Optimal. Leaf size=99 \[ \frac {A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {A c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {A c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3} \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^5} \, dx &=-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {\int \frac {(-4 a B+A c x) \sqrt {a+c x^2}}{x^4} \, dx}{4 a}\\ &=-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac {(A c) \int \frac {\sqrt {a+c x^2}}{x^3} \, dx}{4 a}\\ &=-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac {(A c) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{8 a}\\ &=\frac {A c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac {\left (A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac {A c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac {(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}\\ &=\frac {A c \sqrt {a+c x^2}}{8 a x^2}-\frac {A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac {B \left (a+c x^2\right )^{3/2}}{3 a x^3}+\frac {A 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 = 53, normalized size = 0.54 \begin {gather*} -\frac {\left (a+c x^2\right )^{3/2} \left (a^2 B+A c^2 x^3 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {c x^2}{a}+1\right )\right )}{3 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 91, normalized size = 0.92 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-6 a A-8 a B x-3 A c x^2-8 B c x^3\right )}{24 a x^4}-\frac {A 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 = 0.45, size = 171, normalized size = 1.73 \begin {gather*} \left [\frac {3 \, 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 (8 \, B a c x^{3} + 3 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{2} x^{4}}, -\frac {3 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (8 \, B a c x^{3} + 3 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{2} x^{4}}\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 = 2.70 \begin {gather*} -\frac {A c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a} + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A c^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a c^{\frac {3}{2}} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a c^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{2} c^{\frac {3}{2}} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{3} c^{\frac {3}{2}} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{3} c^{2} - 8 \, B a^{4} c^{\frac {3}{2}}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 107, normalized size = 1.08 \begin {gather*} \frac {A \,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}\, A \,c^{2}}{8 a^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B}{3 a \,x^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 95, normalized size = 0.96 \begin {gather*} \frac {A 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} A c^{2}}{8 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A c}{8 \, a^{2} x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B}{3 \, a x^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.09, size = 75, normalized size = 0.76 \begin {gather*} \frac {A\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{3/2}}-\frac {A\,\sqrt {c\,x^2+a}}{8\,x^4}-\frac {A\,{\left (c\,x^2+a\right )}^{3/2}}{8\,a\,x^4}-\frac {B\,{\left (c\,x^2+a\right )}^{3/2}}{3\,a\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.94, size = 144, normalized size = 1.45 \begin {gather*} - \frac {A a}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 A \sqrt {c}}{8 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {A c^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {A c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 a^{\frac {3}{2}}} - \frac {B \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 x^{2}} - \frac {B c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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