Optimal. Leaf size=147 \[ -\frac {3 B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}}-\frac {8 A c^2 \sqrt {a+c x^2}}{15 a^3 x}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4} \]
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Rubi [A] time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {8 A c^2 \sqrt {a+c x^2}}{15 a^3 x}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}-\frac {3 B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4} \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^6 \sqrt {a+c x^2}} \, dx &=-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {\int \frac {-5 a B+4 A c x}{x^5 \sqrt {a+c x^2}} \, dx}{5 a}\\ &=-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {\int \frac {-16 a A c-15 a B c x}{x^4 \sqrt {a+c x^2}} \, dx}{20 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}-\frac {\int \frac {45 a^2 B c-32 a A c^2 x}{x^3 \sqrt {a+c x^2}} \, dx}{60 a^3}\\ &=-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {\int \frac {64 a^2 A c^2+45 a^2 B c^2 x}{x^2 \sqrt {a+c x^2}} \, dx}{120 a^4}\\ &=-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {8 A c^2 \sqrt {a+c x^2}}{15 a^3 x}+\frac {\left (3 B c^2\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{8 a^2}\\ &=-\frac {A \sqrt {a+c x^2}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {8 A c^2 \sqrt {a+c x^2}}{15 a^3 x}+\frac {\left (3 B 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}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {8 A c^2 \sqrt {a+c x^2}}{15 a^3 x}+\frac {(3 B 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}}{5 a x^5}-\frac {B \sqrt {a+c x^2}}{4 a x^4}+\frac {4 A c \sqrt {a+c x^2}}{15 a^2 x^3}+\frac {3 B c \sqrt {a+c x^2}}{8 a^2 x^2}-\frac {8 A c^2 \sqrt {a+c x^2}}{15 a^3 x}-\frac {3 B 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 = 72, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (A \left (3 a^2-4 a c x^2+8 c^2 x^4\right )+15 B c^2 x^5 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {c x^2}{a}+1\right )\right )}{15 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.52, size = 106, normalized size = 0.72 \begin {gather*} \frac {3 B 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 (-24 a^2 A-30 a^2 B x+32 a A c x^2+45 a B c x^3-64 A c^2 x^4\right )}{120 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 190, normalized size = 1.29 \begin {gather*} \left [\frac {45 \, B \sqrt {a} c^{2} x^{5} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, a^{3} x^{5}}, \frac {45 \, B \sqrt {-a} c^{2} x^{5} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, a^{3} x^{5}}\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 = 1.64 \begin {gather*} \frac {3 \, B c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} - \frac {45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B c^{2} - 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} B a c^{2} - 640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{2} c^{\frac {5}{2}} + 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{3} c^{2} + 320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{3} c^{\frac {5}{2}} - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{4} c^{2} - 64 \, A a^{4} c^{\frac {5}{2}}}{60 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{5} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 129, normalized size = 0.88 \begin {gather*} -\frac {3 B \,c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}-\frac {8 \sqrt {c \,x^{2}+a}\, A \,c^{2}}{15 a^{3} x}+\frac {3 \sqrt {c \,x^{2}+a}\, B c}{8 a^{2} x^{2}}+\frac {4 \sqrt {c \,x^{2}+a}\, A c}{15 a^{2} x^{3}}-\frac {\sqrt {c \,x^{2}+a}\, B}{4 a \,x^{4}}-\frac {\sqrt {c \,x^{2}+a}\, A}{5 a \,x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 117, normalized size = 0.80 \begin {gather*} -\frac {3 \, B c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} - \frac {8 \, \sqrt {c x^{2} + a} A c^{2}}{15 \, a^{3} x} + \frac {3 \, \sqrt {c x^{2} + a} B c}{8 \, a^{2} x^{2}} + \frac {4 \, \sqrt {c x^{2} + a} A c}{15 \, a^{2} x^{3}} - \frac {\sqrt {c x^{2} + a} B}{4 \, a x^{4}} - \frac {\sqrt {c x^{2} + a} A}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 99, normalized size = 0.67 \begin {gather*} \frac {3\,B\,{\left (c\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {5\,B\,\sqrt {c\,x^2+a}}{8\,a\,x^4}-\frac {3\,B\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}-\frac {A\,\sqrt {c\,x^2+a}\,\left (3\,a^2-4\,a\,c\,x^2+8\,c^2\,x^4\right )}{15\,a^3\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.46, size = 408, normalized size = 2.78 \begin {gather*} - \frac {3 A a^{4} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac {2 A a^{3} c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac {3 A a^{2} c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac {12 A a c^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac {8 A c^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac {B}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B \sqrt {c}}{8 a x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {3 B c^{\frac {3}{2}}}{8 a^{2} x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 B c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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