Optimal. Leaf size=82 \[ \frac {(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x}{b^2} \]
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Rubi [A] time = 0.10, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {390, 385, 205} \[ \frac {(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x}{b^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 390
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx &=\int \left (\frac {d^2}{b^2}+\frac {b^2 c^2-a^2 d^2+2 b d (b c-a d) x^2}{b^2 \left (a+b x^2\right )^2}\right ) \, dx\\ &=\frac {d^2 x}{b^2}+\frac {\int \frac {b^2 c^2-a^2 d^2+2 b d (b c-a d) x^2}{\left (a+b x^2\right )^2} \, dx}{b^2}\\ &=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {((b c-a d) (b c+3 a d)) \int \frac {1}{a+b x^2} \, dx}{2 a b^2}\\ &=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-a d) (b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 1.07 \[ \frac {\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 297, normalized size = 3.62 \[ \left [\frac {4 \, a^{2} b^{2} d^{2} x^{3} + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{4 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, \frac {2 \, a^{2} b^{2} d^{2} x^{3} + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{2 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 94, normalized size = 1.15 \[ \frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (b x^{2} + a\right )} a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 129, normalized size = 1.57 \[ \frac {a \,d^{2} x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}+\frac {c^{2} x}{2 \left (b \,x^{2}+a \right ) a}+\frac {c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {c d x}{\left (b \,x^{2}+a \right ) b}+\frac {c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {d^{2} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 95, normalized size = 1.16 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 124, normalized size = 1.51 \[ \frac {d^2\,x}{b^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,\left (b^3\,x^2+a\,b^2\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{\sqrt {a}\,\left (-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.71, size = 236, normalized size = 2.88 \[ \frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log {\left (- \frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log {\left (\frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} + \frac {d^{2} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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