Optimal. Leaf size=111 \[ -\frac {\left (a+b x^2\right )^{3/2} (A-B x)}{2 x^2}-\frac {3 \sqrt {a+b x^2} (a B-A b x)}{2 x}-\frac {3}{2} \sqrt {a} A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {3}{2} a \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {813, 844, 217, 206, 266, 63, 208} \[ -\frac {\left (a+b x^2\right )^{3/2} (A-B x)}{2 x^2}-\frac {3 \sqrt {a+b x^2} (a B-A b x)}{2 x}-\frac {3}{2} \sqrt {a} A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {3}{2} a \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^3} \, dx &=-\frac {(A-B x) \left (a+b x^2\right )^{3/2}}{2 x^2}-\frac {3}{8} \int \frac {(-4 a B-4 A b x) \sqrt {a+b x^2}}{x^2} \, dx\\ &=-\frac {3 (a B-A b x) \sqrt {a+b x^2}}{2 x}-\frac {(A-B x) \left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {3}{16} \int \frac {8 a A b+8 a b B x}{x \sqrt {a+b x^2}} \, dx\\ &=-\frac {3 (a B-A b x) \sqrt {a+b x^2}}{2 x}-\frac {(A-B x) \left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {1}{2} (3 a A b) \int \frac {1}{x \sqrt {a+b x^2}} \, dx+\frac {1}{2} (3 a b B) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {3 (a B-A b x) \sqrt {a+b x^2}}{2 x}-\frac {(A-B x) \left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {1}{4} (3 a A b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )+\frac {1}{2} (3 a b B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {3 (a B-A b x) \sqrt {a+b x^2}}{2 x}-\frac {(A-B x) \left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {3}{2} a \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} (3 a A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=-\frac {3 (a B-A b x) \sqrt {a+b x^2}}{2 x}-\frac {(A-B x) \left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {3}{2} a \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {3}{2} \sqrt {a} A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 90, normalized size = 0.81 \[ \frac {A b \left (a+b x^2\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x^2}{a}+1\right )}{5 a^2}-\frac {a B \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 425, normalized size = 3.83 \[ \left [\frac {3 \, B a \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 3 \, A \sqrt {a} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (B b x^{3} + 2 \, A b x^{2} - 2 \, B a x - A a\right )} \sqrt {b x^{2} + a}}{4 \, x^{2}}, -\frac {6 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 3 \, A \sqrt {a} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (B b x^{3} + 2 \, A b x^{2} - 2 \, B a x - A a\right )} \sqrt {b x^{2} + a}}{4 \, x^{2}}, \frac {6 \, A \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + 3 \, B a \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b x^{3} + 2 \, A b x^{2} - 2 \, B a x - A a\right )} \sqrt {b x^{2} + a}}{4 \, x^{2}}, -\frac {3 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 3 \, A \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (B b x^{3} + 2 \, A b x^{2} - 2 \, B a x - A a\right )} \sqrt {b x^{2} + a}}{2 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 191, normalized size = 1.72 \[ \frac {3 \, A a b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3}{2} \, B a \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {1}{2} \, {\left (B b x + 2 \, A b\right )} \sqrt {b x^{2} + a} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b - 2 \, B a^{3} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 150, normalized size = 1.35 \[ -\frac {3 A \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {3 B a \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}+\frac {3 \sqrt {b \,x^{2}+a}\, B b x}{2}+\frac {3 \sqrt {b \,x^{2}+a}\, A b}{2}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B b x}{a}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A b}{2 a}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B}{a x}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 112, normalized size = 1.01 \[ \frac {3}{2} \, \sqrt {b x^{2} + a} B b x + \frac {3}{2} \, B a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3}{2} \, A \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b x^{2} + a} A b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 91, normalized size = 0.82 \[ A\,b\,\sqrt {b\,x^2+a}-\frac {A\,a\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {3\,A\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2}-\frac {B\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.57, size = 182, normalized size = 1.64 \[ - \frac {3 A \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {A a \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {B \sqrt {a} b x \sqrt {1 + \frac {b x^{2}}{a}}}{2} - \frac {B \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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