Optimal. Leaf size=108 \[ a^{3/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {\left (a+c x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac {1}{2} \sqrt {a+c x^2} (2 a B+3 A c x)+\frac {3}{2} a A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 108, 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 = {813, 815, 844, 217, 206, 266, 63, 208} \begin {gather*} a^{3/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {\left (a+c x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac {1}{2} \sqrt {a+c x^2} (2 a B+3 A c x)+\frac {3}{2} a A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{x^2} \, dx &=-\frac {(3 A-B x) \left (a+c x^2\right )^{3/2}}{3 x}-\frac {1}{2} \int \frac {(-2 a B-6 A c x) \sqrt {a+c x^2}}{x} \, dx\\ &=\frac {1}{2} (2 a B+3 A c x) \sqrt {a+c x^2}-\frac {(3 A-B x) \left (a+c x^2\right )^{3/2}}{3 x}-\frac {\int \frac {-4 a^2 B c-6 a A c^2 x}{x \sqrt {a+c x^2}} \, dx}{4 c}\\ &=\frac {1}{2} (2 a B+3 A c x) \sqrt {a+c x^2}-\frac {(3 A-B x) \left (a+c x^2\right )^{3/2}}{3 x}+\left (a^2 B\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\frac {1}{2} (3 a A c) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {1}{2} (2 a B+3 A c x) \sqrt {a+c x^2}-\frac {(3 A-B x) \left (a+c x^2\right )^{3/2}}{3 x}+\frac {1}{2} \left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\frac {1}{2} (3 a A c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {1}{2} (2 a B+3 A c x) \sqrt {a+c x^2}-\frac {(3 A-B x) \left (a+c x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {\left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c}\\ &=\frac {1}{2} (2 a B+3 A c x) \sqrt {a+c x^2}-\frac {(3 A-B x) \left (a+c x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-a^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.16, size = 105, normalized size = 0.97 \begin {gather*} -a^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {a^2 A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {c x^2}{a}\right )}{x \sqrt {a+c x^2}}+\frac {1}{3} B \sqrt {a+c x^2} \left (4 a+c x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 115, normalized size = 1.06 \begin {gather*} 2 a^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+c x^2} \left (-6 a A+8 a B x+3 A c x^2+2 B c x^3\right )}{6 x}-\frac {3}{2} a A \sqrt {c} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 411, normalized size = 3.81 \begin {gather*} \left [\frac {9 \, A a \sqrt {c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 6 \, B a^{\frac {3}{2}} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B c x^{3} + 3 \, A c x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {c x^{2} + a}}{12 \, x}, -\frac {9 \, A a \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 3 \, B a^{\frac {3}{2}} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (2 \, B c x^{3} + 3 \, A c x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {c x^{2} + a}}{6 \, x}, \frac {12 \, B \sqrt {-a} a x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 9 \, A a \sqrt {c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (2 \, B c x^{3} + 3 \, A c x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {c x^{2} + a}}{12 \, x}, -\frac {9 \, A a \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 6 \, B \sqrt {-a} a x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, B c x^{3} + 3 \, A c x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {c x^{2} + a}}{6 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 124, normalized size = 1.15 \begin {gather*} \frac {2 \, B a^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3}{2} \, A a \sqrt {c} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {2 \, A a^{2} \sqrt {c}}{{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left (8 \, B a + {\left (2 \, B c x + 3 \, A c\right )} x\right )} \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 = 1.17 \begin {gather*} \frac {3 A a \sqrt {c}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2}-B \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {3 \sqrt {c \,x^{2}+a}\, A c x}{2}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A c x}{a}+\sqrt {c \,x^{2}+a}\, B a +\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B}{3}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 88, normalized size = 0.81 \begin {gather*} \frac {3}{2} \, \sqrt {c x^{2} + a} A c x + \frac {3}{2} \, A a \sqrt {c} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - B a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B + \sqrt {c x^{2} + a} B a - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.98, size = 86, normalized size = 0.80 \begin {gather*} \frac {B\,{\left (c\,x^2+a\right )}^{3/2}}{3}-B\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )+B\,a\,\sqrt {c\,x^2+a}-\frac {A\,{\left (c\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {c\,x^2}{a}\right )}{x\,{\left (\frac {c\,x^2}{a}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.07, size = 184, normalized size = 1.70 \begin {gather*} - \frac {A a^{\frac {3}{2}}}{x \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {A \sqrt {a} c x \sqrt {1 + \frac {c x^{2}}{a}}}{2} - \frac {A \sqrt {a} c x}{\sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A a \sqrt {c} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2} - B a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )} + \frac {B a^{2}}{\sqrt {c} x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B a \sqrt {c} x}{\sqrt {\frac {a}{c x^{2}} + 1}} + B c \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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