Optimal. Leaf size=60 \[ -\frac {3 c}{2 a^2 x}+\frac {c}{2 a x \left (a+b x^2\right )}-\frac {3 \sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {21, 296, 331,
211} \begin {gather*} -\frac {3 \sqrt {b} c \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3 c}{2 a^2 x}+\frac {c}{2 a x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 211
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^3} \, dx &=c \int \frac {1}{x^2 \left (a+b x^2\right )^2} \, dx\\ &=\frac {c}{2 a x \left (a+b x^2\right )}+\frac {(3 c) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac {3 c}{2 a^2 x}+\frac {c}{2 a x \left (a+b x^2\right )}-\frac {(3 b c) \int \frac {1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac {3 c}{2 a^2 x}+\frac {c}{2 a x \left (a+b x^2\right )}-\frac {3 \sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.93 \begin {gather*} c \left (-\frac {1}{a^2 x}-\frac {b x}{2 a^2 \left (a+b x^2\right )}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 47, normalized size = 0.78
method | result | size |
default | \(c \left (-\frac {b \left (\frac {x}{2 b \,x^{2}+2 a}+\frac {3 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {1}{a^{2} x}\right )\) | \(47\) |
risch | \(\frac {-\frac {3 b c \,x^{2}}{2 a^{2}}-\frac {c}{a}}{x \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {-a b}\, c \ln \left (-b x +\sqrt {-a b}\right )}{4 a^{3}}-\frac {3 \sqrt {-a b}\, c \ln \left (-b x -\sqrt {-a b}\right )}{4 a^{3}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 52, normalized size = 0.87 \begin {gather*} -\frac {3 \, b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {3 \, b c x^{2} + 2 \, a c}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.92, size = 144, normalized size = 2.40 \begin {gather*} \left [-\frac {6 \, b c x^{2} - 3 \, {\left (b c x^{3} + a c x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 4 \, a c}{4 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac {3 \, b c x^{2} + 3 \, {\left (b c x^{3} + a c x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, a c}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 94, normalized size = 1.57 \begin {gather*} c \left (\frac {3 \sqrt {- \frac {b}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}}}{b} + x \right )}}{4} - \frac {3 \sqrt {- \frac {b}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}}}{b} + x \right )}}{4} + \frac {- 2 a - 3 b x^{2}}{2 a^{3} x + 2 a^{2} b x^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 50, normalized size = 0.83 \begin {gather*} -\frac {3 \, b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {3 \, b c x^{2} + 2 \, a c}{2 \, {\left (b x^{3} + a x\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 48, normalized size = 0.80 \begin {gather*} -\frac {\frac {c}{a}+\frac {3\,b\,c\,x^2}{2\,a^2}}{b\,x^3+a\,x}-\frac {3\,\sqrt {b}\,c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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