\right ) + x^{5} \left (\frac {a^{2} d^{3}}{5} + \frac {6 a b c d^{2}}{5} + \frac {3 b^{2} c^{2} d}{5}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac {b^{2} c^{3}}{3}\right ) + x \left (3 a^{2} c^{2} d + 2 a b c^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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