Optimal. Leaf size=70 \[ -\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 A \sqrt {a+c x^2}}{a^2 x}+\frac {A+B x}{a x \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {823, 807, 266, 63, 208} \[ -\frac {2 A \sqrt {a+c x^2}}{a^2 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {A+B x}{a x \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a x \sqrt {a+c x^2}}-\frac {\int \frac {-2 a A c-a B c x}{x^2 \sqrt {a+c x^2}} \, dx}{a^2 c}\\ &=\frac {A+B x}{a x \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{a^2 x}+\frac {B \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{a}\\ &=\frac {A+B x}{a x \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{a^2 x}+\frac {B \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {A+B x}{a x \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{a^2 x}+\frac {B \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c}\\ &=\frac {A+B x}{a x \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{a^2 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 1.03 \[ -\frac {a (A-B x)+\sqrt {a} B x \sqrt {a+c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+2 A c x^2}{a^2 x \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 169, normalized size = 2.41 \[ \left [\frac {{\left (B c x^{3} + B a x\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (2 \, A c x^{2} - B a x + A a\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c x^{3} + a^{3} x\right )}}, \frac {{\left (B c x^{3} + B a x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, A c x^{2} - B a x + A a\right )} \sqrt {c x^{2} + a}}{a^{2} c x^{3} + a^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 96, normalized size = 1.37 \[ -\frac {\frac {A c x}{a^{2}} - \frac {B}{a}}{\sqrt {c x^{2} + a}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2 \, A \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 80, normalized size = 1.14 \[ -\frac {2 A c x}{\sqrt {c \,x^{2}+a}\, a^{2}}-\frac {B \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {3}{2}}}+\frac {B}{\sqrt {c \,x^{2}+a}\, a}-\frac {A}{\sqrt {c \,x^{2}+a}\, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 68, normalized size = 0.97 \[ -\frac {2 \, A c x}{\sqrt {c x^{2} + a} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {B}{\sqrt {c x^{2} + a} a} - \frac {A}{\sqrt {c x^{2} + a} a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 70, normalized size = 1.00 \[ \frac {B}{a\,\sqrt {c\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {A}{a\,x\,\sqrt {c\,x^2+a}}-\frac {2\,A\,c\,x}{a^2\,\sqrt {c\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.93, size = 235, normalized size = 3.36 \[ A \left (- \frac {1}{a \sqrt {c} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {2 \sqrt {c}}{a^{2} \sqrt {\frac {a}{c x^{2}} + 1}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {c x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} c x^{2}} + \frac {a^{3} \log {\left (\frac {c x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} c x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} c x^{2}} + \frac {a^{2} c x^{2} \log {\left (\frac {c x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} c x^{2}} - \frac {2 a^{2} c x^{2} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} c x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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