Optimal. Leaf size=116 \[ \frac {3 x \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right )}{8 \left (c+d x^2\right )}+\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{5/2}}-\frac {x \left (a+b x^2\right ) (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {413, 385, 205} \[ \frac {3 x \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right )}{8 \left (c+d x^2\right )}+\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{5/2}}-\frac {x \left (a+b x^2\right ) (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=-\frac {(b c-a d) x \left (a+b x^2\right )}{4 c d \left (c+d x^2\right )^2}+\frac {\int \frac {a (b c+3 a d)+b (3 b c+a d) x^2}{\left (c+d x^2\right )^2} \, dx}{4 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )}{4 c d \left (c+d x^2\right )^2}+\frac {3 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) x}{8 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^2 d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )}{4 c d \left (c+d x^2\right )^2}+\frac {3 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) x}{8 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 121, normalized size = 1.04 \[ \frac {x \left (a^2 d^2 \left (5 c+3 d x^2\right )-2 a b c d \left (c-d x^2\right )-b^2 c^2 \left (3 c+5 d x^2\right )\right )}{8 c^2 d^2 \left (c+d x^2\right )^2}+\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 449, normalized size = 3.87 \[ \left [-\frac {2 \, {\left (5 \, b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} - 3 \, a^{2} c d^{4}\right )} x^{3} + {\left (3 \, b^{2} c^{4} + 2 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (3 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (3 \, b^{2} c^{4} d + 2 \, a b c^{3} d^{2} - 5 \, a^{2} c^{2} d^{3}\right )} x}{16 \, {\left (c^{3} d^{5} x^{4} + 2 \, c^{4} d^{4} x^{2} + c^{5} d^{3}\right )}}, -\frac {{\left (5 \, b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} - 3 \, a^{2} c d^{4}\right )} x^{3} - {\left (3 \, b^{2} c^{4} + 2 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (3 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (3 \, b^{2} c^{4} d + 2 \, a b c^{3} d^{2} - 5 \, a^{2} c^{2} d^{3}\right )} x}{8 \, {\left (c^{3} d^{5} x^{4} + 2 \, c^{4} d^{4} x^{2} + c^{5} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 126, normalized size = 1.09 \[ \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{2} d^{2}} - \frac {5 \, b^{2} c^{2} d x^{3} - 2 \, a b c d^{2} x^{3} - 3 \, a^{2} d^{3} x^{3} + 3 \, b^{2} c^{3} x + 2 \, a b c^{2} d x - 5 \, a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 147, normalized size = 1.27 \[ \frac {3 a^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c^{2}}+\frac {a b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{4 \sqrt {c d}\, c d}+\frac {3 b^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, d^{2}}+\frac {\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) x^{3}}{8 c^{2} d}+\frac {\left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) x}{8 c \,d^{2}}}{\left (d \,x^{2}+c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 138, normalized size = 1.19 \[ -\frac {{\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x}{8 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.03, size = 130, normalized size = 1.12 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,c^{5/2}\,d^{5/2}}-\frac {\frac {x\,\left (-5\,a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,c\,d^2}-\frac {x^3\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d-5\,b^2\,c^2\right )}{8\,c^2\,d}}{c^2+2\,c\,d\,x^2+d^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.98, size = 223, normalized size = 1.92 \[ - \frac {\sqrt {- \frac {1}{c^{5} d^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- c^{3} d^{2} \sqrt {- \frac {1}{c^{5} d^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c^{5} d^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log {\left (c^{3} d^{2} \sqrt {- \frac {1}{c^{5} d^{5}}} + x \right )}}{16} + \frac {x^{3} \left (3 a^{2} d^{3} + 2 a b c d^{2} - 5 b^{2} c^{2} d\right ) + x \left (5 a^{2} c d^{2} - 2 a b c^{2} d - 3 b^{2} c^{3}\right )}{8 c^{4} d^{2} + 16 c^{3} d^{3} x^{2} + 8 c^{2} d^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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