Optimal. Leaf size=97 \[ \frac {B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {2 A c \sqrt {a+c x^2}}{3 a^2 x}-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {835, 807, 266, 63, 208} \begin {gather*} \frac {2 A c \sqrt {a+c x^2}}{3 a^2 x}+\frac {B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 \sqrt {a+c x^2}} \, dx &=-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {\int \frac {-3 a B+2 A c x}{x^3 \sqrt {a+c x^2}} \, dx}{3 a}\\ &=-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2}+\frac {\int \frac {-4 a A c-3 a B c x}{x^2 \sqrt {a+c x^2}} \, dx}{6 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2}+\frac {2 A c \sqrt {a+c x^2}}{3 a^2 x}-\frac {(B c) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{2 a}\\ &=-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2}+\frac {2 A c \sqrt {a+c x^2}}{3 a^2 x}-\frac {(B c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2}+\frac {2 A c \sqrt {a+c x^2}}{3 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 )}{2 a}\\ &=-\frac {A \sqrt {a+c x^2}}{3 a x^3}-\frac {B \sqrt {a+c x^2}}{2 a x^2}+\frac {2 A c \sqrt {a+c x^2}}{3 a^2 x}+\frac {B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 73, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+c x^2} \left (\frac {-2 a A-3 a B x+4 A c x^2}{x^3}+\frac {3 B c \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )}{\sqrt {\frac {c x^2}{a}+1}}\right )}{6 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 80, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-2 a A-3 a B x+4 A c x^2\right )}{6 a^2 x^3}-\frac {B c \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 142, normalized size = 1.46 \begin {gather*} \left [\frac {3 \, B \sqrt {a} c x^{3} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (4 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt {c x^{2} + a}}{12 \, a^{2} x^{3}}, -\frac {3 \, B \sqrt {-a} c x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (4 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt {c x^{2} + a}}{6 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 151, normalized size = 1.56 \begin {gather*} -\frac {B c \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B c + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{2} c - 4 \, A a^{2} c^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{3} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 87, normalized size = 0.90 \begin {gather*} \frac {B c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}+\frac {2 \sqrt {c \,x^{2}+a}\, A c}{3 a^{2} x}-\frac {\sqrt {c \,x^{2}+a}\, B}{2 a \,x^{2}}-\frac {\sqrt {c \,x^{2}+a}\, A}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 75, normalized size = 0.77 \begin {gather*} \frac {B c \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} + \frac {2 \, \sqrt {c x^{2} + a} A c}{3 \, a^{2} x} - \frac {\sqrt {c x^{2} + a} B}{2 \, a x^{2}} - \frac {\sqrt {c x^{2} + a} A}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 66, normalized size = 0.68 \begin {gather*} \frac {B\,c\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {B\,\sqrt {c\,x^2+a}}{2\,a\,x^2}-\frac {A\,\sqrt {c\,x^2+a}\,\left (a-2\,c\,x^2\right )}{3\,a^2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.79, size = 97, normalized size = 1.00 \begin {gather*} - \frac {A \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a x^{2}} + \frac {2 A c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{2}} - \frac {B \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{2 a x} + \frac {B c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{2 a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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