Optimal. Leaf size=17 \[ \frac {\left (a+b x^2\right )^2}{c+d x} \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1590} \[ \frac {\left (a+b x^2\right )^2}{c+d x} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (-a d+4 b c x+3 b d x^2\right )}{(c+d x)^2} \, dx &=\frac {\left (a+b x^2\right )^2}{c+d x}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 62, normalized size = 3.65 \[ \frac {a^2 d^4+2 a b d^2 \left (c^2+c d x+d^2 x^2\right )+b^2 \left (c^4+c^3 d x+d^4 x^4\right )}{d^4 (c+d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 78, normalized size = 4.59 \[ \frac {b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} + b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4} + {\left (b^{2} c^{3} d + 2 \, a b c d^{3}\right )} x}{d^{5} x + c d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 111, normalized size = 6.53 \[ \frac {{\left (b^{2} - \frac {4 \, b^{2} c}{d x + c} + \frac {6 \, b^{2} c^{2}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a b d^{2}}{{\left (d x + c\right )}^{2}}\right )} {\left (d x + c\right )}^{3}}{d^{4}} + \frac {\frac {b^{2} c^{4} d^{3}}{d x + c} + \frac {2 \, a b c^{2} d^{5}}{d x + c} + \frac {a^{2} d^{7}}{d x + c}}{d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 76, normalized size = 4.47 \[ \frac {\left (b \,d^{2} x^{3}-b c d \,x^{2}+2 a \,d^{2} x +b \,c^{2} x \right ) b}{d^{3}}-\frac {-a^{2} d^{4}-2 a b \,c^{2} d^{2}-b^{2} c^{4}}{\left (d x +c \right ) d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 82, normalized size = 4.82 \[ \frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{d^{5} x + c d^{4}} + \frac {b^{2} d^{2} x^{3} - b^{2} c d x^{2} + {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} x}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 85, normalized size = 5.00 \[ x\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )+\frac {b^2\,x^3}{d}+\frac {a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4}{d\,\left (x\,d^4+c\,d^3\right )}-\frac {b^2\,c\,x^2}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.37, size = 73, normalized size = 4.29 \[ - \frac {b^{2} c x^{2}}{d^{2}} + \frac {b^{2} x^{3}}{d} + x \left (\frac {2 a b}{d} + \frac {b^{2} c^{2}}{d^{3}}\right ) + \frac {a^{2} d^{4} + 2 a b c^{2} d^{2} + b^{2} c^{4}}{c d^{4} + d^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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