Optimal. Leaf size=127 \[ \frac {(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} \sqrt {b}}+\frac {c (5 b c-6 a d)}{3 a^3 x}+\frac {x \left (3 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{6 a^3 \left (a+b x^2\right )}-\frac {c^2}{3 a x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 125, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {462, 456, 453, 205} \[ \frac {x \left (\frac {b c (5 b c-6 a d)}{a^2}+3 d^2\right )}{6 a \left (a+b x^2\right )}+\frac {c (5 b c-6 a d)}{3 a^3 x}+\frac {(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} \sqrt {b}}-\frac {c^2}{3 a x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 453
Rule 456
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x^4 \left (a+b x^2\right )^2} \, dx &=-\frac {c^2}{3 a x^3 \left (a+b x^2\right )}+\frac {\int \frac {-c (5 b c-6 a d)+3 a d^2 x^2}{x^2 \left (a+b x^2\right )^2} \, dx}{3 a}\\ &=-\frac {c^2}{3 a x^3 \left (a+b x^2\right )}+\frac {\left (3 d^2+\frac {b c (5 b c-6 a d)}{a^2}\right ) x}{6 a \left (a+b x^2\right )}-\frac {\int \frac {\frac {2 c (5 b c-6 a d)}{a}-\left (\frac {5 b^2 c^2}{a^2}-\frac {6 b c d}{a}+3 d^2\right ) x^2}{x^2 \left (a+b x^2\right )} \, dx}{6 a}\\ &=\frac {c (5 b c-6 a d)}{3 a^3 x}-\frac {c^2}{3 a x^3 \left (a+b x^2\right )}+\frac {\left (3 d^2+\frac {b c (5 b c-6 a d)}{a^2}\right ) x}{6 a \left (a+b x^2\right )}+\frac {((b c-a d) (5 b c-a d)) \int \frac {1}{a+b x^2} \, dx}{2 a^3}\\ &=\frac {c (5 b c-6 a d)}{3 a^3 x}-\frac {c^2}{3 a x^3 \left (a+b x^2\right )}+\frac {\left (3 d^2+\frac {b c (5 b c-6 a d)}{a^2}\right ) x}{6 a \left (a+b x^2\right )}+\frac {(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 107, normalized size = 0.84 \[ \frac {x (a d-b c)^2}{2 a^3 \left (a+b x^2\right )}-\frac {2 c (a d-b c)}{a^3 x}-\frac {c^2}{3 a^2 x^3}+\frac {\left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 356, normalized size = 2.80 \[ \left [-\frac {4 \, a^{3} b c^{2} - 6 \, {\left (5 \, a b^{3} c^{2} - 6 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} - 4 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d\right )} x^{2} + 3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{12 \, {\left (a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )}}, -\frac {2 \, a^{3} b c^{2} - 3 \, {\left (5 \, a b^{3} c^{2} - 6 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d\right )} x^{2} - 3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{6 \, {\left (a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 112, normalized size = 0.88 \[ \frac {{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (b x^{2} + a\right )} a^{3}} + \frac {6 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 161, normalized size = 1.27 \[ \frac {d^{2} x}{2 \left (b \,x^{2}+a \right ) a}+\frac {d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {b c d x}{\left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {b^{2} c^{2} x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{2} c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {2 c d}{a^{2} x}+\frac {2 b \,c^{2}}{a^{3} x}-\frac {c^{2}}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.44, size = 118, normalized size = 0.93 \[ \frac {3 \, {\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \, {\left (5 \, a b c^{2} - 6 \, a^{2} c d\right )} x^{2}}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} + \frac {{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 146, normalized size = 1.15 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (a\,d-5\,b\,c\right )}{\sqrt {a}\,\left (a^2\,d^2-6\,a\,b\,c\,d+5\,b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-5\,b\,c\right )}{2\,a^{7/2}\,\sqrt {b}}-\frac {\frac {c^2}{3\,a}-\frac {x^4\,\left (a^2\,d^2-6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{2\,a^3}+\frac {c\,x^2\,\left (6\,a\,d-5\,b\,c\right )}{3\,a^2}}{b\,x^5+a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.00, size = 248, normalized size = 1.95 \[ - \frac {\sqrt {- \frac {1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log {\left (- \frac {a^{4} \sqrt {- \frac {1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log {\left (\frac {a^{4} \sqrt {- \frac {1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac {- 2 a^{2} c^{2} + x^{4} \left (3 a^{2} d^{2} - 18 a b c d + 15 b^{2} c^{2}\right ) + x^{2} \left (- 12 a^{2} c d + 10 a b c^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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