Optimal. Leaf size=67 \[ \frac {(b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}-\frac {x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac {d x}{b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 388, 205} \begin {gather*} -\frac {x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac {(b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {d x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 455
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}-\frac {\int \frac {-b c+a d-2 b d x^2}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {d x}{b^2}-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}+\frac {(b c-3 a d) \int \frac {1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {d x}{b^2}-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}+\frac {(b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 68, normalized size = 1.01 \begin {gather*} -\frac {(3 a d-b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}-\frac {x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac {d x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 202, normalized size = 3.01 \begin {gather*} \left [\frac {4 \, a b^{2} d x^{3} + {\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\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^{2} c - 3 \, a^{2} b d\right )} x}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac {2 \, a b^{2} d x^{3} + {\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (a b^{2} c - 3 \, a^{2} b d\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 58, normalized size = 0.87 \begin {gather*} \frac {d x}{b^{2}} + \frac {{\left (b c - 3 \, a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} - \frac {b c x - a d x}{2 \, {\left (b x^{2} + a\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 82, normalized size = 1.22 \begin {gather*} \frac {a d x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {c x}{2 \left (b \,x^{2}+a \right ) b}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {d x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.32, size = 60, normalized size = 0.90 \begin {gather*} -\frac {{\left (b c - a d\right )} x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {d x}{b^{2}} + \frac {{\left (b c - 3 \, a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 59, normalized size = 0.88 \begin {gather*} \frac {x\,\left (\frac {a\,d}{2}-\frac {b\,c}{2}\right )}{b^3\,x^2+a\,b^2}+\frac {d\,x}{b^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,a\,d-b\,c\right )}{2\,\sqrt {a}\,b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 114, normalized size = 1.70 \begin {gather*} \frac {x \left (a d - b c\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a b^{5}}} \left (3 a d - b c\right ) \log {\left (- a b^{2} \sqrt {- \frac {1}{a b^{5}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a b^{5}}} \left (3 a d - b c\right ) \log {\left (a b^{2} \sqrt {- \frac {1}{a b^{5}}} + x \right )}}{4} + \frac {d x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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