Optimal. Leaf size=109 \[ -\frac {c \sqrt {a+c x^2} (2 A-3 B x)}{2 x}-\frac {\left (a+c x^2\right )^{3/2} (2 A+3 B x)}{6 x^3}+A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {3}{2} \sqrt {a} B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {811, 813, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {c \sqrt {a+c x^2} (2 A-3 B x)}{2 x}-\frac {\left (a+c x^2\right )^{3/2} (2 A+3 B x)}{6 x^3}+A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {3}{2} \sqrt {a} B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 811
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{x^4} \, dx &=-\frac {(2 A+3 B x) \left (a+c x^2\right )^{3/2}}{6 x^3}-\frac {\int \frac {(-4 a A c-6 a B c x) \sqrt {a+c x^2}}{x^2} \, dx}{4 a}\\ &=-\frac {c (2 A-3 B x) \sqrt {a+c x^2}}{2 x}-\frac {(2 A+3 B x) \left (a+c x^2\right )^{3/2}}{6 x^3}+\frac {\int \frac {12 a^2 B c+8 a A c^2 x}{x \sqrt {a+c x^2}} \, dx}{8 a}\\ &=-\frac {c (2 A-3 B x) \sqrt {a+c x^2}}{2 x}-\frac {(2 A+3 B x) \left (a+c x^2\right )^{3/2}}{6 x^3}+\frac {1}{2} (3 a B c) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\left (A c^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=-\frac {c (2 A-3 B x) \sqrt {a+c x^2}}{2 x}-\frac {(2 A+3 B x) \left (a+c x^2\right )^{3/2}}{6 x^3}+\frac {1}{4} (3 a B c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\left (A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {c (2 A-3 B x) \sqrt {a+c x^2}}{2 x}-\frac {(2 A+3 B x) \left (a+c x^2\right )^{3/2}}{6 x^3}+A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {1}{2} (3 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )\\ &=-\frac {c (2 A-3 B x) \sqrt {a+c x^2}}{2 x}-\frac {(2 A+3 B x) \left (a+c x^2\right )^{3/2}}{6 x^3}+A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {3}{2} \sqrt {a} B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 92, normalized size = 0.84 \begin {gather*} \frac {B c \left (a+c x^2\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {c x^2}{a}+1\right )}{5 a^2}-\frac {a A \sqrt {a+c x^2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^2}{a}\right )}{3 x^3 \sqrt {\frac {c x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 113, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-2 a A-3 a B x-8 A c x^2+6 B c x^3\right )}{6 x^3}-A c^{3/2} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )+3 \sqrt {a} B c \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 426, normalized size = 3.91 \begin {gather*} \left [\frac {6 \, A c^{\frac {3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 9 \, 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 (6 \, B c x^{3} - 8 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt {c x^{2} + a}}{12 \, x^{3}}, -\frac {12 \, A \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 9 \, 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 (6 \, B c x^{3} - 8 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt {c x^{2} + a}}{12 \, x^{3}}, \frac {9 \, B \sqrt {-a} c x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 3 \, A c^{\frac {3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (6 \, B c x^{3} - 8 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt {c x^{2} + a}}{6 \, x^{3}}, -\frac {6 \, A \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 9 \, B \sqrt {-a} c x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (6 \, B c x^{3} - 8 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt {c x^{2} + a}}{6 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 211, normalized size = 1.94 \begin {gather*} \frac {3 \, B a c \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - A c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \sqrt {c x^{2} + a} B c + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a c + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a c^{\frac {3}{2}} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{2} c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{3} c + 8 \, A a^{3} c^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 174, normalized size = 1.60 \begin {gather*} A \,c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )-\frac {3 B \sqrt {a}\, c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {\sqrt {c \,x^{2}+a}\, A \,c^{2} x}{a}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{2} x}{3 a^{2}}+\frac {3 \sqrt {c \,x^{2}+a}\, B c}{2}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B c}{2 a}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} A c}{3 a^{2} x}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{2 a \,x^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 136, normalized size = 1.25 \begin {gather*} \frac {\sqrt {c x^{2} + a} A c^{2} x}{a} + A c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - \frac {3}{2} \, B \sqrt {a} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {c x^{2} + a} B c + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B c}{2 \, a} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c}{3 \, a x} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B}{2 \, a x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.33, size = 202, normalized size = 1.85 \begin {gather*} - \frac {A \sqrt {a} c}{x \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 x^{2}} - \frac {A c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3} + A c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )} - \frac {A c^{2} x}{\sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {3 B \sqrt {a} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{2} - \frac {B a \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} + \frac {B a \sqrt {c}}{x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B c^{\frac {3}{2}} x}{\sqrt {\frac {a}{c x^{2}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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