Optimal. Leaf size=122 \[ -\frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3} \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^5 \sqrt {a+c x^2}} \, dx &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {\int \frac {-4 a B+3 A c x}{x^4 \sqrt {a+c x^2}} \, dx}{4 a}\\ &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {\int \frac {-9 a A c-8 a B c x}{x^3 \sqrt {a+c x^2}} \, dx}{12 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {\int \frac {16 a^2 B c-9 a A c^2 x}{x^2 \sqrt {a+c x^2}} \, dx}{24 a^3}\\ &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}+\frac {\left (3 A c^2\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{8 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}+\frac {\left (3 A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{16 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}+\frac {(3 A c) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{8 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}-\frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 60, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (3 A c^2 x^3 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {c x^2}{a}+1\right )+a B \left (a-2 c x^2\right )\right )}{3 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 91, normalized size = 0.75 \begin {gather*} \frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {\sqrt {a+c x^2} \left (-6 a A-8 a B x+9 A c x^2+16 B c x^3\right )}{24 a^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 171, normalized size = 1.40 \begin {gather*} \left [\frac {9 \, A \sqrt {a} c^{2} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (16 \, B a c x^{3} + 9 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{3} x^{4}}, \frac {9 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (16 \, B a c x^{3} + 9 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{3} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 241, normalized size = 1.98 \begin {gather*} \frac {3 \, A c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} - \frac {9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A c^{2} - 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a c^{2} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{2} c^{\frac {3}{2}} - 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 64 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{3} c^{\frac {3}{2}} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{3} c^{2} - 16 \, B a^{4} c^{\frac {3}{2}}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 108, normalized size = 0.89 \begin {gather*} -\frac {3 A \,c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}+\frac {2 \sqrt {c \,x^{2}+a}\, B c}{3 a^{2} x}+\frac {3 \sqrt {c \,x^{2}+a}\, A c}{8 a^{2} x^{2}}-\frac {\sqrt {c \,x^{2}+a}\, B}{3 a \,x^{3}}-\frac {\sqrt {c \,x^{2}+a}\, A}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 96, normalized size = 0.79 \begin {gather*} -\frac {3 \, A c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {2 \, \sqrt {c x^{2} + a} B c}{3 \, a^{2} x} + \frac {3 \, \sqrt {c x^{2} + a} A c}{8 \, a^{2} x^{2}} - \frac {\sqrt {c x^{2} + a} B}{3 \, a x^{3}} - \frac {\sqrt {c x^{2} + a} A}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 86, normalized size = 0.70 \begin {gather*} \frac {3\,A\,{\left (c\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {5\,A\,\sqrt {c\,x^2+a}}{8\,a\,x^4}-\frac {3\,A\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}-\frac {B\,\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 = 5.71, size = 153, normalized size = 1.25 \begin {gather*} - \frac {A}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {A \sqrt {c}}{8 a x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {3 A c^{\frac {3}{2}}}{8 a^{2} x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 A c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 a^{\frac {5}{2}}} - \frac {B \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a x^{2}} + \frac {2 B c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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