Optimal. Leaf size=122 \[ \frac {B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}+\frac {B c \sqrt {a+c x^2}}{8 a x^2}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4} \]
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Rubi [A] time = 0.09, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {835, 807, 266, 47, 63, 208} \begin {gather*} \frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}+\frac {B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}+\frac {B c \sqrt {a+c x^2}}{8 a x^2}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+c x^2}}{x^6} \, dx &=-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {\int \frac {(-5 a B+2 A c x) \sqrt {a+c x^2}}{x^5} \, dx}{5 a}\\ &=-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4}+\frac {\int \frac {(-8 a A c-5 a B c x) \sqrt {a+c x^2}}{x^4} \, dx}{20 a^2}\\ &=-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4}+\frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac {(B c) \int \frac {\sqrt {a+c x^2}}{x^3} \, dx}{4 a}\\ &=-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4}+\frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac {(B c) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{8 a}\\ &=\frac {B c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4}+\frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac {\left (B c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac {B c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4}+\frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac {(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}\\ &=\frac {B c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{5 a x^5}-\frac {B \left (a+c x^2\right )^{3/2}}{4 a x^4}+\frac {2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}+\frac {B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 62, normalized size = 0.51 \begin {gather*} -\frac {\left (a+c x^2\right )^{3/2} \left (a A \left (3 a-2 c x^2\right )+5 B c^2 x^5 \, _2F_1\left (\frac {3}{2},3;\frac {5}{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 = 0.54, size = 106, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-24 a^2 A-30 a^2 B x-8 a A c x^2-15 a B c x^3+16 A c^2 x^4\right )}{120 a^2 x^5}-\frac {B c^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{4 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 190, normalized size = 1.56 \begin {gather*} \left [\frac {15 \, 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 (16 \, A c^{2} x^{4} - 15 \, B a c x^{3} - 8 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, a^{2} x^{5}}, -\frac {15 \, B \sqrt {-a} c^{2} x^{5} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (16 \, A c^{2} x^{4} - 15 \, B a c x^{3} - 8 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, a^{2} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 267, normalized size = 2.19 \begin {gather*} -\frac {B c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a} + \frac {15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B c^{2} + 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} B a c^{2} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} A a c^{\frac {5}{2}} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{2} c^{\frac {5}{2}} - 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{3} c^{2} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{3} c^{\frac {5}{2}} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{4} c^{2} - 16 \, A a^{4} c^{\frac {5}{2}}}{60 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{5} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 1.03 \begin {gather*} \frac {B \,c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+a}\, B \,c^{2}}{8 a^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B c}{8 a^{2} x^{2}}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A c}{15 a^{2} x^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B}{4 a \,x^{4}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A}{5 a \,x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 114, normalized size = 0.93 \begin {gather*} \frac {B c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {\sqrt {c x^{2} + a} B c^{2}}{8 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B c}{8 \, a^{2} x^{2}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c}{15 \, a^{2} x^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B}{4 \, a x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 95, normalized size = 0.78 \begin {gather*} \frac {B\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{3/2}}-\frac {B\,\sqrt {c\,x^2+a}}{8\,x^4}-\frac {B\,{\left (c\,x^2+a\right )}^{3/2}}{8\,a\,x^4}-\frac {A\,\sqrt {c\,x^2+a}\,\left (3\,a^2+a\,c\,x^2-2\,c^2\,x^4\right )}{15\,a^2\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.25, size = 173, normalized size = 1.42 \begin {gather*} - \frac {A \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {A c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a x^{2}} + \frac {2 A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{2}} - \frac {B a}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 B \sqrt {c}}{8 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {B c^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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