Optimal. Leaf size=108 \[ a^{3/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac {1}{2} \sqrt {a+b x^2} (2 a B+3 A b x)+\frac {3}{2} a A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A] time = 0.09, 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+b x^2}}{\sqrt {a}}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac {1}{2} \sqrt {a+b x^2} (2 a B+3 A b x)+\frac {3}{2} a A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b 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+b x^2\right )^{3/2}}{x^2} \, dx &=-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}-\frac {1}{2} \int \frac {(-2 a B-6 A b x) \sqrt {a+b x^2}}{x} \, dx\\ &=\frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\int \frac {-4 a^2 b B-6 a A b^2 x}{x \sqrt {a+b x^2}} \, dx}{4 b}\\ &=\frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {1}{2} (3 a A b) \int \frac {1}{\sqrt {a+b x^2}} \, dx+\left (a^2 B\right ) \int \frac {1}{x \sqrt {a+b x^2}} \, dx\\ &=\frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {1}{2} (3 a A b) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} \left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {\left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=\frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-a^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.18, size = 105, normalized size = 0.97 \begin {gather*} -a^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {a^2 A \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \sqrt {a+b x^2}}+\frac {1}{3} B \sqrt {a+b x^2} \left (4 a+b 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 {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+b x^2} \left (-6 a A+8 a B x+3 A b x^2+2 b B x^3\right )}{6 x}-\frac {3}{2} a A \sqrt {b} \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 411, normalized size = 3.81 \begin {gather*} \left [\frac {9 \, A a \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 6 \, B a^{\frac {3}{2}} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x}, -\frac {9 \, A a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 3 \, B a^{\frac {3}{2}} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x}, \frac {12 \, B \sqrt {-a} a x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + 9 \, A a \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x}, -\frac {9 \, A a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 6 \, B \sqrt {-a} a x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b 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 = 0.60, size = 124, normalized size = 1.15 \begin {gather*} \frac {2 \, B a^{2} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3}{2} \, A a \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {2 \, A a^{2} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{6} \, \sqrt {b x^{2} + a} {\left (8 \, B a + {\left (2 \, B b x + 3 \, A b\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 126, normalized size = 1.17 \begin {gather*} \frac {3 A a \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}-B \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {3 \sqrt {b \,x^{2}+a}\, A b x}{2}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A b x}{a}+\sqrt {b \,x^{2}+a}\, B a +\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B}{3}-\frac {\left (b \,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 = 1.39, size = 88, normalized size = 0.81 \begin {gather*} \frac {3}{2} \, \sqrt {b x^{2} + a} A b x + \frac {3}{2} \, A a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - B a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B + \sqrt {b x^{2} + a} B a - \frac {{\left (b 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 = 1.88, size = 86, normalized size = 0.80 \begin {gather*} \frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{3}-B\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+B\,a\,\sqrt {b\,x^2+a}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,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 = 13.29, size = 184, normalized size = 1.70 \begin {gather*} - \frac {A a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {A \sqrt {a} b x \sqrt {1 + \frac {b x^{2}}{a}}}{2} - \frac {A \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2} - B a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B a \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + B b \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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