Optimal. Leaf size=63 \[ \frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{5/2}}-\frac {b x (b c-2 a d)}{d^2}+\frac {b^2 x^3}{3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {390, 205} \[ -\frac {b x (b c-2 a d)}{d^2}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{5/2}}+\frac {b^2 x^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {b (b c-2 a d)}{d^2}+\frac {b^2 x^2}{d}+\frac {b^2 c^2-2 a b c d+a^2 d^2}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {b (b c-2 a d) x}{d^2}+\frac {b^2 x^3}{3 d}+\frac {(b c-a d)^2 \int \frac {1}{c+d x^2} \, dx}{d^2}\\ &=-\frac {b (b c-2 a d) x}{d^2}+\frac {b^2 x^3}{3 d}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.94 \[ \frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{5/2}}+\frac {b x \left (6 a d-3 b c+b d x^2\right )}{3 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 179, normalized size = 2.84 \[ \left [\frac {2 \, b^{2} c d^{2} x^{3} - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x}{6 \, c d^{3}}, \frac {b^{2} c d^{2} x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x}{3 \, c d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 72, normalized size = 1.14 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{2}} + \frac {b^{2} d^{2} x^{3} - 3 \, b^{2} c d x + 6 \, a b d^{2} x}{3 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 95, normalized size = 1.51 \[ \frac {b^{2} x^{3}}{3 d}+\frac {a^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}}-\frac {2 a b c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d}+\frac {b^{2} c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{2}}+\frac {2 a b x}{d}-\frac {b^{2} c x}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 68, normalized size = 1.08 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{2}} + \frac {b^{2} d x^{3} - 3 \, {\left (b^{2} c - 2 \, a b d\right )} x}{3 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 90, normalized size = 1.43 \[ \frac {b^2\,x^3}{3\,d}-x\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^2}{\sqrt {c}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{\sqrt {c}\,d^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 172, normalized size = 2.73 \[ \frac {b^{2} x^{3}}{3 d} + x \left (\frac {2 a b}{d} - \frac {b^{2} c}{d^{2}}\right ) - \frac {\sqrt {- \frac {1}{c d^{5}}} \left (a d - b c\right )^{2} \log {\left (- \frac {c d^{2} \sqrt {- \frac {1}{c d^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{c d^{5}}} \left (a d - b c\right )^{2} \log {\left (\frac {c d^{2} \sqrt {- \frac {1}{c d^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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