Optimal. Leaf size=75 \[ -\frac {\sqrt {a+c x^2} (A-B x)}{x}+A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\sqrt {a} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {813, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {\sqrt {a+c x^2} (A-B x)}{x}+A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\sqrt {a} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+c x^2}}{x^2} \, dx &=-\frac {(A-B x) \sqrt {a+c x^2}}{x}-\frac {1}{2} \int \frac {-2 a B-2 A c x}{x \sqrt {a+c x^2}} \, dx\\ &=-\frac {(A-B x) \sqrt {a+c x^2}}{x}+(a B) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+(A c) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=-\frac {(A-B x) \sqrt {a+c x^2}}{x}+\frac {1}{2} (a B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+(A c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {(A-B x) \sqrt {a+c x^2}}{x}+A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {(a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c}\\ &=-\frac {(A-B x) \sqrt {a+c x^2}}{x}+A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\sqrt {a} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 99, normalized size = 1.32 \begin {gather*} \frac {\sqrt {a+c x^2} (B x-A)}{x}+\frac {\sqrt {a} A \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a+c x^2}}-\sqrt {a} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 92, normalized size = 1.23 \begin {gather*} \frac {\sqrt {a+c x^2} (B x-A)}{x}-A \sqrt {c} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )+2 \sqrt {a} B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 333, normalized size = 4.44 \begin {gather*} \left [\frac {A \sqrt {c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + B \sqrt {a} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt {c x^{2} + a} {\left (B x - A\right )}}{2 \, x}, -\frac {2 \, A \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - B \sqrt {a} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, \sqrt {c x^{2} + a} {\left (B x - A\right )}}{2 \, x}, \frac {2 \, B \sqrt {-a} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + A \sqrt {c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, \sqrt {c x^{2} + a} {\left (B x - A\right )}}{2 \, x}, -\frac {A \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - B \sqrt {-a} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - \sqrt {c x^{2} + a} {\left (B x - A\right )}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 102, normalized size = 1.36 \begin {gather*} \frac {2 \, B a \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - A \sqrt {c} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \sqrt {c x^{2} + a} B + \frac {2 \, A a \sqrt {c}}{{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 97, normalized size = 1.29 \begin {gather*} A \sqrt {c}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )-B \sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {\sqrt {c \,x^{2}+a}\, A c x}{a}+\sqrt {c \,x^{2}+a}\, B -\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 59, normalized size = 0.79 \begin {gather*} A \sqrt {c} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - B \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \sqrt {c x^{2} + a} B - \frac {\sqrt {c x^{2} + a} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.80, size = 89, normalized size = 1.19 \begin {gather*} B\,\sqrt {c\,x^2+a}-\frac {A\,\sqrt {c\,x^2+a}}{x}-B\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )-\frac {A\,\sqrt {c}\,\mathrm {asin}\left (\frac {\sqrt {c}\,x\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {c\,x^2}{a}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.56, size = 124, normalized size = 1.65 \begin {gather*} - \frac {A \sqrt {a}}{x \sqrt {1 + \frac {c x^{2}}{a}}} + A \sqrt {c} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )} - \frac {A c x}{\sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} - B \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )} + \frac {B a}{\sqrt {c} x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B \sqrt {c} x}{\sqrt {\frac {a}{c x^{2}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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